Goal: to verify that Z7’(z2,t)=0
instead we will show that z2*Z7(z2,t)=0

ω=(Sqrt[((4-11 t+8 (t^2))/t)]);

q0=(16 (((-1+t))^3) (t^2) (-1+5 t-6 (t^2)+(t^3)));
pp1=(-6+25 t+14 (t^2)-266 (t^3)+648 (t^4)-752 (t^5)+464 (t^6)-144 (t^7)+16 (t^8));
pp2=(2t-1)(2-3 t-8 (t^2)+20 (t^3)-14 (t^4)+4 (t^5));
pp3=2(2t-1)^2t(10-41 t-48 (t^2)+590 (t^3)-1482 (t^4)+1870 (t^5)-1328 (t^6)+526 (t^7)-104 (t^8)+8 (t^9));
pp4=2 (2t-1)^2 t^2((19-124 t+322 (t^2)-422 (t^3)+292 (t^4)-100 (t^5)+12 (t^6)));

z2s=(-6+25 t+14 (t^2)-266 (t^3)+648 (t^4)-752 (t^5)+464 (t^6)-144 (t^7)+16 (t^8)+(-1+2 t) ωω (2-3 t-8 (t^2)+20 (t^3)-14 (t^4)+4 (t^5)+(Sqrt[2]) √(t (10-41 t-48 (t^2)+590 (t^3)-1482 (t^4)+1870 (t^5)-1328 (t^6)+526 (t^7)-104 (t^8)+8 (t^9)+t (19-124 t+322 (t^2)-422 (t^3)+292 (t^4)-100 (t^5)+12 (t^6)) ωω))))/(16 (((-1+t))^3) (t^2) (-1+5 t-6 (t^2)+(t^3)))

q=(2t-1)^10;

