Mathematics » Rational Expressions and Equations » Multiply and Divide Rational Expressions

Dividing Rational Expressions

Dividing Rational Expressions

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

The above image has three columns. It shows the steps to divide rational expressions. Step one is to rewrite the division as the product of the first rational expression and the reciprocal of the second for x plus 9 divided by 6 minus x divided by x squared minus 81 divided by x minus 6. “Flip” the second fraction and change the division sign to multiplication to get x plus 9 divided by 6 minus x times x minus 6 divided by x squared minus 81.Step two is to factor the numerators and denominators completely. Factor x squared minus 81 to get x plus 9 divided by 6 minus x times x minus 6 divided by x minus 9 times x plus 9.Step three is to multiply the numerators and denominators to get x plus 9 times x minus 6 divided by 6 minus x times x minus 9 times x plus 9.Step four is to simplify by dividing out common factors. Divide out the common factors x plus 9, x minus 6 from the numerator and 6 minus x and x plus 9 from the denominator. Remember opposites divide to negative 1. This simplifies to negative 1 divided by x minus 9.

Divide rational expressions.

  1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
  2. Factor the numerators and denominators completely.
  3. Multiply the numerators and denominators together.
  4. 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..

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