Module 4: Rational Expressions, Functions, and Equations
4.2 Multiplying and Dividing Rational Expressions
Learning Outcomes
Multiply and simplify rational expressions.
Divide and simplify rational expressions.
Just as we can multiply and divide fractions, we can multiply and divide rational expressions. In fact, we use the same processes for multiplying and dividing rational expressions as we use for multiplying and dividing fractions. The process is the same even though the expressions look different!
Multiply Rational Expressions
Remember that there are two ways to multiply fractions.
One way is to multiply the numerators and the denominators separately, and then simplify the product, as shown here. Notice we factor out the greatest common factor (between the numerator and denominator) for efficiency, though we could have worked with the prime factorization as well.
Although both methods result in the same product, it usually makes more sense to simplify fractions before multiplying.
The same two approaches can be applied to rational expressions. Our first two examples apply both techniques to one expression. After that we will let you decide which works best for you. You can factor out any common factors, but finding the greatest one will take fewer steps.
Example
For the following expression, multiply, and state the product in simplest form.
[latex] \begin{align} \frac{5a^2}{14}\cdot \frac{7}{10a^3} &= \frac{5\cdot a^2}{7\cdot 2}\cdot \frac{7}{5\cdot 2\cdot a^2 \cdot a} && {\color{blue}\textsf{factor the numerators and denominators; look for the GCFs}}\\[5pt] &= \frac{{\color{red}\cancel{\color{black}{5}}}\cdot {\color{red}\cancel{\color{black}{a^2}}}\cdot 1}{{\color{red}\cancel{\color{black}{7}}}\cdot 2}\cdot \frac{{\color{red}\cancel{\color{black}{7}}}\cdot 1}{{\color{red}\cancel{\color{black}{5}}}\cdot 2\cdot {\color{red}\cancel{\color{black}{a^2}}} \cdot a} && {\color{blue}\textsf{divide out common factors}}\\[5pt] &= \frac{1}{4a} && {\color{blue}\textsf{simplify}} \end{align} [/latex]
Both methods produced the same answer whereas the second method appeared faster and easier.
Remember that when working with rational expressions, we should get into the habit of identifying the restricted values of the variable, any values that would result in division by [latex]0[/latex]. This will be very important later when we solve rational equations. In the example above, [latex] \displaystyle \frac{5{{a}^{2}}}{14}\cdot \frac{7}{10{{a}^{3}}}[/latex], the only restricted value of [latex] a [/latex] is [latex] 0 [/latex]. When [latex]a=0[/latex], the denominator of the rational expression [latex]\dfrac{7}{10a^3}[/latex] equals [latex]0[/latex], which will make the rational expression undefined.
Rational expressions might contain multi-term polynomials. To multiply such rational expressions, the best approach is to first factor the polynomials and then look for common factors to divide out. Multiplying the numerators and denominators accordingly before factoring will often create complicated polynomials and then we will have to factor these polynomials anyway! For this reason, it is easier to multiply by factoring and then simplifying. Just take it step by step like in the examples below.
Example
For the following expression, state the restricted values of the variable, multiply, and state the product in simplest form.
Since [latex] a=0 [/latex] makes the denominator of the first rational expression equal to [latex] 0 [/latex] and [latex] a=-1 [/latex] makes the denominator of the second rational expression equal to [latex] 0 [/latex], the restricted values of [latex] a [/latex] are [latex] 0 [/latex] and [latex] -1 [/latex]. Now,
Since [latex] 2t^2-t-10=(2t-5)(t+2) [/latex], using the Zero-Product Property, we acknowledge that [latex] t=\dfrac{5}{2} [/latex] and [latex] t=-2 [/latex] make the denominator of the first rational expression equal to [latex] 0 [/latex]. Also, [latex] t^2+2t=t(t+2) [/latex], so [latex] t=0 [/latex] and [latex] t=-2 [/latex] make the denominator of the second rational expression equal to [latex] 0 [/latex]. Hence, the restricted values of [latex] t [/latex] are [latex] \dfrac{5}{2} [/latex], [latex] 0 [/latex], and [latex] -2 [/latex]. Now,
[latex] \begin{align} \frac{t^2+4t+4}{2t^2-t-10}\cdot\frac{t+5}{t^2+2t} &= \frac{(t+2)\cdot (t+2)}{(2t-5)\cdot (t+2)}\cdot\frac{(t+5)}{t\cdot (t+2)} && {\color{blue}\textsf{factor the numerators and denominators}}\\[5pt] &= \frac{{\color{red}\cancel{\color{black}{(t+2)}}}\cdot {\color{red}\cancel{\color{black}{(t+2)}}}\cdot1}{(2t-5)\cdot {\color{red}\cancel{\color{black}{(t+2)}}}}\cdot\frac{(t+5)}{t\cdot {\color{red}\cancel{\color{black}{(t+2)}}}} && {\color{blue}\textsf{divide out common factors}}\\[5pt] &= \frac{t+5}{t(2t-5)} && {\color{blue}\textsf{multiply and simplify}} \end{align} [/latex]
The final answer may also be written in expanded (distributed) form [latex]\dfrac{t+5}{2t^2-5t}[/latex].