p61=16t^5(-10+27 t+288 (t^2)-2022 (t^3)+5825 (t^4)-9477 (t^5)+9336 (t^6)-5494 (t^7)+1712 (t^8)-122 (t^9)-80 (t^10)+16 (t^11));
p62=(-16 t^6)(-25+206 t-726 (t^2)+1459 (t^3)-1927 (t^4)+1870 (t^5)-1394 (t^6)+724 (t^7)-210 (t^8)+24 (t^9));
p41=(-4t^3)(50-687 t+4203 (t^2)-15695 (t^3)+41282 (t^4)-82204 (t^5)+125852 (t^6)-144150 (t^7)+117216 (t^8)-62520 (t^9)+18496 (t^10)-1080 (t^11)-960 (t^12)+192 (t^13));
p42=4t^3(10-121 t+677 (t^2)-2521 (t^3)+7132 (t^4)-15412 (t^5)+24800 (t^6)-29742 (t^7)+27184 (t^8)-18840 (t^9)+9072 (t^10)-2520 (t^11)+288 (t^12));
p21=(-2+40 t-360 (t^2)+2245 (t^3)-11098 (t^4)+42924 (t^5)-126314 (t^6)+282304 (t^7)-483968 (t^8)+638896 (t^9)-640864 (t^10)+470400 (t^11)-234144 (t^12)+65792 (t^13)-2784 (t^14)-3840 (t^15)+768 (t^16));
p22=-t^3(65-774 t+4344 (t^2)-15694 (t^3)+41224 (t^4)-82096 (t^5)+124784 (t^6)-144768 (t^7)+127840 (t^8)-83808 (t^9)+37824 (t^10)-10080 (t^11)+1152 (t^12));
p01=(1-t) (1-19 t+161 (t^2)-924 (t^3)+4066 (t^4)-13822 (t^5)+35840 (t^6)-70584 (t^7)+105400 (t^8)-118168 (t^9)+96704 (t^10)-54208 (t^11)+17760 (t^12)-1440 (t^13)-1024 (t^14)+256 (t^15));
p02=(t^3) (25-290 t+1636 (t^2)-6010 (t^3)+15992 (t^4)-32064 (t^5)+48928 (t^6)-56632 (t^7)+49024 (t^8)-30752 (t^9)+13120 (t^10)-3360 (t^11)+384 (t^12));
p11=8t^3(t-1)(-125+4005 t-60169 (t^2)+570158 (t^3)-3862166 (t^4)+20045945 (t^5)-83223823 (t^6)+284217439 (t^7)-813613881 (t^8)+1978202656 (t^9)-4124892528 (t^10)+7432364016 (t^11)-11645707296 (t^12)+15960414880 (t^13)-19242695952 (t^14)+20530254512 (t^15)-19487643408 (t^16)+16504985216 (t^17)-12441863808 (t^18)+8263236224 (t^19)-4749495424 (t^20)+2309017088 (t^21)-926839040 (t^22)+300215040 (t^23)-76503296 (t^24)+14585856 (t^25)-1794048 (t^26)+73728 (t^27)+8192 (t^28));
p12=8t^3(1-t)(-25+765 t-11171 (t^2)+105634 (t^3)-736638 (t^4)+4055851 (t^5)-18297419 (t^6)+69029409 (t^7)-220386161 (t^8)+599953820 (t^9)-1399432680 (t^10)+2804818368 (t^11)-4835265904 (t^12)+7165985152 (t^13)-9113381872 (t^14)+9916520688 (t^15)-9197207856 (t^16)+7238388416 (t^17)-4811313152 (t^18)+2688147584 (t^19)-1256283520 (t^20)+488377856 (t^21)-156990208 (t^22)+41689856 (t^23)-9309952 (t^24)+1799168 (t^25)-280576 (t^26)+24576 (t^27));
p31=8 t^3(t-1)(-125+4005 t-59369 (t^2)+553998 (t^3)-3730470 (t^4)+19565545 (t^5)-83537119 (t^6)+297885767 (t^7)-901975729 (t^8)+2350136384 (t^9)-5333296784 (t^10)+10652922784 (t^11)-18880321184 (t^12)+29852879392 (t^13)-42255474560 (t^14)+53630116864 (t^15)-60971031168 (t^16)+61759492352 (t^17)-55128623872 (t^18)+42657332224 (t^19)-28037451008 (t^20)+15314942464 (t^21)-6803466496 (t^22)+2405327616 (t^23)-657957120 (t^24)+131469312 (t^25)-16441344 (t^26)+663552 (t^27)+73728 (t^28));
p32=8 t^3(1-t)(-25+765 t-11171 (t^2)+107634 (t^3)-787518 (t^4)+4665067 (t^5)-22973819 (t^6)+95297577 (t^7)-336259225 (t^8)+1016942748 (t^9)-2647457192 (t^10)+5940364352 (t^11)-11479425312 (t^12)+19070848736 (t^13)-27171006848 (t^14)+33097766976 (t^15)-34341234112 (t^16)+30212895232 (t^17)-22415990528 (t^18)+13930544128 (t^19)-7190272256 (t^20)+3053491200 (t^21)-1060255488 (t^22)+303176448 (t^23)-73578240 (t^24)+15455232 (t^25)-2525184 (t^26)+221184 (t^27));
p51=128t^8(t-1)(-1500+28960 t-257939 (t^2)+1404604 (t^3)-5211955 (t^4)+13795890 (t^5)-25855137 (t^6)+29614248 (t^7)+3063112 (t^8)-113012293 (t^9)+333031055 (t^10)-652968101 (t^11)+993241608 (t^12)-1221269400 (t^13)+1224164696 (t^14)-994694744 (t^15)+647101024 (t^16)-331790224 (t^17)+131533872 (t^18)-39147024 (t^19)+8229120 (t^20)-1046016 (t^21)+41472 (t^22)+4608 (t^23));
p52=128 t^8(1-t)(-100+80 t+20039 (t^2)-260728 (t^3)+1786957 (t^4)-8269540 (t^5)+28531279 (t^6)-77414974 (t^7)+170435054 (t^8)-309998835 (t^9)+470441223 (t^10)-598385907 (t^11)+638090092 (t^12)-567729488 (t^13)+417123064 (t^14)-249429768 (t^15)+119783328 (t^16)-46124976 (t^17)+14587824 (t^18)-3960432 (t^19)+919872 (t^20)-157824 (t^21)+13824 (t^22));
p71=2048 t^13(t-1)(-25+56 t+2520 (t^2)-25565 (t^3)+120032 (t^4)-330197 (t^5)+533239 (t^6)-339915 (t^7)-575358 (t^8)+1899107 (t^9)-2748394 (t^10)+2566194 (t^11)-1675233 (t^12)+781067 (t^13)-257689 (t^14)+57232 (t^15)-7392 (t^16)+288 (t^17)+32 (t^18));
p72=2048 t^13(1-t)(-1-50 t+626 (t^2)-2319 (t^3)-1948 (t^4)+47711 (t^5)-201331 (t^6)+485563 (t^7)-779658 (t^8)+878711 (t^9)-712736 (t^10)+425072 (t^11)-193639 (t^12)+72003 (t^13)-23071 (t^14)+6068 (t^15)-1096 (t^16)+96 (t^17));

