Internal problem ID [6743]
Internal file name [OUTPUT/5991_Sunday_June_05_2022_04_12_11_PM_19657200/index.tex]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS.
EXERCISES 8.2. Page 346
Problem number: 16.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "system of linear ODEs"
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{3} \left (t \right )-\frac {9 x_{4} \left (t \right )}{5}\\ x_{2}^{\prime }\left (t \right )&=\frac {51 x_{2} \left (t \right )}{10}-x_{4} \left (t \right )+3 x_{5} \left (t \right )\\ x_{3}^{\prime }\left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )-3 x_{3} \left (t \right )\\ x_{4}^{\prime }\left (t \right )&=x_{2} \left (t \right )-\frac {31 x_{3} \left (t \right )}{10}+4 x_{4} \left (t \right )\\ x_{5}^{\prime }\left (t \right )&=-\frac {14 x_{1} \left (t \right )}{5}+\frac {3 x_{4} \left (t \right )}{2}-x_{5} \left (t \right ) \end {align*}
In this method, we will assume we have found the matrix exponential \(e^{A t}\) allready. There are different methods to determine this but will not be shown here. This is a system of linear ODE’s given as Warning. Unable to find the matrix exponential.
This is a system of linear ODE’s given as \begin {align*} \vec {x}'(t) &= A\, \vec {x}(t) \end {align*}
Or \begin {align*} \left [\begin {array}{c} x_{1}^{\prime }\left (t \right ) \\ x_{2}^{\prime }\left (t \right ) \\ x_{3}^{\prime }\left (t \right ) \\ x_{4}^{\prime }\left (t \right ) \\ x_{5}^{\prime }\left (t \right ) \end {array}\right ] &= \left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ]\, \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \left (t \right ) \\ x_{5} \left (t \right ) \end {array}\right ] \end {align*}
The first step is find the homogeneous solution. We start by finding the eigenvalues of \(A\). This is done by solving the following equation for the eigenvalues \(\lambda \) \begin {align*} \operatorname {det} \left ( A- \lambda I \right ) &= 0 \end {align*}
Expanding gives \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ]-\lambda \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) &= 0 \end {align*}
Therefore \begin {align*} \operatorname {det} \left (\left [\begin {array}{ccccc} 1-\lambda & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\lambda & 0 & -1 & 3 \\ 1 & 2 & -3-\lambda & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\lambda & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1-\lambda \end {array}\right ]\right ) &= 0 \end {align*}
Which gives the characteristic equation \begin {align*} \lambda ^{5}-\frac {61}{10} \lambda ^{4}-\frac {89}{10} \lambda ^{3}+\frac {3511}{50} \lambda ^{2}+\frac {20779}{500} \lambda -\frac {81879}{500}&=0 \end {align*}
The roots of the above are the eigenvalues. \begin {align*} \lambda _1 &= \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\\ \lambda _2 &= \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\\ \lambda _3 &= \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\\ \lambda _4 &= \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\\ \lambda _5 &= \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) \end {align*}
This table summarises the above result
| eigenvalue | algebraic multiplicity | type of eigenvalue |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\) | \(1\) | real eigenvalue |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\) | \(1\) | real eigenvalue |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\) | \(1\) | real eigenvalue |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\) | \(1\) | real eigenvalue |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\) | \(1\) | real eigenvalue |
Now the eigenvector for each eigenvalue are found.
Considering the eigenvalue \(\lambda _{1} = \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ] - \left (\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\right ) \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccccc} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) & 0 & -1 & 3 \\ 1 & 2 & -3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 1&2&-3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {R_{1}}{1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}+\frac {14 R_{1}}{5 \left (1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {2 R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{729}-\frac {8885}{729}&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}-\frac {R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{729}-\frac {8885}{729}&0\\ 0&0&-{\frac {31}{10}}&\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{2187}-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{2187}-\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{2187}-\frac {4669 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{4374}+\frac {26381}{4374}&-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{1458}-\frac {8885}{1458}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}+\frac {31 R_{3}}{10 \left (\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{16495}-\frac {684446}{82475}\right ) R_{3}}{\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{755802519705}+\frac {918467832062531}{7558025197050}&\frac {29168608000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{50386834647}-\frac {123898658800 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{50386834647}-\frac {454634516000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{50386834647}+\frac {1190776829513 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{50386834647}+\frac {2942009880017}{50386834647}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{755802519705}+\frac {918467832062531}{7558025197050}\right ) R_{4}}{\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{251934173235}+\frac {171587005558547}{5038683464700}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \text {Expression too large to display} \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{5}\}\) and the leading variables are \(\{v_{1}, v_{2}, v_{3}, v_{4}\}\). Let \(v_{5} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{2} = \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ] - \left (\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\right ) \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccccc} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) & 0 & -1 & 3 \\ 1 & 2 & -3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1 \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 1&2&-3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {R_{1}}{1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}+\frac {14 R_{1}}{5 \left (1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {2 R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{729}-\frac {8885}{729}&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}-\frac {R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{729}-\frac {8885}{729}&0\\ 0&0&-{\frac {31}{10}}&\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{2187}-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{2187}-\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{2187}-\frac {4669 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{4374}+\frac {26381}{4374}&-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{1458}-\frac {8885}{1458}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}+\frac {31 R_{3}}{10 \left (\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{82475}+\frac {7397139}{824750}&-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )-1&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{16495}-\frac {684446}{82475}\right ) R_{3}}{\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{755802519705}+\frac {918467832062531}{7558025197050}&\frac {29168608000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{50386834647}-\frac {123898658800 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{50386834647}-\frac {454634516000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{50386834647}+\frac {1190776829513 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{50386834647}+\frac {2942009880017}{50386834647}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{755802519705}+\frac {918467832062531}{7558025197050}\right ) R_{4}}{\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{251934173235}+\frac {171587005558547}{5038683464700}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \text {Expression too large to display} \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{5}\}\) and the leading variables are \(\{v_{1}, v_{2}, v_{3}, v_{4}\}\). Let \(v_{5} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{3} = \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ] - \left (\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\right ) \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccccc} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) & 0 & -1 & 3 \\ 1 & 2 & -3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 1&2&-3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {R_{1}}{1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}+\frac {14 R_{1}}{5 \left (1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {2 R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{729}-\frac {8885}{729}&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}-\frac {R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{729}-\frac {8885}{729}&0\\ 0&0&-{\frac {31}{10}}&\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{2187}-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{2187}-\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{2187}-\frac {4669 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{4374}+\frac {26381}{4374}&-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{1458}-\frac {8885}{1458}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}+\frac {31 R_{3}}{10 \left (\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{16495}-\frac {684446}{82475}\right ) R_{3}}{\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{755802519705}+\frac {918467832062531}{7558025197050}&\frac {29168608000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{50386834647}-\frac {123898658800 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{50386834647}-\frac {454634516000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{50386834647}+\frac {1190776829513 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{50386834647}+\frac {2942009880017}{50386834647}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{755802519705}+\frac {918467832062531}{7558025197050}\right ) R_{4}}{\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{251934173235}+\frac {171587005558547}{5038683464700}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \text {Expression too large to display} \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{5}\}\) and the leading variables are \(\{v_{1}, v_{2}, v_{3}, v_{4}\}\). Let \(v_{5} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{4} = \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ] - \left (\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\right ) \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccccc} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) & 0 & -1 & 3 \\ 1 & 2 & -3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 1&2&-3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {R_{1}}{1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}+\frac {14 R_{1}}{5 \left (1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {2 R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{729}-\frac {8885}{729}&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}-\frac {R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{729}-\frac {8885}{729}&0\\ 0&0&-{\frac {31}{10}}&\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{2187}-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{2187}-\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{2187}-\frac {4669 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{4374}+\frac {26381}{4374}&-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{1458}-\frac {8885}{1458}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}+\frac {31 R_{3}}{10 \left (\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{16495}-\frac {684446}{82475}\right ) R_{3}}{\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{755802519705}+\frac {918467832062531}{7558025197050}&\frac {29168608000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{50386834647}-\frac {123898658800 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{50386834647}-\frac {454634516000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{50386834647}+\frac {1190776829513 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{50386834647}+\frac {2942009880017}{50386834647}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{755802519705}+\frac {918467832062531}{7558025197050}\right ) R_{4}}{\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{251934173235}+\frac {171587005558547}{5038683464700}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \text {Expression too large to display} \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{5}\}\) and the leading variables are \(\{v_{1}, v_{2}, v_{3}, v_{4}\}\). Let \(v_{5} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] Considering the eigenvalue \(\lambda _{5} = \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\)
We need to solve \(A \vec {v} = \lambda \vec {v}\) or \((A-\lambda I) \vec {v} = \vec {0}\) which becomes \begin {align*} \left (\left [\begin {array}{ccccc} 1 & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10} & 0 & -1 & 3 \\ 1 & 2 & -3 & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4 & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1 \end {array}\right ] - \left (\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\right ) \left [\begin {array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end {array}\right ]\right ) \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ]\\ \left [\begin {array}{ccccc} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) & 0 & 2 & -\frac {9}{5} & 0 \\ 0 & \frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) & 0 & -1 & 3 \\ 1 & 2 & -3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) & 0 & 0 \\ 0 & 1 & -\frac {31}{10} & 4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) & 0 \\ -\frac {14}{5} & 0 & 0 & \frac {3}{2} & -1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) \end {array}\right ] \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ]&=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \end {align*}
