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Study Guides > ALGEBRA / TRIG I

Multiplying and Dividing Rational Expressions

Learning Outcomes

  • Multiply rational expressions
  • Divide rational expressions
  Just as you can multiply and divide fractions, you can multiply and divide rational expressions. In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions. The process is the same even though the expressions look different!
Multiply and Divide Multiply and Divide

Multiply Rational Expressions

Remember that there are two ways to multiply numeric fractions. One way is to multiply the numerators and the denominators and then simplify the product, as shown here.

[latex] \displaystyle \frac{4}{5}\cdot \frac{9}{8}=\frac{36}{40}=\frac{3\cdot 3\cdot 2\cdot 2}{5\cdot 2\cdot 2\cdot 2}=\frac{3\cdot 3\cdot \cancel{2}\cdot\cancel{2}}{5\cdot \cancel{2}\cdot\cancel{2}\cdot 2}=\frac{3\cdot 3}{5\cdot 2}\cdot 1=\frac{9}{10}[/latex]

A second way is to factor and simplify the fractions before performing the multiplication.

[latex]\frac{4}{5}\cdot\frac{9}{8}=\frac{2\cdot2}{5}\cdot\frac{3\cdot3}{2\cdot2\cdot2}=\frac{\cancel{2}\cdot\cancel{2}\cdot3\cdot3}{\cancel{2}\cdot5\cdot\cancel{2}\cdot2}=1\cdot\frac{3\cdot3}{5\cdot2}=\frac{9}{10}[/latex]

Notice that both methods result in the same product. In some cases you may find it easier to multiply and then simplify, while in others it may make more sense to simplify fractions before multiplying. The same two approaches can be applied to rational expressions. In the following examples, both techniques are shown. First, let’s multiply and then simplify.

Example

Multiply.[latex] \displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}[/latex] State the product in simplest form.

Answer: Multiply the numerators, and then multiply the denominators.

[latex]\frac{5a^{2}}{14}\cdot\frac{7}{10a^{3}}=\frac{35a^{2}}{140a^{3}}[/latex]

Simplify by finding common factors in the numerator and denominator. Simplify the common factors.

[latex]\large\begin{array}{l}\frac{35a^{2}}{140a^{3}}=\frac{5\cdot7\cdot{a}^{2}}{5\cdot7\cdot2\cdot2\cdot{a}^{2}\cdot{a}}\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\frac{\cancel{5}\cdot\cancel{7}\cdot\cancel{{a}^{2}}}{\cancel{5}\cdot\cancel{7}\cdot2\cdot2\cdot\cancel{{a}^{2}}\cdot{a}}\\\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\normalsize1\cdot\large\frac{1}{4a}\end{array}[/latex]

Simplify.

[latex] \displaystyle \frac{1}{4a}[/latex]

Answer

[latex-display] \displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}=\frac{1}{4a}[/latex][latex] \displaystyle [/latex-display]

Okay, that worked. But this time let’s simplify first, then multiply. When using this method, it helps to look for the greatest common factor. You can factor out any common factors, but finding the greatest one will take fewer steps.

Example

Multiply.  [latex]\frac{5a^{2}}{14}\cdot\frac{7}{10a^{3}}[/latex] State the product in simplest form.

Answer: Factor the numerators and denominators. Look for the greatest common factors.

[latex] \displaystyle \frac{5\cdot {{a}^{2}}}{7\cdot 2}\cdot \frac{7}{5\cdot 2\cdot {{a}^{2}}\cdot a}[/latex]

Simplify common factors, then multiply.

[latex]\large\begin{array}{c}\frac{5\cdot {{a}^{2}}}{7\cdot 2}\cdot \frac{7}{5\cdot 2\cdot {{a}^{2}}\cdot a}\\\\=\frac{\cancel{5}\cdot\cancel{{a}^{2}}}{\cancel{7}\cdot 2}\cdot \frac{\cancel{7}}{\cancel{5}\cdot 2\cdot\cancel{{a}^{2}}\cdot a}\\\\=\frac{1\cdot1\cdot1}{2\cdot2\cdot{a}}=\frac{1}{4a}\end{array}[/latex]

Answer

[latex-display]\frac{5a^{2}}{14}\cdot\frac{7}{10a^{3}}=\frac{1}{4a}[/latex-display]

Both methods produced the same answer. Also, remember that when working with rational expressions, you should get into the habit of identifying any values for the variables that would result in division by [latex]0[/latex]. These excluded values must be eliminated from the domain, the set of all possible values of the variable. In the example above, [latex] \displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}[/latex], the domain is all real numbers where a is not equal to 0. When [latex]a=0[/latex], the denominator of the fraction [latex]\frac{7}{10a^{3}}[/latex] equals 0, which will make the fraction undefined. Some rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, the best approach is to first factor the polynomials and then look for common factors. (Multiplying the terms before factoring will often create complicated polynomials…and then you will have to factor these polynomials anyway! For this reason, it is easier to factor, simplify, and then multiply.) Just take it step by step, like in the examples below.

