Chapter A.3, Problem 24E

Solution

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

  $(x+5)(x+3)$ .

Given:

The given rational expression is $\frac{x-4}{x+5}=\frac{x^2-x-12}{?}$ .

Calculation:

The given rational expression is $\frac{x-4}{x+5}=\frac{x^2-x-12}{?}$ .

As given the two rational expressions are equal.

  $\begin{array}{c}\frac{x-4}{x+5}=\frac{x^2-x-12}{?}\\ \frac{(x-4)}{(x+5)(x^2-x-12)}=\frac{1}{?}\\ solve{x}^2-x-12=x^2-(4-3)x-12\\ =x^2-4x+3x-12\\ =x(x-4)+3(x-4)\\ =(x+3)(x-4)\\ putthevalueabove\\ \frac{(x-4)}{(x+5)(x^2-x-12)}=\frac{1}{?}\\ \frac{(x-4)}{(x+5)(x+3)(x-4)}=\frac{1}{?}\\ \frac{1}{(x+5)(x+3)}=\frac{1}{?}\\ ?=(x+5)(x+3)\end{array}$

So the rational expression will be $\frac{x-4}{x+5}=\frac{x^2-x-12}{(x+5)(x+3)}$ .

Thus the missing denominator (?) is = $(x+5)(x+3)$ .