(*reference https://mathematica.stackexchange.com/questions/311157/does-mathematica-have-weaker-symbolic-computation-especially-symbolic-polynomia*)

TIMING only (see end of notebook for OUTPUT only)

Example 1

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Expand[Times@@(x-Tuples[{-1,1},Length@#].#&[Sqrt@{2,3,5,7,11}])];
Factor[%,Extension->{Sqrt[2],Sqrt[3],Sqrt[5]}];//AbsoluteTiming

Example 2

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ClearSystemCache[];
MinimalPolynomial[2^(1/2)+2^(1/3)+2^(1/5)+2^(1/7),x];//AbsoluteTiming

Example 3 (*Aborted after 10 minutes. Did not want to wait more...*)

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Minimize[Sqrt[2 a^2-8 a+10]+Sqrt[b^2-6 b+10]+Sqrt[2 a^2-2 a b+b^2],{a,b}];//AbsoluteTiming
Minimize[Sqrt[x^2+(y-1)^2]+Sqrt[(x-1)^2+y^2]+Sqrt[(x+1)^2+y^2],{x,y}];//AbsoluteTiming

Example 4 a

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Expand[Sum[i x[i],{i,1,12}]^12];//AbsoluteTiming

Example 4 b

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Expand[(2x+3y+4z)^50 (5x+6y+7z)^51 (8x+9y+10z)^52];//AbsoluteTiming

Example 5

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PolynomialMod[(-2826*x^12-8790+x^6*(6631-4230*y)-8419*y+5267*y^2+8182*y^3-3940*y^4-2386*y^5+x*(-6026*y^5+1622*y^4-6940*y^3-1813*y^2-6030*y-7923))^2*(((9756*x^30-6880*y^5+3699*y^4+1340*y^3-9362*y^2+2794*y+5059)-2578)+x*(-3125*y^5+3550*y^4-6425*y^3+127*y^2+7206*y-899)),-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]//Factor[#,Modulus->-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]&;//AbsoluteTiming

Example 6

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PolynomialGCD[2140562698718732140814186601134305108361857809490-2784610594523673638484774509153686569751130624770 x+3858489651140979026284638768361566754023704101963 x^2-286771157043724591851751736007270812641162273605 x^3+1040070357631236974043798352460064492096759420031 x^4+527961813592576990225122032458485764714353948184 x^5-223657724243176368425265508444964464071309890719 x^6-3733131181727340801070133191959530969316170794800 x^30+10745497692902796595974970291342011573796136255136 x^31-7905771579741612232706547483087384561886791433680 x^32+1395228177175535320672887161804454665548111711824 x^33-5230961365618779655379147055541516478070736328364 x^34-3706780992584927498391542297913941634360275844528 x^35-11678244373494087080209876359797173892928099938592 x^60-5514859882567035785577006631456739844414574289760 x^61-21295212021140539667882396747720054113720182080880 x^62-21789643204816141848070620182749478757088623737712 x^63-4720988527381715377218961193451142849496347654320 x^64-27444066359527368757788068859219000443573584940160 x^90-33565578213248720198933124767949411665123563970688 x^91-26098925852445627183784692986363859867173544281664 x^92-7417015118303371350476354681923558088324668065600 x^93-26712940739577401605838204365261893824578131755520 x^120-21489656182324275196520668674864724640109707593728 x^121-3537371331357954585225002235363286524220979284224 x^122-7370705120862668705991302948457148370726768590848 x^150-3449161085875451377293133065291672477113785516032 x^151+454778382410716370295091942238975349531795652608 x^180,(-2826*x^12-8790+x^6*(6631-4230*y)-8419*y+5267*y^2+8182*y^3-3940*y^4-2386*y^5+x*(-6026*y^5+1622*y^4-6940*y^3-1813*y^2-6030*y-7923))^2*(((9756*x^30-6880*y^5+3699*y^4+1340*y^3-9362*y^2+2794*y+5059)-2578)+x*(-3125*y^5+3550*y^4-6425*y^3+127*y^2+7206*y-899))//Expand,Modulus->-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]//AbsoluteTiming

