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Section 8.3 Multiplication and Division of Rational Expressions

In the last section, we learned about rational functions, including how to find their domain and how to simplify them. In this section, we will learn how to multiply and divide rational expressions.

Subsection 8.3.1 Multiplication and Division of Rational Functions and Expressions

Recall the property for multiplying fractions 1.2.15, which states that the product of two fractions is equal to the product of their numerators divided by the product of their denominators. We will use this same property for multiplying rational expressions.

When multiplying fractions, one approach is to multiply the numerator and denominator, and then simplify the fraction that results by determining the greatest common factor in both the numerator and denominator, like this:

\begin{align*} \frac{14}{9}\cdot\frac{3}{10}\amp=\frac{14\cdot 3}{9\cdot 10}\\ \amp=\frac{42}{90}\\ \amp=\frac{7\cdot \cancelhighlight{6}}{15\cdot \cancelhighlight{6}}\\ \amp=\frac{7}{15} \end{align*}

This approach works great when we can easily identify that \(6\) is the greatest common factor in both \(42\) and \(90\text{.}\) But in more complicated instances, it isn't always an easy approach. It also won't work particularly well when we have \((x+2)\) instead of \(2\) as a factor, as we'll see shortly.

Another approach to multiplying and simplifying fractions involves utilizing the prime factorization of each the numerator and denominator, like this:

\begin{align*} \frac{14}{9}\cdot\frac{3}{10}\amp=\frac{2\cdot 7}{3^2}\cdot \frac{3}{2\cdot 5}\\ \amp=\frac{\cancelhighlight{2} \cdot 7 \cdot \cancelhighlight{3}}{\cancelhighlight{3} \cdot 3 \cdot \cancelhighlight{2}\cdot5}\\ \amp=\frac{7}{15} \end{align*}

The method for multiplying and simplifying rational expressions is nearly identical, as shown here:

\begin{align*} \frac{x^2+9x+14}{x^2+6x+9}\cdot \frac{x+3}{x^2+7x+10}\amp=\frac{(x+2)(x+7)}{(x+3)^2}\cdot \frac{x+3}{(x+2)(x+5)}\\ \amp=\frac{\cancelhighlight{(x+2)}(x+7)\cancelhighlight{(x+3)}}{\cancelhighlight{(x+3)}(x+3)\cancelhighlight{(x+2)}(x+5)}\\ \amp=\frac{(x+7)}{(x+3)(x+5)}, \text{ for } x\neq -2 \end{align*}

This process will be used for both multiplying and dividing rational expressions. The main distinctions in various examples will be in the factoring methods required.

Example 8.3.1.

Multiply the rational expressions: \(\frac{x^2-4x}{x^2-4}\cdot\frac{4-4x+x^2}{20-x-x^2}\text{.}\)

Explanation

Note that to factor the second rational expression, we'll want to re-write the terms in descending order for both the numerator and denominator. In the denominator, we'll first factor out \(-1\) as the leading term is \(-x^2\text{.}\)

\begin{align*} \frac{x^2-4x}{x^2-4}\cdot\frac{4-4x+x^2}{20-x-x^2}\amp=\frac{x^2-4x}{x^2-4}\cdot\frac{x^2-4x+4}{-x^2-x+20}\\ \amp=\frac{x^2-4x}{x^2-4}\cdot\frac{x^2-4x+4}{-(x^2+x-20)}\\ \amp=\frac{x\cancelhighlight{(x-4)}}{(x+2)\cancelhighlight{(x-2)}} \cdot\frac{(x-2)\cancelhighlight{(x-2)}}{-(x+5)\cancelhighlight{(x-4)}}\\ \amp=-\frac{x(x-2)}{(x+2)(x+5)}, \text{ for } x\neq 2,x\neq 4 \end{align*}
Example 8.3.2.

Multiply the rational expressions: \(\frac{p^2q^4}{3r}\cdot\frac{9r^2}{pq^2}\text{.}\) Note that this book ignores domain restrictions on multivariable expressions.

Explanation

We won't need to factor anything in this example, and can simply multiply across and then simplify.

\begin{align*} \frac{p^2q^4}{3r}\cdot\frac{9r^2}{pq^2}\amp=\frac{p^2q^4\cdot9r^2}{3r\cdot pq^2}\\ \amp=\frac{pq^2\cdot 3r}{1}\\ \amp=3pq^2r \end{align*}

We can divide rational expressions using the property for dividing fractions 1.2.17, which simply requires that we change dividing by an expression to multiplying by its reciprocal.

Remark 8.3.3.

Keep in mind that the result of dividing rational expressions will only be defined over the domain of the original expressions. So, because we invert the second expression in the division process, it's important that we state the explicit domain restrictions of that expression in the inversion step.

Let's look at a few examples.

Example 8.3.4.

