Question

Which rational expression should be added to $\frac{x^4-3x+1}{x+3}$ to get $\frac{x^2+1}{x-2}$?

Answer 1

$P\left(x\right)=\frac{x^4-3x+1}{x+3}$
Let $Q\left(x\right)$ be added to $P\left(x\right)$ to get $\frac{x^2+1}{x+2}$
$P\left(x\right)+Q\left(x\right)=\frac{x^2+1}{x+2}$
$\frac{x^4-3x+1}{x+3}+Q\left(x\right)=\frac{x^2+1}{x+2}$
$Q\left(x\right)=\frac{x^2+1}{x+2}-\frac{x^4-3x+1}{x+3}$
$Q\left(x\right)=\frac{(x^2+1)(x+3)-(x+2)(x^4-3x+1)}{(x+3)(x+3)}$$
$Q\left(x\right)=\frac{x^3+3x^2+x+3-(x^5-3x^2+x+2x^4-6x+2)}{(x+2)(x+3)}$
$Q\left(x\right)=\frac{x^3+3x^2+x+3-x^5-2x^4+3x^2+5x-2}{(x+2)(x+3)}$
$Q\left(x\right)=\frac{-x^5-2x^4+x^3+6x^2+6x+1}{(x+2)(x+3)}$
$Q\left(x\right)=\frac{-x^5-2x^4+x^3+6x^2+6x+1}{(x+2)(x+3)}$