Question

divide the rational expressions. write your answer in simplest form. $\frac{7x - 35}{x^{2}-3x - 18}div\frac{-4x + 12}{x^{2}-9}$
$-\frac{7(x - 5)}{4(x - 6)}$
$\frac{7(x - 5)}{4(x + 6)}$
$\frac{7(x + 5)}{4(x - 6)}$
$-\frac{4(x - 6)}{7(x - 5)}$
steps: 1. multiply by reciprocal 2. factor all terms 3. reduce common terms 3. write in simplest factored form

Explanation:

Step1: Change division to multiplication

$\frac{7x - 35}{x^{2}-3x - 18}\div\frac{-4x + 12}{x^{2}-9}=\frac{7x - 35}{x^{2}-3x - 18}\times\frac{x^{2}-9}{-4x + 12}$

Step2: Factor the expressions

• Factor $7x - 35$: $7x-35 = 7(x - 5)$
• Factor $x^{2}-3x - 18$: $x^{2}-3x - 18=(x - 6)(x+3)$
• Factor $-4x + 12$: $-4x + 12=-4(x - 3)$
• Factor $x^{2}-9$: $x^{2}-9=(x + 3)(x - 3)$

So the expression becomes $\frac{7(x - 5)}{(x - 6)(x + 3)}\times\frac{(x + 3)(x - 3)}{-4(x - 3)}$

Step3: Cancel out common terms

Cancel out the common terms $(x + 3)$ and $(x - 3)$
We get $\frac{7(x - 5)}{(x - 6)}\times\frac{1}{-4}=-\frac{7(x - 5)}{4(x - 6)}$

Answer:

$-\frac{7(x - 5)}{4(x - 6)}$