Divide the rational expressions: \(\frac{x+2}{x+5}\div \frac{x+2}{x-3}\text{.}\)

Explanation
\begin{align*} \frac{x+2}{x+5}\div \frac{x+2}{x-3}\amp=\frac{\cancelhighlight{x+2}}{x+5}\cdot \frac{x-3}{\cancelhighlight{x+2}}, \text{ for }x\neq 3\\ \amp=\frac{x-3}{x+5}, \text{ for }x\neq -2, x\neq 3 \end{align*}
Example 8.3.5.

Simplify the rational expression using division: \(\frac{\frac{3x-6}{2x+10}}{\frac{x^2-4}{3x+15}}\text{.}\)

Explanation

To begin, we'll note that the larger fraction bar is denoting division, so we will use multiplication by the reciprocal. After that, we'll factor each expression and cancel any common factors.

\begin{align*} \frac{\frac{3x-6}{2x+10}}{\frac{x^2-4}{3x+15}}\amp=\frac{3x-6}{2x+10}\div\frac{x^2-4}{3x+15}\\ \amp=\frac{3x-6}{2x+10}\cdot\frac{3x+15}{x^2-4}, \text{ for }x\neq -5\\ \amp=\frac{3\cancelhighlight{(x-2)}}{2\cancelhighlight{(x+5)}}\cdot\frac{3\cancelhighlight{(x+5)}}{(x+2)\cancelhighlight{(x-2)}}\\ \amp=\frac{3\cdot 3}{2(x+2)}, \text{ for }x\neq 2\\ \amp=\frac{9}{2x+4}, \text{ for }x\neq -5, x\neq 2 \end{align*}
Example 8.3.6.

Divide the rational expressions: \(\frac{x^2-5x-14}{x^2+7x+10}\div\frac{x-7}{x+4}\text{.}\)

Explanation
\begin{align*} \frac{x^2-5x-14}{x^2+7x+10}\div\frac{x-7}{x+4}\amp=\frac{x^2-5x-14}{x^2+7x+10}\cdot\frac{x+4}{x-7}, \text{ for }x\neq -4\\ \amp=\frac{\cancelhighlight{(x-7)}\cancelhighlight{(x+2)}}{(x+5)\cancelhighlight{(x+2)}}\cdot\frac{x+4}{\cancelhighlight{x-7}}, \text{ for }x\neq -4\\ \amp=\frac{x+4}{x+5}, \text{ for }x\neq -4, x\neq -2, x\neq 7 \end{align*}
Example 8.3.7.

Divide the rational expressions: \((p^4-16)\div\frac{p^4-2p^3}{2p}\text{.}\)

Explanation
\begin{align*} (p^4-16)\div\frac{p^4-2p^3}{2p}\amp=\frac{p^4-16}{1}\cdot\frac{2p}{p^4-2p^3}, \text{ for }p\neq 0\\ \amp=\frac{(p^2+4)(p+2)\cancelhighlight{(p-2)}}{1}\cdot\frac{2p}{p^3\cancelhighlight{(p-2)}}\\ \amp=\frac{2(p^2+4)(p+2)}{p^2}, \text{ for } p\neq 2 \end{align*}

Note that in our final answer, we do not need to explicitly state that \(p\) can't equal zero, as it is implicitly evident in our final rational expression.

Example 8.3.8.

Divide the rational expressions: \(\frac{3x^2}{x^2-9y^2}\div\frac{6x^3}{x^2-2xy-15y^2}\text{.}\) Note that this book ignores domain restrictions on multivariable expressions.

Explanation
\begin{align*} \frac{3x^2}{x^2-9y^2}\div\frac{6x^3}{x^2-2xy-15y^2}\amp=\frac{3x^2}{x^2-9y^2}\cdot\frac{x^2-2xy-15y^2}{6x^3}\\ \amp=\frac{3x^2}{\cancelhighlight{(x+3y)}(x-3y)}\cdot\frac{\cancelhighlight{(x+3y)}(x-5y)}{6x^3}\\ \amp=\frac{1}{x-3y}\cdot\frac{x-5y}{2x}\\ \amp=\frac{x-5y}{2x(x-3y)} \end{align*}
Example 8.3.9.

Divide the rational expressions: \(\frac{m^2n^2-3mn-4}{2mn}\div(m^2n^2-16)\text{.}\) Note that this book ignores domain restrictions on multivariable expressions.

Explanation
\begin{align*} \frac{m^2n^2-3mn-4}{2mn}\div(m^2n^2-16)\amp=\frac{m^2n^2-3mn-4}{2mn}\cdot\frac{1}{m^2n^2-16}\\ \amp=\frac{\cancelhighlight{(mn-4)}(mn+1)}{2mn}\cdot\frac{1}{(mn+4)\cancelhighlight{(mn-4)}}\\ \amp=\frac{mn+1}{2mn}\cdot\frac{1}{mn+4}\\ \amp=\frac{mn+1}{2mn(mn+4)} \end{align*}

Exercises 8.3.2 Exercises

Multiplying and Dividing Rational Expressions with One Variable
1.

