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

Multiplying Rational Expressions

Multiplying Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Multiplication of Rational Expressions

If \(p,q,r,s\) are polynomials where \(q\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}s\ne 0\), then

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

To multiply rational expressions, multiply the numerators and multiply the denominators.

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

Example

Multiply: \(\frac{10}{28}·\frac{8}{15}.\)

Solution

 .
Multiply the numerators and denominators..
Look for common factors, and then remove them..
Simplify..

Remember, throughout this tutorial, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, \(x\ne 0\) and \(y\ne 0\).

Example

Mulitply: \(\frac{2x}{3{y}^{2}}·\frac{6x{y}^{3}}{{x}^{2}y}.\)

Solution

 .
Multiply..
Factor the numerator and denominator completely, and then remove common factors..
Simplify..

Example: How to Multiply Rational Expressions

Mulitply: \(\frac{2x}{{x}^{2}+x+12}·\frac{{x}^{2}-9}{6{x}^{2}}.\)

Solution

The above image has three columns and three rows to show how to multiply rational expressions. Step one is to factor each numerator and denominator completely. Factor x squared minus 9 and x squared plus x plus 12. The rational equation is 2x divided by x squared plus x plus 12 times x squared minus 9 divided by 6x squared, then to 2x divided by x minus 3 times x minus 4 times x minus 3 times x plus 3 divided by 6x squared.Step 2 is to multiply the numerators and denominators. It is helpful to multiply the monomials first. Multiply 2x times x minus 3 times x plus 3 divided by 6x squared times x minus 3 times x minus 4.Step 3 is to divide out the common factors, canceling out 2, x, and x minus 3 in the numerator and 2, x and x minus 3 in the denominator. Leave the denominator in factored form to get x plus 3 divided by 3x times x minus 4.

Multiply a rational expression.

  1. Factor each numerator and denominator completely.
  2. Multiply the numerators and denominators.
  3. Simplify by dividing out common factors.

Example

Multiply: \(\frac{{n}^{2}-7n}{{n}^{2}+2n+1}·\frac{n+1}{2n}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{n}^{2}-7n}{{n}^{2}+2n+1}·\frac{n+1}{2n}\hfill \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)}{\left(n+1\right)\left(n+1\right)}·\frac{n+1}{2n}\hfill \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n\left(n-7\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)2n}\hfill \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\require{cancel}\cancel{n}\left(n-7\right)\require{cancel}\cancel{\left(n+1\right)}}{\left(n+1\right)\require{cancel}\cancel{\left(n+1\right)}2\require{cancel}\cancel{n}}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{n-7}{2\left(n+1\right)}\hfill \end{array}\)

Example

Multiply: \(\frac{16-4x}{2x-12}·\frac{{x}^{2}-5x-6}{{x}^{2}-16}.\)

Solution

\(\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{16-4x}{2x-12}·\frac{{x}^{2}-5x-6}{{x}^{2}-16}\hfill \\ \\ \text{Factor each numerator and denominator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(4-x\right)}{2\left(x-6\right)}·\frac{\left(x-6\right)\left(x+1\right)}{\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \begin{array}{c}\text{Multiply the numerators and the}\hfill \\ \text{denominators.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(4-x\right)\left(x-6\right)\left(x+1\right)}{2\left(x-6\right)\left(x-4\right)\left(x+4\right)}\hfill \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\left(-1\right)\frac{\require{cancel}\cancel{2}·2\require{cancel}\cancel{\left(4-x\right)}\require{cancel}\cancel{\left(x-6\right)}\left(x+1\right)}{\require{cancel}\cancel{2}\require{cancel}\cancel{\left(x-6\right)}\require{cancel}\cancel{\left(x-4\right)}\left(x+4\right)}\hfill \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-\frac{2\left(x+1\right)}{\left(x+4\right)}\hfill \end{array}\)

Example

Multiply: \(\frac{2x-6}{{x}^{2}-8x+15}·\frac{{x}^{2}-25}{2x+10}.\)

Solution

 .
Factor each numerator and denominator..
Multiply the numerators and denominators..
Remove common factors..
Simplify..

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