Chapter A.3, Problem 26E

Solution

Find the missing numerator or denominator so that the two rational expressions are equal.

  $(x^2+x-6)(x+3)$ .

Given:

The given rational expression is $\frac{?}{x^2-9}=\frac{x^2+x-6}{x-3}$ .

Calculation:

The given rational expression is $\frac{?}{x^2-9}=\frac{x^2+x-6}{x-3}$ .

As given the two rational expressions are equal.

  $\begin{array}{c}\frac{?}{x^2-9}=\frac{x^2+x-6}{x-3}\\ \frac{?}{x^2-3^2}=\frac{x^2+x-6}{x-3}\\ \frac{?}{(x+3)(x-3)}=\frac{x^2+x-6}{x-3}\\ \frac{?}{(x+3)}=\frac{x^2+x-6}{1}\\ \frac{?}{1}=\frac{(x^2+x-6)(x+3)}{1}\\ ?=(x^2+x-6)(x+3)\end{array}$

So the rational expression will be $\frac{(x^2+x-6)(x+3)}{x^2-9}=\frac{x^2+x-6}{x-3}$ .

Thus the missing numerator (?) is = $(x^2+x-6)(x+3)$ .