Example 7

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f=-10-8 x+x^2+3 x^3-6 x^4-8 x^5-2 x^6+7 x^7+2 x^8+x^33+5 x^34+9 x^35-x^36-8 x^37-x^47-2 x^48+x^49+(6+x-9 x^3+7 x^4-10 x^5-5 x^6+9 x^7+6 x^8+x^9+2 x^10-10 x^11+3 x^12-5 x^38-6 x^39-7 x^40+10 x^41+x^42+6 x^43-9 x^44-3 x^45+9 x^46+6 x^47+3 x^48) y+(1+3 x-9 x^2-9 x^3+9 x^4-8 x^5+5 x^6-x^7+9 x^35-7 x^36+10 x^37+6 x^41-6 x^42-4 x^43+7 x^44-8 x^46-4 x^47) y^2+(2-6 x+9 x^2+3 x^4+9 x^5-x^6+3 x^7-10 x^8+3 x^9+10 x^38+9 x^39-5 x^40+5 x^41) y^8+(-2+8 x-3 x^2+6 x^3-2 x^4-3 x^5-4 x^7-x^8+2 x^9-2 x^10-x^36-7 x^37+9 x^38-7 x^39-6 x^40) y^9+(6-10 x-2 x^2-9 x^3+2 x^4+10 x^5-2 x^7-4 x^8+7 x^9-4 x^35+6 x^36) y^13+(-7+x+10 x^2-7 x^3-6 x^4-9 x^5+3 x^6-7 x^7) y^42+(3+3 x-9 x^2+7 x^4+2 x^5-8 x^6) y^43+(-1+x-4 x^2-6 x^3-x^4+3 x^5) y^44+(1+2 x+3 x^2+6 x^3-7 x^4) y^45+(9-9 x-7 x^2+9 x^3) y^46+(-6-9 x+5 x^2) y^47+(-3-2 x) y^48+6 y^49;
Discriminant[f,y];//AbsoluteTiming

Example 8

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f=32 x^10+400 x^9 y+176 x^10 y+2240 x^8 y^2+2272 x^9 y^2+648 x^10 y^2+7400 x^7 y^3+12776 x^8 y^3+6484 x^9 y^3-48 x^10 y^3+15970 x^6 y^4+41192 x^7 y^4+28070 x^8 y^4-1020 x^9 y^4-550 x^10 y^4+23525 x^5 y^5+84091 x^6 y^5+68643 x^7 y^5-6112 x^8 y^5-4289 x^9 y^5-149 x^10 y^5+23955 x^4 y^6+112892 x^5 y^6+103672 x^6 y^6-17447 x^7 y^6-13730 x^8 y^6-786 x^9 y^6+81 x^10 y^6+16650 x^3 y^7+99786 x^4 y^7+98987 x^5 y^7-27472 x^6 y^7-23093 x^7 y^7-1538 x^8 y^7+504 x^9 y^7+66 x^10 y^7+7560 x^2 y^8+56052 x^3 y^8+58329 x^4 y^8-24561 x^5 y^8-21524 x^6 y^8-1313 x^7 y^8+1171 x^8 y^8+318 x^9 y^8+36 x^10 y^8+2025 x y^9+18171 x^2 y^9+19386 x^3 y^9-11718 x^4 y^9-10543 x^5 y^9-396 x^6 y^9+1204 x^7 y^9+508 x^8 y^9+117 x^9 y^9+11 x^10 y^9+243 y^10+2592 x y^10+2781 x^2 y^10-2322 x^3 y^10-2121 x^4 y^10+17 x^5 y^10+461 x^6 y^10+266 x^7 y^10+91 x^8 y^10+16 x^9 y^10+x^10 y^10;
Solve[{f,D[f,x],D[f,y]}==0,{x,y}]//AbsoluteTiming

Example 9

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CoefficientList[(a1 x1+a10 x10+a11 x11+a2 x2+a3 x3+a4 x4+a5 x5+a6 x6+a7 x7+a8 x8+a9 x9)^11//Expand,{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11}];//AbsoluteTiming

Example 10

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ClearSystemCache[];
f=Function[{x,y},Evaluate@Expand[(x+y+1)^60]];
a=ToNumberField@Root[-12992+40546 #1-59434 #1^2-28068 #1^3+10027 #1^4+74288 #1^5+95220 #1^6&,1];
f[55424+10799 a+32924 a^2-94123 a^3-57476 a^4+16513 a^5,49641-47101 a-37888 a^2-93639 a^3-63032 a^4-33804 a^5]//AbsoluteTiming

data={{"Problem number","Mathematica time (sec)","Maple time (sec)"},{1,7.060114`,0.694},{2,3.6512579`,0.190},{3,Infinity,1.118},{4a,2.3486905`,0.227},{4b,0.3114615`,0.113},{5,40.5864421`,0.137},{6,207.386756`,0.023},{7,13.2426646`,1.805},{8,41.1657519`,0.088},{9,4.8441381`,1.198},{10,9.1800838`,0.651}};
Grid[data,Frame->All]

