Question

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

Answer 1

Let p(x) be the required rational expression.

Given that Given that $\frac{x^3-1}{x^2+2}+p\left(x\right)=\frac{2x^3-x^2+3}{x^2+2}$

$p\left(x\right)=\frac{2x^3-x^2+3}{x^2+2}-\frac{x^3-1}{x^2+2}$

$=\frac{2x^3-x^2+3-x^3+1}{x^2+2}=\frac{x^3-x^2+4}{x^2+2}$