Example

Multiply.  [latex] \displaystyle \frac{{{a}^{2}}-a-2}{5a}\cdot \frac{10a}{a+1}\,\,,\,\,\,\,\,\,a\,\ne \,\,-1\,,\,\,0[/latex] State the product in simplest form.

Answer:

Factor the numerators and denominators.

[latex]\frac{\left(a-2\right)\left(a+1\right)}{5\cdot{a}}\cdot\frac{5\cdot2\cdot{a}}{\left(a+1\right)}[/latex]

Simplify common factors:

[latex]\large\begin{array}{c}\frac{\left(a-2\right)\cancel{\left(a+1\right)}}{\cancel{5}\cdot{\cancel{a}}}\cdot\frac{\cancel{5}\cdot2\cdot{\cancel{a}}}{\cancel{\left(a+1\right)}}\\\\=\frac{a-2}{1}\cdot\frac{2}{1}\end{array}[/latex]

Multiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.

[latex]\begin{array}{c}\frac{\left(a-2\right)}{1}\cdot\frac{2}{1}\\\\=2\left(a-2\right)\end{array}[/latex]

Answer

[latex-display] \displaystyle \frac{{{a}^{2}}-a-2}{5a}\cdot \frac{10a}{a+1}=2a-4[/latex-display]

Example

Multiply.  [latex]\frac{a^{2}+4a+4}{2a^{2}-a-10}\cdot\frac{a+5}{a^{2}+2a},\,\,\,a\neq-2,0,\frac{5}{2}[/latex] State the product in simplest form.

Answer: Factor the numerators and denominators.

[latex]\frac{\left(a+2\right)\left(a+2\right)}{\left(2a-5\right)\left(a+2\right)}\cdot\frac{a+5}{a\left(a+2\right)}[/latex]

Simplify common factors.

[latex]\large\frac{\cancel{\left(a+2\right)}\cancel{\left(a+2\right)}}{\left(2a-5\right)\cancel{\left(a+2\right)}}\cdot\frac{a+5}{a\cancel{\left(a+2\right)}}[/latex]

Multiply simplified rational expressions. This expression can be left with the denominator in factored form or multiplied out.

[latex]\frac{1}{\left(2a-5\right)}\cdot\frac{a+5}{a}=\frac{a+5}{a\left(2a-5\right)}[/latex]

Answer

[latex-display]\frac{a^{2}+4a+4}{2a^{2}-a-10}\cdot\frac{a+5}{a^{2}+2a}=\frac{a+5}{a\left(2a-5\right)}[/latex-display]

Note that in the answer above, you cannot simplify the rational expression any further. It may be tempting to express the [latex]5[/latex]’s in the numerator and denominator as the fraction [latex]\frac{5}{5}[/latex], but these [latex]5[/latex]’s are terms because they are being added or subtracted. Remember that only common factors, not terms, can be regrouped to form factors of [latex]1[/latex]! In the following video we present another example of multiplying rational expressions. https://www.youtube.com/watch?v=Hj6gF1SNttk&feature=youtu.be

Try It

[ohm_question]1458[/ohm_question]

Divide Rational Expressions

You've seen that you multiply rational expressions as you multiply numeric fractions. It should come as no surprise that you also divide rational expressions the same way you divide numeric fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression. Let’s begin by recalling division of numerical fractions.

[latex]\frac{2}{3}\div\frac{5}{9}=\frac{2}{3}\cdot\frac{9}{5}=\frac{18}{15}=\frac{6}{5}[/latex]

Use the same process to divide rational expressions. You can think of division as multiplication by the reciprocal, and then use what you know about multiplication to simplify.
Reciprocal Architecture Reciprocal Architecture
You do still need to think about the domain, specifically the variable values that would make either denominator equal zero. But there's a new consideration this time—because you divide by multiplying by the reciprocal of one of the rational expressions, you also need to find the values that would make the numerator of that expression equal zero. Have a look.

