Question

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

Answer 1

Let the required rational expression be $P\left(x\right)$.

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

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

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

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

$\Rightarrow P\left(x\right)=\frac{x^3+5x-2}{(x+1)(x-1)}$