Question Number 104181 by mathocean1 last updated on 19/Jul/20

Given  P(x)=x^4 +2x^3 −41x^2 +42x+360  Determinate Q(x) a quadratic poly−  nom such that:  P(x)=(Q(x))^2 −42(Q(x))+360

Answered by mr W last updated on 19/Jul/20

let Q(x)=x^2 +bx+c  P(x)=(Q(x))^2 −42(Q(x))+360  =x^4 +2bx^3 +(b^2 +2c)x^2 +2bcx+c^2                          −42x^2            −42x−42c+360  ⇒c^2 −42c+360=360 ⇒c=42  ⇒2bc−42=42 ⇒bc=42 ⇒b=1  ⇒b^2 +2c−42=41=41 ok!  ⇒2b=2=2 ok!    ⇒Q(x)=x^2 +x+42

Commented bymathocean1 last updated on 19/Jul/20

thank you sir!

Commented bymathocean1 last updated on 19/Jul/20

sir can we also admit that the  general form for quadratic  polynoms is Q(x)=ax^2 +bx+c ?

Commented bymr W last updated on 19/Jul/20

yes, but here it′s obvious that a=1.