Now forward elimination is applied to solve for the eigenvector \(\vec {v}\). The augmented matrix is \[ \left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 1&2&-3-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \] \begin {align*} R_{3} = R_{3}-\frac {R_{1}}{1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&0\\ -{\frac {14}{5}}&0&0&{\frac {3}{2}}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}+\frac {14 R_{1}}{5 \left (1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&2&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&-\frac {90 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {459 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {1260 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {25299 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {440001}{164950}&0&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{3} = R_{3}-\frac {2 R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{729}-\frac {8885}{729}&0\\ 0&1&-{\frac {31}{10}}&4-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}-\frac {R_{2}}{\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{729}-\frac {8885}{729}&0\\ 0&0&-{\frac {31}{10}}&\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{2187}-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{2187}-\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{2187}-\frac {4669 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{4374}+\frac {26381}{4374}&-\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {1750 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{1458}-\frac {8885}{1458}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{4} = R_{4}+\frac {31 R_{3}}{10 \left (\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}\right )} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {684446}{82475}&\frac {252 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {6426 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16495}-\frac {3528 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {354186 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{82475}+\frac {7397139}{824750}&-1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (-\frac {280 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}+\frac {1428 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}+\frac {3920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}-\frac {78708 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{16495}-\frac {684446}{82475}\right ) R_{3}}{\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{755802519705}+\frac {918467832062531}{7558025197050}&\frac {29168608000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{50386834647}-\frac {123898658800 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{50386834647}-\frac {454634516000 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{50386834647}+\frac {1190776829513 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{50386834647}+\frac {2942009880017}{50386834647}&0 \end {array} \right ] \end {align*}
\begin {align*} R_{5} = R_{5}-\frac {\left (\frac {162168785240 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{151160503941}-\frac {766665001004 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{151160503941}-\frac {2531996273212 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{151160503941}+\frac {39237476310704 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{755802519705}+\frac {918467832062531}{7558025197050}\right ) R_{4}}{\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{251934173235}+\frac {171587005558547}{5038683464700}} &\Longrightarrow \hspace {5pt}\left [\begin {array}{@{}ccccc!{\ifdefined \HCode |\else \color {red}\vline width 0.6pt\fi }c@{}} 1-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&2&-{\frac {9}{5}}&0&0\\ 0&\frac {51}{10}-\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )&0&-1&3&0\\ 0&0&\frac {100 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{3299}-\frac {510 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{3299}-\frac {1400 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{3299}+\frac {2323 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{3299}-\frac {596}{16495}&\frac {627920 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{7214913}+\frac {179083 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{7214913}-\frac {8790880 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{7214913}-\frac {60194938 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{36074565}+\frac {503298563}{360745650}&-\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{729}+\frac {250 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{729}+\frac {3500 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{729}+\frac {295 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{729}-\frac {8885}{729}&0\\ 0&0&0&\frac {14024369740 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{50386834647}-\frac {124603878473 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{100773669294}-\frac {226318787525 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{50386834647}+\frac {3011640382564 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{251934173235}+\frac {171587005558547}{5038683464700}&\frac {1707585125 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{4}}{16795611549}-\frac {10246240625 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{3}}{16795611549}-\frac {24711469450 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )^{2}}{16795611549}+\frac {205729497305 \operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )}{33591223098}+\frac {453749641865}{33591223098}&0\\ 0&0&0&0&0&0 \end {array} \right ] \end {align*}
Therefore the system in Echelon form is \[ \text {Expression too large to display} \left [\begin {array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \\ v_{5} \end {array}\right ] = \left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end {array}\right ] \] The free variables are \(\{v_{5}\}\) and the leading variables are \(\{v_{1}, v_{2}, v_{3}, v_{4}\}\). Let \(v_{5} = t\). Now we start back substitution. Solving the above equation for the leading variables in terms of free variables gives equation \(\text {Expression too large to display}\)
Hence the solution is \[ \text {Expression too large to display} \] Since there is one free Variable, we have found one eigenvector associated with this eigenvalue. The above can be written as \[ \text {Expression too large to display} \text {Expression too large to display} \] Let \(t = 1\) the eigenvector becomes \[ \text {Expression too large to display} \] Which is normalized to \[ \text {Expression too large to display} \] The following table gives a summary of this result. It shows for each eigenvalue the algebraic multiplicity \(m\), and its geometric multiplicity \(k\) and the eigenvectors associated with the eigenvalue. If \(m>k\) then the eigenvalue is defective which means the number of normal linearly independent eigenvectors associated with this eigenvalue (called the geometric multiplicity \(k\)) does not equal the algebraic multiplicity \(m\), and we need to determine an additional \(m-k\) generalized eigenvectors for this eigenvalue.
