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### Factor a quadratic in a quartic polynomial

Hi, I have done this calculation using a very tedious way and have checked that it is correct. Can I possibly perform this using Sage only.

I have a polynomial : $$D=M^2-(A/(2p^4))M+(B/(16p^4))$$ where $$A=18p^{10} - 54p^9 + 59p^8 + 130p^7 - 209p^6 - 98p^5 + 407p^4 + 362p^3 + 49p^2 - 16p + 8$$ and $$B=9( p + 1 )^2(p^4 - 2p^3 + 2p^2 + 2p + 1)(4p^8 - 52p^7 + 373p^6 + 68p^5 - 445p^4 + 72p^3 + 163p^2 - 48*p + 9)$$.

I have checked using multiple software that the factorization of D using the quartic in $p$ $$v^2= p^4-2p^3+5p^2+8p+4$$ gives $$[M-((A+2Fv)/(4p^4))][M-((A-2Fv)/(4p^4))]$$ where $$F=9p^8-18p^7-7p^6+45p^5-21p^4-74p^3-18p^2+6p-2$$.

Can someone help me obtain the same result by using Sage only. Thank you.

### Factor a quadratic in a quartic polynomial

Hi, I have done this calculation using a very tedious way and have checked that it is correct. Can I possibly perform this using Sage only.

I have a polynomial : $$D=M^2-(A/(2 D=M^2-(A/(2p^4))M+(B/(16p^4))$$ M+(B/(16*p^4))$where $$A=18 A=18p^{10} - 54p^9 + 59p^8 + 130p^7 - 209p^6 - 98p^5 + 407p^4 + 362p^3 + 49p^2 - 16p + 8$$ p + 8$

and $$B=9 B=9( p + 1 )^2(p^4 - 2p^3 + 2p^2 + 2p + 1)(4p^8 - 52p^7 + 373p^6 + 68p^5 - 445p^4 + 72p^3 + 163p^2 - 48*p + 9)$$.48p + 9)$. I have checked using multiple software that the factorization of D using the quartic in$p$$$v^2= v^2= p^4-2p^3+5p^2+8p+4$$ p^2+8*p+4$ gives $$[M-((A+2 [M-((A+2Fv)/(4p^4))][M-((A-2Fv)/(4p^4))]$$ v)/(4*p^4))]$where $$F=9 F=9p^8-18p^7-7p^6+45p^5-21p^4-74p^3-18p^2+6p-2$$.p-2$.

Can someone help me obtain the same result by using Sage only. Thank you.

### Factor a quadratic in a quartic polynomial

Hi, I have done this calculation using a very tedious way and have checked that it is correct. Can I possibly perform this using Sage only.

I have a polynomial :

D=M^2-(A/(2*p^4))*M+(B/(16*p^4))


$D=M^2-(A/(2p^4))M+(B/(16*p^4))$ where

$A=18p^{10} A=18*p^{10} - 54p^9 54*p^9 + 59p^8 59*p^8 + 130p^7 130*p^7 - 209p^6 209*p^6 - 98p^5 98*p^5 + 407p^4 407*p^4 + 362p^3 362*p^3 + 49p^2 49*p^2 - 16p 16*p + 8$8


and

$B=9( B=9*( p + 1 )^2(p^4 )^2*(p^4 - 2p^3 2*p^3 + 2p^2 2*p^2 + 2p 2*p + 1)(4p^8 1)*(4*p^8 - 52p^7 52*p^7 + 373p^6 373*p^6 + 68p^5 68*p^5 - 445p^4 445*p^4 + 72p^3 72*p^3 + 163p^2 163*p^2 - 48p + 9)$.48*p+ 9).


I have checked using multiple software that the factorization of D using the quartic in $p$

$v^2= p^4-2p^3+5p^2+8*p+4$ v^2= p^4-2*p^3+5*p^2+8*p+4 gives

$[M-((A+2Fv)/(4p^4))][M-((A-2Fv)/(4*p^4))]$ [M-((A+2*F*v)/(4*p^4))]*[M-((A-2*F*v)/(4*p^4))] where

$F=9p^8-18p^7-7p^6+45p^5-21p^4-74p^3-18p^2+6p-2$.F=9*p^8-18*p^7-7*p^6+45*p^5-21*p^4-74*p^3-18*p^2+6*p-2.

Can someone help me obtain the same result by using Sage only. Thank you.