CAUTION!
DON’T EVEN THINK ABOUT simplifying the [latex] t [/latex]s or [latex] 5 [/latex]s between the numerator and denominator in the final answer in the example above!
In the following video, we present another example of multiplying rational expressions.
The following example involves a difference of cubes.
Example
For the following expression, state the restricted values of the variable, multiply, and simplify.
Since [latex] x^2=0 [/latex] only if [latex] x=0 [/latex], and [latex] 4x^2-1=0 [/latex], equivalent to [latex] (2x)^2-1^2=0 [/latex], equivalent to [latex] (2x-1)(2x+1)=0 [/latex], results in [latex] x=\dfrac{1}{2} [/latex] or [latex] x=-\dfrac{1}{2} [/latex] by the Zero-Factor Property, the restricted values of [latex] x [/latex] are [latex] 0 [/latex], [latex] \dfrac{1}{2} [/latex], and [latex] -\dfrac{1}{2} [/latex].
Now, referring to Sec. 3.7 how to factor a difference of cubes, we have
You have seen that we multiply rational expressions as we multiply fractions. It should come as no surprise that we also divide rational expressions the same way we divide fractions. Recall that to divide means to multiply by the reciprocal. 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.
Use the same process to divide rational expressions. Once division is presented as multiplication by the reciprocal, use what you know about multiplication to simplify.
We still need to think about the restricted values of the variable. There is a new consideration this time – because we divide by a rational expression, we also need to find the values that would make the numerator of that expression equal zero. Have a look.
Example
For the following expression, state the restricted values of the variable, divide, and simplify.
For the restricted values of [latex] x [/latex], first notice the two denominators, [latex]9[/latex] and [latex]27[/latex]. These never equal zero. Since we divide by [latex]\dfrac{15x^3}{27}[/latex], the numerator of that rational expression cannot equal zero either. Hence, [latex] x\ne 0 [/latex] and the only restricted value of [latex] x [/latex] is [latex] 0 [/latex]. Now,
[latex] \begin{align} \frac{5x^2}{9}\div\frac{15x^3}{27} &= \frac{5x^2}{9}\cdot\frac{27}{15x^3} && {\color{blue}\textsf{rewrite division as multiplication by the reciprocal}}\\[5pt] &= \frac{5\cdot x^2}{9}\cdot\frac{9\cdot 3}{5\cdot 3 \cdot x^2\cdot x} && {\color{blue}\textsf{factor the numerators and denominators; look for the GCFs}}\\[5pt] &= \frac{{\color{red}\cancel{\color{black}{5}}}\cdot {\color{red}\cancel{\color{black}{x^2}}}\cdot 1}{{\color{red}\cancel{\color{black}{9}}}\cdot 1}\cdot\frac{{\color{red}\cancel{\color{black}{9}}}\cdot {\color{red}\cancel{\color{black}{3}}}\cdot 1}{{\color{red}\cancel{\color{black}{5}}}\cdot {\color{red}\cancel{\color{black}{3}}} \cdot {\color{red}\cancel{\color{black}{x^2}}}\cdot x} && {\color{blue}\textsf{divide out common factors}}\\[5pt] &= \frac{1}{x} && {\color{blue}\textsf{multiply and simplify}} \end{align} [/latex]
Example
For the following expression, state the restricted values of the variable, divide, and simplify.