| multiplicity
| ||||
| eigenvalue | algebraic \(m\) | geometric \(k\) | defective? | eigenvectors |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\) | \(1\) | \(1\) | No | \(\text {Expression too large to display}\) |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\) | \(1\) | \(1\) | No | \(\text {Expression too large to display}\) |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\) | \(1\) | \(1\) | No | \(\text {Expression too large to display}\) |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\) | \(1\) | \(1\) | No | \(\text {Expression too large to display}\) |
| \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\) | \(1\) | \(1\) | No | \(\text {Expression too large to display}\) |
Now that we found the eigenvalues and associated eigenvectors, we will go over each eigenvalue and generate the solution basis. The only problem we need to take care of is if the eigenvalue is defective. Since eigenvalue \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right )\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{1}(t) &= \vec {v}_{1} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) t}\\ &= \text {Expression too large to display} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =1\right ) t} \end {align*}
Since eigenvalue \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right )\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{2}(t) &= \vec {v}_{2} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) t}\\ &= \text {Expression too large to display} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =2\right ) t} \end {align*}
Since eigenvalue \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right )\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{3}(t) &= \vec {v}_{3} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) t}\\ &= \text {Expression too large to display} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =3\right ) t} \end {align*}
Since eigenvalue \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right )\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{4}(t) &= \vec {v}_{4} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) t}\\ &= \text {Expression too large to display} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =4\right ) t} \end {align*}
Since eigenvalue \(\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right )\) is real and distinct then the corresponding eigenvector solution is \begin {align*} \vec {x}_{5}(t) &= \vec {v}_{5} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) t}\\ &= \text {Expression too large to display} e^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} =5\right ) t} \end {align*}
Therefore the final solution is \begin {align*} \vec {x}_h(t) &= c_{1} \vec {x}_{1}(t) + c_{2} \vec {x}_{2}(t) + c_{3} \vec {x}_{3}(t) + c_{4} \vec {x}_{4}(t) + c_{5} \vec {x}_{5}(t) \end {align*}
Which is written as \begin {align*} \left [\begin {array}{c} x_{1} \left (t \right ) \\ x_{2} \left (t \right ) \\ x_{3} \left (t \right ) \\ x_{4} \left (t \right ) \\ x_{5} \left (t \right ) \end {array}\right ] &= c_{1} \text {Expression too large to display} + c_{2} \text {Expression too large to display} + c_{3} \text {Expression too large to display} + c_{4} \text {Expression too large to display} + c_{5} \text {Expression too large to display} \end {align*}
Which becomes \begin {align*} \text {Expression too large to display} \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 1389
dsolve([diff(x__1(t),t)=x__1(t)+2*x__3(t)-18/10*x__4(t),diff(x__2(t),t)=51/10*x__2(t)-x__4(t)+3*x__5(t),diff(x__3(t),t)=x__1(t)+2*x__2(t)-3*x__3(t),diff(x__4(t),t)=x__2(t)-31/10*x__3(t)+4*x__4(t),diff(x__5(t),t)=-28/10*x__1(t)+15/10*x__4(t)-x__5(t)],singsol=all)
\begin{align*} x_{1} \left (t \right ) &= \frac {1334393 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{1052220}+\frac {3565 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}-\frac {29663 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {49145 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}+\frac {20174057 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{10522200} \\ x_{2} \left (t \right ) &= -\frac {16027 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{420888}+\frac {10385 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {210025 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {371003 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{2104440}+\frac {92973853 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{21044400} \\ x_{3} \left (t \right ) &= -\frac {33875 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}-\frac {2585 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}-\frac {79853 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444}+\frac {1675 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{105222}+\frac {5714563 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{2104440} \\ x_{4} \left (t \right ) &= \moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}} \\ x_{5} \left (t \right ) &= \frac {527 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{4} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{140296}-\frac {511531 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{3} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{4208880}+\frac {203195 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right )^{2} {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{420888}+\frac {7637447 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) {\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{7014800}-\frac {943531891 \left (\moverset {5}{\munderset {\textit {\_a} &=1}{\sum }}{\mathrm e}^{\operatorname {RootOf}\left (500 \textit {\_Z}^{5}-3050 \textit {\_Z}^{4}-4450 \textit {\_Z}^{3}+35110 \textit {\_Z}^{2}+20779 \textit {\_Z} -81879, \operatorname {index} &=\textit {\_a} \right ) t} \textit {\_C}_{\textit {\_a}}\right )}{210444000} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.059 (sec). Leaf size: 2839
DSolve[{x1'[t]==x1[t]+2*x3[t]-18/10*x4[t],x2'[t]==51/10*x2[t]-x4[t]+3*x5[t],x3'[t]==x1[t]+2*x2[t]-3*x3[t],x4'[t]==x2[t]-31/10*x3[t]+4*x4[t],x5'[t]==-28/10*x1[t]+15/10*x4[t]-x5[t]},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions -> True]
Too large to display