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Dividing Rational Expressions

The method of dividing rational expressions is same as the method of dividing fractions . That is, to divide a rational expression by another rational expression, multiply the first rational expression by the reciprocal of the second rational expression.

For all rational expressions $\frac{a}{b}$ and $\frac{c}{d}$ with $b \neq 0$ , $c \neq 0$ , and $d \neq 0$ , $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{a d}{b c}$ .

Example

Divide and then simplify.

$\frac{x^{2} - 4}{x + 6} \div \frac{x + 2}{2 x + 12}$

Write as multiplication by the reciprocal.

The reciprocal of $\frac{x + 2}{2 x + 12}$ is $\frac{2 x + 12}{x + 2}$ .

$\frac{x^{2} - 4}{x + 6} \div \frac{x + 2}{2 x + 12} = \frac{x^{2} - 4}{x + 6} \cdot \frac{2 x + 12}{x + 2}$

Now multiply the numerators and the denominators.

$= \frac{(x^{2} - 4) (2 x + 12)}{(x + 6) (x + 2)}$

Factor the terms in the numerator.

$= \frac{(x + 2) (x - 2) \cdot 2 (x + 6)}{(x + 6) (x + 2)}$

Divide out the common factors.

$= \frac{(x + 2) (x - 2) \cdot 2 (x + 6)}{(x + 6) (x + 2)}$

Simplify.

$\begin{array}{l} = \frac{2 (x - 2)}{1} \\ = 2 x - 4 \end{array}$