RootReduce[1/q{Sqrt[p71+p72 ωω],p61+p62 ωω,-Sqrt[p51+p52 ωω],p41+p42 ωω,Sqrt[p31+p32 ωω],p21+p22 ωω,-Sqrt[p11+p12 ωω],p01+p02 ωω}.Table[(-Sqrt[z2s])^j,{j,7,0,-1}]/.ωω->ω/.{t->4/5}]

RootReduce[{7Sqrt[p71+p72 ωω],6(p61+p62 ωω),-5Sqrt[p51+p52 ωω],4(p41+p42 ωω),3Sqrt[p31+p32 ωω],2(p21+p22 ωω),-Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,6,0,-1}]/.ωω->ω/.{t->4/5}]

RootReduce[{7Sqrt[p71+p72 ωω],6(p61+p62 ωω),-5Sqrt[p51+p52 ωω],4(p41+p42 ωω),3Sqrt[p31+p32 ωω],2(p21+p22 ωω),-Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,1,-1}]/.ωω->ω/.{t->4/5}]

RootReduce[{7Sqrt[p71+p72 ωω],6(p61+p62 ωω),-5Sqrt[p51+p52 ωω],4(p41+p42 ωω),3Sqrt[p31+p32 ωω],2(p21+p22 ωω),-Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,6,0,-1}]/.ωω->ω/.{t->1/(11-4Sqrt[6])}]

The first idea is again consider z2 Z7’(z2)

RootReduce[6(p61+p62 ωω)z2s^3+4(p41+p42 ωω)z2s^2+2(p21+p22 ωω)z2s/.ωω->ω/.t->4/5]

RootReduce[{-7Sqrt[p71+p72 ωω],+5Sqrt[p51+p52 ωω],-3Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[{-7Sqrt[p71+p72 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,7,-2}]/.ωω->ω/.{t->4/5}]

qq1=Factor[Resultant[(q0 zs-pp1-pp2 ωω)^2-ωω^2(pp3+pp4 ωω),t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq2=Factor[Resultant[y-2(p21+p22 ωω)zs-4(p41+p42 ωω)zs^2-6(p61+p62 ωω)zs^3,t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq3=Collect[Factor[Resultant[qq1,qq2,zs]][[5]],y]

Solve[%==0/.t->4/5]

RootReduce[{-7Sqrt[p71+p72 ωω],+5Sqrt[p51+p52 ωω],-3Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,1,-2}]/.ωω->ω/.{t->4/5}]

qq4=Factor[Resultant[y1^2-1(p11+p12 ωω)zs,t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq4b=Factor[Resultant[qq4,qq1,zs]][[5]]

RootReduce[Solve[%==0/.t->4/5]]

qq5=Factor[Resultant[y3^2-9(p31+p32 ωω)zs^3,t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq5b=Factor[Resultant[qq5,qq1,zs]][[6]]

RootReduce[Solve[%==0/.t->4/5]]

qq45a=Factor[Resultant[y13-(y1+y3),qq4b,y1]];

qq45b=Factor[Resultant[qq45a,qq5b,y3]][[2]]

