# 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 $b\ne 0$ , $c\ne 0$ , and $d\ne 0$ , $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}=\frac{ad}{bc}$ .

Example

Divide and then simplify.

$\frac{{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}÷\frac{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}{2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12}$

Write as multiplication by the reciprocal.

The reciprocal of $\frac{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}{2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12}$ is $\frac{2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}$ .

$\frac{{x}^{2}-4}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}÷\frac{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}{2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12}=\frac{{x}^{2}-4}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6}\cdot \frac{2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}$

Now multiply the numerators and the denominators.

$=\frac{\left({x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4\right)\left(2x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}12\right)}{\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6\right)\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\right)}$

Factor the terms in the numerator.

$=\frac{\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\right)\left(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\right)\cdot 2\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6\right)}{\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}6\right)\left(x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2\right)}$

Divide out the common factors.

$=\frac{\overline{){\left(}{x}\text{\hspace{0.17em}}{+}\text{\hspace{0.17em}}{2}{\right)}}\left(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\right)\cdot 2\overline{){\left(}{x}\text{\hspace{0.17em}}{+}\text{\hspace{0.17em}}{6}{\right)}}}{\overline{){\left(}{x}\text{\hspace{0.17em}}{+}\text{\hspace{0.17em}}{6}{\right)}}\overline{){\left(}{x}\text{\hspace{0.17em}}{+}\text{\hspace{0.17em}}{2}{\right)}}}$

Simplify.

$\begin{array}{l}=\frac{2\left(x\text{\hspace{0.17em}}-\text{\hspace{0.17em}}2\right)}{1}\\ =2x-4\end{array}$