Alg1.2.14

What rational expression should be subtracted from $\frac{2x^2+2x-7}{x^2+x-6}\,$ to get $\frac{x-1}{x-2}\,$

Let the expression to be subtracted be $\frac{p(x)}{q(x)}\,$

Hence as per the given

$\frac{2x^2+2x-7}{x^2+x-6}-\frac{p(x)}{q(x)}=\frac{x-1}{x-2}\,$

$\frac{2x^2+2x-7}{x^2+x-6}-\frac{x-1}{x-2}=\frac{p(x)}{q(x)}\,$

$\frac{2x^2+2x-7}{(x+3)(x-2)}-\frac{x-1}{x-2}=\frac{p(x)}{q(x)}\,$

$\frac{2x^2+2x-7-(x-1)(x+3)}{(x+3)(x-2)}=\frac{p(x)}{q(x)}\,$

$\frac{x^2-4}{(x+3)(x-2)}=\frac{(x+2)(x-2)}{(x+3)(x-2)}=\frac{x+2}{x+3}\,$

Therefore,the required expression is $\frac{x+2}{x+3}\,$