For the restricted values of [latex] x [/latex], first notice that [latex] x=-2 [/latex] makes the denominator of the first rational expression, [latex] x+2 [/latex], equal to zero. Since [latex] x^2+5x+6=(x+3)(x+2) [/latex], using the Zero-Product Property, we acknowledge that [latex] x=-3 [/latex] and [latex] x=-2 [/latex] make the denominator of the second rational expression equal to zero. Since we divide by [latex]\dfrac{6x^4}{x^2+5x+6}[/latex], the numerator of that rational expression, [latex] 6x^4 [/latex], cannot equal zero either. Hence, [latex] x\ne 0 [/latex]. Summarizing, the restricted values of [latex] x [/latex] are [latex] -3 [/latex], [latex] -2 [/latex], and [latex] 0 [/latex]. Now,
[latex] \begin{align} \frac{3x^2}{x+2}\div\frac{6x^4}{x^2+5x+6} &= \frac{3x^2}{x+2}\cdot\frac{x^2+5x+6}{6x^4} && {\color{blue}\textsf{rewrite division as multiplication by the reciprocal}}\\[5pt] &= \frac{3\cdot x^2}{(x+2)}\cdot\frac{(x+3)\cdot (x+2)}{3\cdot 2\cdot x^2\cdot x^2} && {\color{blue}\textsf{factor the numerators and denominators; look for the GCFs}}\\[5pt] &= \frac{{\color{red}\cancel{\color{black}{3}}}\cdot {\color{red}\cancel{\color{black}{x^2}}}\cdot 1}{{\color{red}\cancel{\color{black}{(x+2)}}}\cdot 1}\cdot\frac{(x+3)\cdot {\color{red}\cancel{\color{black}{(x+2)}}}}{{\color{red}\cancel{\color{black}{3}}}\cdot 2\cdot {\color{red}\cancel{\color{black}{x^2}}}\cdot x^2} && {\color{blue}\textsf{divide out common factors}}\\[5pt] &= \frac{x+3}{2x^2} && {\color{blue}\textsf{multiply and simplify}} \end{align} [/latex]
In the video that follows, we present another example of dividing rational expressions.
We conclude with an example involving both multiplication and division of rational expressions.
Example
For the following expression, state the restricted values of the variable, perform the operations, and simplify.
For the restricted values of [latex] x [/latex], we acknowledge that in addition to restricting the zeros of all denominators, we must prevent division by zero when dividing by [latex] \dfrac{5x}{x^2-4} [/latex] by restricting the zeros of the numerator of that expression. We solve four equations:
[latex] 2x+4=0 [/latex], resulting in [latex] x=-2 [/latex],
[latex] x^2-4=0 [/latex], equivalent to [latex] (x-2)(x+2)=0 [/latex], resulting in [latex] x=2 [/latex] or [latex] x=-2 [/latex] by the Zero-Factor Property,
[latex] x-2=0 [/latex], resulting in [latex] x=2 [/latex], and
[latex] 5x=0 [/latex], resulting in [latex] x=0 [/latex].
Summarizing, the restricted values of [latex] x [/latex] are [latex] -2 [/latex], [latex] 2 [/latex], and [latex] 0 [/latex].
Remembering that to divide means to multiply by the reciprocal, we have
Rational expressions are multiplied and divided the same way as fractions. To multiply, first factor the numerators and denominators of the rational expressions acknowledging greatest common factors. Next, divide out common factors between the numerators and denominators. Then, multiply and simplify. To divide, first rewrite the division as multiplication by the reciprocal. After that, the steps are the same as those for multiplication.
When expressing a product or quotient, it is important to remember to identify restricted values of the variable. It will be very important when solving rational equations.
Licenses and Attributions
CC licensed content, Original
Screenshot: Multiply and Divide. Provided by: Lumen Learning. License: CC BY: Attribution
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
Multiply Rational Expressions and Give the Domain. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/Hj6gF1SNttk. License: CC BY: Attribution
Unit 15: Rational Expressions, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: http://nrocnetwork.org/dm-opentext. License: CC BY: Attribution
