Question

Find polynomials p(x) and q(x) such that
$P\left(x\right)+q\left(x\right)=x^2-4x+1andP\left(x\right)-q\left(x\right)=x^2+5x-2$

Answer 1

Given

$p\left(x\right)+q\left(x\right)=x^2-4x+1$ __ (1)

$p\left(x\right)-q\left(x\right)=x^2+5x-2$ __ (2)

$\left(1\right)+\left(2\right)\Rightarrow 2p\left(x\right)=2x^2+x-1$

$p\left(x\right)=x^2+\frac{x}{2}-\frac{1}{2}$

from (1)

$q\left(x\right)=x^2-4x+1-p\left(x\right)$

$=x^2-4x+1-x^2-\frac{x}{2}+\frac{1}{2}$

$=\frac{-9x}{2}+\frac{3}{2}$

$\therefore P\left(x\right)=x^2+\frac{x}{2}-\frac{1}{2}$

$Q\left(x\right)=\frac{-9x}{2}+\frac{3}{2}$