OUTPUT only (no timing)

Example 1

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Expand[Times@@(x-Tuples[{-1,1},Length@#].#&[Sqrt@{2,3,5,7,11}])];
Factor[%,Extension->{Sqrt[2],Sqrt[3],Sqrt[5]}]

Example 2

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ClearSystemCache[];
MinimalPolynomial[2^(1/2)+2^(1/3)+2^(1/5)+2^(1/7),x]

Example 3 (*Aborted after 10 minutes. Did not want to wait more...*)

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Minimize[Sqrt[2 a^2-8 a+10]+Sqrt[b^2-6 b+10]+Sqrt[2 a^2-2 a b+b^2],{a,b}];//AbsoluteTiming
Minimize[Sqrt[x^2+(y-1)^2]+Sqrt[(x-1)^2+y^2]+Sqrt[(x+1)^2+y^2],{x,y}];//AbsoluteTiming

Example 4 a

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Expand[Sum[i x[i],{i,1,12}]^12]

Example 4 b

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Expand[(2x+3y+4z)^50 (5x+6y+7z)^51 (8x+9y+10z)^52]

Example 5

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PolynomialMod[(-2826*x^12-8790+x^6*(6631-4230*y)-8419*y+5267*y^2+8182*y^3-3940*y^4-2386*y^5+x*(-6026*y^5+1622*y^4-6940*y^3-1813*y^2-6030*y-7923))^2*(((9756*x^30-6880*y^5+3699*y^4+1340*y^3-9362*y^2+2794*y+5059)-2578)+x*(-3125*y^5+3550*y^4-6425*y^3+127*y^2+7206*y-899)),-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]//Factor[#,Modulus->-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]&

Example 6

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PolynomialGCD[2140562698718732140814186601134305108361857809490-2784610594523673638484774509153686569751130624770 x+3858489651140979026284638768361566754023704101963 x^2-286771157043724591851751736007270812641162273605 x^3+1040070357631236974043798352460064492096759420031 x^4+527961813592576990225122032458485764714353948184 x^5-223657724243176368425265508444964464071309890719 x^6-3733131181727340801070133191959530969316170794800 x^30+10745497692902796595974970291342011573796136255136 x^31-7905771579741612232706547483087384561886791433680 x^32+1395228177175535320672887161804454665548111711824 x^33-5230961365618779655379147055541516478070736328364 x^34-3706780992584927498391542297913941634360275844528 x^35-11678244373494087080209876359797173892928099938592 x^60-5514859882567035785577006631456739844414574289760 x^61-21295212021140539667882396747720054113720182080880 x^62-21789643204816141848070620182749478757088623737712 x^63-4720988527381715377218961193451142849496347654320 x^64-27444066359527368757788068859219000443573584940160 x^90-33565578213248720198933124767949411665123563970688 x^91-26098925852445627183784692986363859867173544281664 x^92-7417015118303371350476354681923558088324668065600 x^93-26712940739577401605838204365261893824578131755520 x^120-21489656182324275196520668674864724640109707593728 x^121-3537371331357954585225002235363286524220979284224 x^122-7370705120862668705991302948457148370726768590848 x^150-3449161085875451377293133065291672477113785516032 x^151+454778382410716370295091942238975349531795652608 x^180,(-2826*x^12-8790+x^6*(6631-4230*y)-8419*y+5267*y^2+8182*y^3-3940*y^4-2386*y^5+x*(-6026*y^5+1622*y^4-6940*y^3-1813*y^2-6030*y-7923))^2*(((9756*x^30-6880*y^5+3699*y^4+1340*y^3-9362*y^2+2794*y+5059)-2578)+x*(-3125*y^5+3550*y^4-6425*y^3+127*y^2+7206*y-899))//Expand,Modulus->-84568-10331 y-91710 y^2-79947 y^3+38747 y^4-97554 y^5+55518 y^6]