Example

Identify the domain of the expression.  [latex]\frac{5x^{2}}{9}\div\frac{15x^{3}}{27}[/latex]

Answer: State the Domain: Find excluded values. Notice the two denominators, [latex]9[/latex] and [latex]27[/latex], can never equal [latex]0[/latex]. Because [latex]15x^{3}[/latex] becomes the denominator in the reciprocal of [latex] \displaystyle \frac{15{{x}^{3}}}{27}[/latex], you must find the values of x that would make [latex]15x^{3}[/latex] equal 0.

[latex]\begin{array}{c}15x^{3}=0\\x=0\,\text{is an excluded value}.\end{array}[/latex]

Answer

The domain is all real numbers except  [latex]0[/latex].

Knowing how to find the domain may seem unimportant here, but it will help you when you learn how to solve rational equations. To divide, multiply by the reciprocal.

Example

Divide.  [latex]\frac{5x^{2}}{9}\div\frac{15x^{3}}{27}[/latex] State the quotient in simplest form.

Answer: Rewrite division as multiplication by the reciprocal.

[latex]\frac{5x^{2}}{9}\cdot\frac{27}{15x^{3}}[/latex]

Factor the numerators and denominators.

[latex]\frac{5\cdot{x}\cdot{x}}{3\cdot3}\cdot\frac{3\cdot3\cdot3}{5\cdot3\cdot{x}\cdot{x}\cdot{x}}[/latex]

Simplify common factors. Simplify.

[latex]\large\begin{array}{c}\frac{\cancel{5}\cdot{\cancel{x}}\cdot{\cancel{x}}}{\cancel{3}\cdot\cancel{3}}\cdot\frac{\cancel{3}\cdot\cancel{3}\cdot\cancel{3}}{\cancel{5}\cdot\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}\cdot{x}}\\\\=\frac{1}{x}\end{array}[/latex]

Answer

[latex-display] \displaystyle \frac{5{{x}^{2}}}{9}\div \frac{15{{x}^{3}}}{27}=\frac{1}{x},x\ne 0[/latex-display]

Example

Divide.  [latex]\frac{3x^{2}}{x+2}\div\frac{6x^{4}}{\left(x^{2}+5x+6\right)}[/latex] State the quotient in simplest form, and express the domain of the expression.

Answer: Determine the excluded values that make the denominators and the numerator of the divisor equal to [latex]0[/latex].

[latex]\begin{array}{r}\left(x+2\right)=0\,\,\,\,\,\\x=-2\\\left({{x}^{2}}+5x+6 \right)=0\,\,\,\,\,\\\left(x+3\right)\left(x+2\right)=0\,\,\,\,\,\\x=-3\,\,\,\,\text{or}\,\,\,\,-2\\6x^{4}=0\,\,\,\,\,\\x=0\,\,\,\,\,\end{array}[/latex]

Domain is all real numbers except [latex]0[/latex], [latex]−2[/latex], and [latex]−3[/latex]. Rewrite division as multiplication by the reciprocal.

[latex]\frac{3x^{2}}{x+2}\cdot\frac{\left(x^{2}+5x+6\right)}{6x^{4}}[/latex]

Factor the numerators and denominators.

[latex]\frac{3\cdot{x}\cdot{x}}{x+2}\cdot\frac{\left(x+2\right)\left(x+3\right)}{2\cdot3\cdot{x}\cdot{x}\cdot{x}\cdot{x}}[/latex]

Simplify common factors

[latex]\large\frac{\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}}{\cancel{x+2}}\cdot\frac{\cancel{\left(x+2\right)}\left(x+3\right)}{2\cdot\cancel{3}\cdot{\cancel{x}}\cdot{\cancel{x}}\cdot{x}\cdot{x}}[/latex]

Simplify.

[latex]\frac{(x+3)}{2{{x}^{2}}}[/latex]

Answer

[latex] \displaystyle \frac{3{{x}^{2}}}{x+2}\div \frac{6{{x}^{4}}}{({{x}^{2}}+5x+6)}=\frac{x+3}{2{{x}^{2}}}[/latex]. The domain is all real numbers except [latex]0[/latex], [latex]−2[/latex], and [latex]−3[/latex].

Notice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions. In the video that follows, we present another example of dividing rational expressions. https://www.youtube.com/watch?v=B1tigfgs268&feature=youtu.be

Try It

[ohm_question]38288[/ohm_question]

Summary

Rational expressions are multiplied and divided the same way as numeric fractions. To multiply, first find the greatest common factors of the numerator and denominator. Next, cancel common factors by regrouping the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication. When expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.  

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  • Screenshot: Multiply and Divide. Provided by: Lumen Learning License: CC BY: Attribution.
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  • Multiply Rational Expressions and Give the Domain. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
  • Divide Rational Expressions and Give the Domain. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.

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  • Unit 15: Rational Expressions, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.