## Dividing Rational Expressions

Contents

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Remember, the **reciprocal** of \(\frac{a}{b}\) is \(\frac{b}{a}\). To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

### Division of Rational Expressions

If \(p,q,r,s\) are polynomials where \(q\ne 0,r\ne 0,s\ne 0\), then

\(\frac{p}{q}÷\frac{r}{s}=\frac{p}{q}·\frac{s}{r}\)

To divide rational expressions multiply the first fraction by the reciprocal of the second.

### Example: How to Divide Rational Expressions

Divide: \(\frac{x+9}{6-x}÷\frac{{x}^{2}-81}{x-6}.\)

### Solution

### Divide rational expressions.

- Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Factor the numerators and denominators completely.
- Multiply the numerators and denominators together.
- Simplify by dividing out common factors.

## Example

Divide: \(\frac{3{n}^{2}}{{n}^{2}-4n}÷\frac{9{n}^{2}-45n}{{n}^{2}-7n+10}.\)

### Solution

Rewrite the division as the product of the first rationalexpression and the reciprocal of the second. | |

Factor the numerators and denominatorsand then multiply. | |

Simplify by dividing out common factors. | |

Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

## Example

Divide: \(\frac{2{x}^{2}+5x-12}{{x}^{2}-16}÷\frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{2{x}^{2}+5x-12}{{x}^{2}-16}÷\frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}\hfill \\ \\ \begin{array}{c}\text{Rewrite the division as multiplication of}\hfill \\ \text{the first expression by the reciprocal of}\hfill \\ \text{the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2{x}^{2}+5x-12}{{x}^{2}-16}·\frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}\hfill \\ \\ \begin{array}{c}\text{Factor the numerators and denominators}\hfill \\ \text{and then multiply.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(2x-3\right)\left(x+4\right)\left(x-4\right)\left(x-4\right)}{\left(x-4\right)\left(x+4\right)\left(2x-3\right)\left(x-5\right)}\hfill \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(2x-3\right)}\require{cancel}\cancel{\left(x+4\right)}\require{cancel}\cancel{\left(x-4\right)}\left(x-4\right)}{\require{cancel}\cancel{\left(x-4\right)}\require{cancel}\cancel{\left(x+4\right)}\require{cancel}\cancel{\left(2x-3\right)}\left(x-5\right)}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{x-4}{x-5}\hfill \end{array}\)

## Example

Divide: \(\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}÷\frac{{p}^{2}-{q}^{2}}{6}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}÷\frac{{p}^{2}-{q}^{2}}{6}\hfill \\ \\ \begin{array}{c}\text{Rewrite the division as a multiplication}\hfill \\ \text{of the first expression times the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}·\frac{6}{{p}^{2}-{q}^{2}}\hfill \\ \\ \begin{array}{c}\text{Factor the numerators and denominators}\hfill \\ \text{and then multiply.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(p+q\right)\left({p}^{2}-pq+{q}^{2}\right)6}{2\left({p}^{2}+pq+{q}^{2}\right)\left(p-q\right)\left(p+q\right)}\hfill \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(p+q\right)}\left({p}^{2}-pq+{q}^{2}\right){\require{cancel}\cancel{6}}^{3}}{\require{cancel}\cancel{2}\left({p}^{2}+pq+{q}^{2}\right)\left(p-q\right)\require{cancel}\cancel{\left(p+q\right)}}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3\left({p}^{2}-pq+{q}^{2}\right)}{\left(p-q\right)\left({p}^{2}+pq+{q}^{2}\right)}\hfill \end{array}\)

Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide \(\frac{3}{5}÷4\), we first write 4 as a fraction so that we can find its reciprocal.

\(\begin{array}{c}\frac{3}{5}÷4\hfill \\ \frac{3}{5}÷\frac{4}{1}\hfill \\ \frac{3}{5}·\frac{1}{4}\hfill \end{array}\)

We do the same thing when we divide rational expressions.

## Example

Divide: \(\frac{{a}^{2}-{b}^{2}}{3ab}÷\left({a}^{2}+2ab+{b}^{2}\right).\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{a}^{2}-{b}^{2}}{3ab}÷\left({a}^{2}+2ab+{b}^{2}\right)\hfill \\ \\ \text{Write the second expression as a fraction.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{a}^{2}-{b}^{2}}{3ab}÷\frac{{a}^{2}+2ab+{b}^{2}}{1}\hfill \\ \\ \begin{array}{c}\text{Rewrite the division as the first}\hfill \\ \text{expression times the reciprocal of the}\hfill \\ \text{second expression.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{a}^{2}-{b}^{2}}{3ab}·\frac{1}{{a}^{2}+2ab+{b}^{2}}\hfill \\ \\ \begin{array}{c}\text{Factor the numerators and the}\hfill \\ \text{denominators, and then multiply.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(a-b\right)\left(a+b\right)·1}{3ab·\left(a+b\right)\left(a+b\right)}\hfill \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(a-b\right)\require{cancel}\cancel{\left(a+b\right)}}{3ab·\require{cancel}\cancel{\left(a+b\right)}\left(a+b\right)}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(a-b\right)}{3ab\left(a+b\right)}\hfill \end{array}\)

Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

## Example

Divide: \(\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.\)

### Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}\hfill \\ \\ \text{Rewrite with a division sign.}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{6{x}^{2}-7x+2}{4x-8}÷\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}\hfill \\ \\ \begin{array}{c}\text{Rewrite as product of first times}\hfill \\ \text{reciprocal of second.}\hfill \end{array}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{6{x}^{2}-7x+2}{4x-8}·\frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}\hfill \\ \\ \begin{array}{c}\text{Factor the numerators and the}\hfill \\ \text{denominators, and then multiply.}\hfill \end{array}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{\left(2x-1\right)\left(3x-2\right)\left(x-2\right)\left(x-3\right)}{4\left(x-2\right)\left(2x-1\right)\left(x-3\right)}\hfill \\ \\ \text{Simplify by dividing out common factors.}\hfill & & & \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{\left(2x-1\right)}\left(3x-2\right)\require{cancel}\cancel{\left(x-2\right)}\require{cancel}\cancel{\left(x-3\right)}}{4\require{cancel}\cancel{\left(x-2\right)}\require{cancel}\cancel{\left(2x-1\right)}\require{cancel}\cancel{\left(x-3\right)}}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3x-2}{4}\hfill \end{array}\)

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

## Example

Divide: \(\frac{3x-6}{4x-4}·\frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}÷\frac{2x+12}{8x+16}.\)

### Solution

Rewrite the division as multiplicationby the reciprocal. | |

Factor the numerators and the denominators,and then multiply. | |

Simplify by dividing out common factors. | |

Simplify. |