Solve[%==0/.t->4/5]

RootReduce[{-3Sqrt[p31+p32 ωω],Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,3,1,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[{-3Sqrt[p31+p32 ωω]}.Table[(-Sqrt[z2s])^j,{j,3,3,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[{Sqrt[p11+p12 ωω]}.Table[(-Sqrt[z2s])^j,{j,1,1,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[zs/.Solve[qq4==0/.{t->4/5,y1->(NumericalApproximation[Root, (RowDefault["-0.0211", FEExpressionsNumericalApproximationElider]), Root[-46587212248392566935943088757112056944-9053298431907805277210427978515625000000 #1+564493410196292603574693202972412109375000 ((#1)^2)+53487501361360045848414301872253417968750000 ((#1)^3)+542101086242752217003726400434970855712890625 ((#1)^4)&,2], ])}]]

RootReduce[zs/.Solve[qq5==0/.{t->4/5,y3->(NumericalApproximation[Root, (RowDefault["0.161", FEExpressionsNumericalApproximationElider]), Root[-698829018774823681853138957878220267979748100919-423234698843559591441610683213065660156250000000000 #1-15132057181764819782021162223815917968750000000000000 ((#1)^2)+109409727737680077552795410156250000000000000000000000 ((#1)^3)+9094947017729282379150390625000000000000000000000000 ((#1)^4)&,4], ])}]]

Solve[qq45b[[1]]==0/.{t->4/5,zs->(NumericalApproximation[Root, (RowDefault["0.356", FEExpressionsNumericalApproximationElider]), Root[11176506961+101644115456 #1-2975270928384 ((#1)^2)+7145894445056 ((#1)^3)+451240001536 ((#1)^4)&,4], ])},Reals]

qq6=Factor[Resultant[y5^2-25(p51+p52 ωω)zs^5,t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq6b=Factor[Resultant[qq6,qq1,zs]][[5]]

RootReduce[Solve[%==0/.t->4/5]]

qq7=Factor[Resultant[y7^2-49(p71+p72 ωω)zs^7,t ωω^2-(4-11 t+8 (t^2)),ωω]][[-1]];

qq7b=Factor[Resultant[qq7,qq1,zs]][[6]]

RootReduce[Solve[%==0/.t->4/5]]

RootReduce[{5Sqrt[p51+p52 ωω]}.Table[(-Sqrt[z2s])^j,{j,5,5,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[{-7Sqrt[p71+p72 ωω]}.Table[(-Sqrt[z2s])^j,{j,7,7,-2}]/.ωω->ω/.{t->4/5}]

RootReduce[(NumericalApproximation[Root, (RowDefault["0.0450", FEExpressionsNumericalApproximationElider]), Root[-2325056565274010875774964383918770540755079-19723395805168432074062820751669324218750000000000 #1-43314256243503330925614455986022949218750000000000000 ((#1)^2)+970855382586643099784851074218750000000000000000000000 ((#1)^3)+9094947017729282379150390625000000000000000000000000 ((#1)^4)&,4], ])+(NumericalApproximation[Root, (RowDefault["-0.172", FEExpressionsNumericalApproximationElider]), Root[-20042318286305529576256608102417097094636439-221467518400310951414854459514378906250000000000 #1+33668193874905802730861375808715820312500000000000 ((#1)^2)+203116586949676275253295898437500000000000000000000 ((#1)^3)+909494701772928237915039062500000000000000000000 ((#1)^4)&,2], ])]

qq67a=Factor[Resultant[y57-(y5+y7),qq6b,y5]];

qq67b=Factor[Resultant[qq67a,qq7b,y7]][[4]]

RootReduce[Solve[%==0/.t->4/5]]

qq8=Resultant[qq67b,qq45b/.{y13->y-y57},y57]

qq8b=Factor[qq8]

Solve[qq8b[[4]]==0/.t->4/5,Reals]

qq8c=qq8b[[4]]

Expand[qq3-qq8c]