Example 7

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f=-10-8 x+x^2+3 x^3-6 x^4-8 x^5-2 x^6+7 x^7+2 x^8+x^33+5 x^34+9 x^35-x^36-8 x^37-x^47-2 x^48+x^49+(6+x-9 x^3+7 x^4-10 x^5-5 x^6+9 x^7+6 x^8+x^9+2 x^10-10 x^11+3 x^12-5 x^38-6 x^39-7 x^40+10 x^41+x^42+6 x^43-9 x^44-3 x^45+9 x^46+6 x^47+3 x^48) y+(1+3 x-9 x^2-9 x^3+9 x^4-8 x^5+5 x^6-x^7+9 x^35-7 x^36+10 x^37+6 x^41-6 x^42-4 x^43+7 x^44-8 x^46-4 x^47) y^2+(2-6 x+9 x^2+3 x^4+9 x^5-x^6+3 x^7-10 x^8+3 x^9+10 x^38+9 x^39-5 x^40+5 x^41) y^8+(-2+8 x-3 x^2+6 x^3-2 x^4-3 x^5-4 x^7-x^8+2 x^9-2 x^10-x^36-7 x^37+9 x^38-7 x^39-6 x^40) y^9+(6-10 x-2 x^2-9 x^3+2 x^4+10 x^5-2 x^7-4 x^8+7 x^9-4 x^35+6 x^36) y^13+(-7+x+10 x^2-7 x^3-6 x^4-9 x^5+3 x^6-7 x^7) y^42+(3+3 x-9 x^2+7 x^4+2 x^5-8 x^6) y^43+(-1+x-4 x^2-6 x^3-x^4+3 x^5) y^44+(1+2 x+3 x^2+6 x^3-7 x^4) y^45+(9-9 x-7 x^2+9 x^3) y^46+(-6-9 x+5 x^2) y^47+(-3-2 x) y^48+6 y^49;
Discriminant[f,y]

Example 8

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f=32 x^10+400 x^9 y+176 x^10 y+2240 x^8 y^2+2272 x^9 y^2+648 x^10 y^2+7400 x^7 y^3+12776 x^8 y^3+6484 x^9 y^3-48 x^10 y^3+15970 x^6 y^4+41192 x^7 y^4+28070 x^8 y^4-1020 x^9 y^4-550 x^10 y^4+23525 x^5 y^5+84091 x^6 y^5+68643 x^7 y^5-6112 x^8 y^5-4289 x^9 y^5-149 x^10 y^5+23955 x^4 y^6+112892 x^5 y^6+103672 x^6 y^6-17447 x^7 y^6-13730 x^8 y^6-786 x^9 y^6+81 x^10 y^6+16650 x^3 y^7+99786 x^4 y^7+98987 x^5 y^7-27472 x^6 y^7-23093 x^7 y^7-1538 x^8 y^7+504 x^9 y^7+66 x^10 y^7+7560 x^2 y^8+56052 x^3 y^8+58329 x^4 y^8-24561 x^5 y^8-21524 x^6 y^8-1313 x^7 y^8+1171 x^8 y^8+318 x^9 y^8+36 x^10 y^8+2025 x y^9+18171 x^2 y^9+19386 x^3 y^9-11718 x^4 y^9-10543 x^5 y^9-396 x^6 y^9+1204 x^7 y^9+508 x^8 y^9+117 x^9 y^9+11 x^10 y^9+243 y^10+2592 x y^10+2781 x^2 y^10-2322 x^3 y^10-2121 x^4 y^10+17 x^5 y^10+461 x^6 y^10+266 x^7 y^10+91 x^8 y^10+16 x^9 y^10+x^10 y^10;
Solve[{f,D[f,x],D[f,y]}==0,{x,y}]

Example 9

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CoefficientList[(a1 x1+a10 x10+a11 x11+a2 x2+a3 x3+a4 x4+a5 x5+a6 x6+a7 x7+a8 x8+a9 x9)^11//Expand,{x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11}]; (*too slow when output*)

Example 10

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ClearSystemCache[];
f=Function[{x,y},Evaluate@Expand[(x+y+1)^60]];
a=ToNumberField@Root[-12992+40546 #1-59434 #1^2-28068 #1^3+10027 #1^4+74288 #1^5+95220 #1^6&,1];
f[55424+10799 a+32924 a^2-94123 a^3-57476 a^4+16513 a^5,49641-47101 a-37888 a^2-93639 a^3-63032 a^4-33804 a^5]