Select all correct equations:

  • \(\displaystyle 6 \cdot \frac{x}{y} = \frac{6 x}{y}\)

  • \(\displaystyle 6 \cdot \frac{x}{y} = \frac{6 x}{6 y}\)

  • \(\displaystyle -\frac{x}{y} = \frac{-x}{-y}\)

  • \(\displaystyle -\frac{x}{y} = \frac{x}{-y}\)

  • \(\displaystyle 6 \cdot \frac{x}{y} = \frac{x}{6 y}\)

  • \(\displaystyle -\frac{x}{y} = \frac{-x}{y}\)

2.

Select all correct equations:

  • \(\displaystyle -\frac{x}{y} = \frac{-x}{y}\)

  • \(\displaystyle 7 \cdot \frac{x}{y} = \frac{7 x}{7 y}\)

  • \(\displaystyle 7 \cdot \frac{x}{y} = \frac{x}{7 y}\)

  • \(\displaystyle -\frac{x}{y} = \frac{x}{-y}\)

  • \(\displaystyle -\frac{x}{y} = \frac{-x}{-y}\)

  • \(\displaystyle 7 \cdot \frac{x}{y} = \frac{7 x}{y}\)

3.

Simplify the following expressions, and if applicable, write the restricted domain.

\(\displaystyle{ -{\frac{r^{3}}{r+3}} \cdot {r^{2}} =}\)

\(\displaystyle{ -{\frac{r^{3}}{r+3}} \cdot {\frac{1}{r^{2}}} =}\)

4.

Simplify the following expressions, and if applicable, write the restricted domain.

\(\displaystyle{ -{\frac{t^{6}}{t+6}} \cdot {t^{4}} =}\)

\(\displaystyle{ -{\frac{t^{6}}{t+6}} \cdot {\frac{1}{t^{4}}} =}\)

5.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{t^{2}-6t+8}{t-1}} \cdot {\frac{4t-4}{t-2}} =}\)

6.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{x^{2}+2x-15}{x+1}} \cdot {\frac{6x+6}{x+5}} =}\)

7.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{x^{2}-4x}{x^{2}-4}} \cdot {\frac{x^{2}-2x}{x^{2}-12x+32}} =}\)

8.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{y^{2}-4y}{y^{2}-4}} \cdot {\frac{y^{2}-2y}{y^{2}-2y-8}} =}\)

9.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{20y+80}{54+9y-9y^{2}}} \cdot {\frac{y^{2}-6y+9}{5y^{2}+20y}} =}\)

10.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{9r-27}{4+2r-2r^{2}}} \cdot {\frac{r^{2}-4r+4}{3r^{2}-9r}} =}\)

11.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{8r^{2}+r-9}{30r^{2}-75r}} \cdot {\frac{15r-6r^{2}}{64r^{2}-81}} =}\)

12.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{7r^{2}-9r+2}{24r^{4}-20r^{3}}} \cdot {\frac{10r^{3}-12r^{4}}{49r^{2}-4}} =}\)

13.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{t}{t+12}} \div {4t^{2}} =}\)

14.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{t}{t+10}} \div {2t^{5}} =}\)

15.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{15x} \div {\frac{3}{x^{5}}} =}\)

16.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{20x} \div {\frac{4}{x^{4}}} =}\)

17.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{({3y+18}) \div ({9y+54}) =}\)

18.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{({5y+30}) \div ({10y+60}) =}\)

19.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{4r^{2}-81}{2r^{2}+19r+45}} \div ({9-2r}) =}\)

20.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{9r^{2}-100}{3r^{2}+13r+10}} \div ({10-3r}) =}\)

21.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{r^{4}}{r^{2}+5r}} \div {\frac{1}{r^{2}-r-30}} =}\)

22.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{t^{2}}{t^{2}-4t}} \div {\frac{1}{t^{2}+t-20}} =}\)

23.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{9m+7}{m}}}{{\frac{m-4}{m}}}}\) =

24.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{6x-6}{x}}}{{\frac{x+2}{x}}}}\) =

25.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{z}{\left(z-1\right)^{2}}}}{{\frac{7z}{z^{2}-1}}}=}\)

26.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{\frac{{\frac{c}{\left(c-8\right)^{2}}}}{{\frac{2c}{c^{2}-64}}}=}\)

27.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{x^{2}-2x}{x^{2}-1}} \div {\frac{x^{2}-4}{x^{2}+2x-3}}={}}\)

28.

Simplify this expression, and if applicable, write the restricted domain.

\(\displaystyle{{\frac{x^{2}+2x}{x^{2}-25}} \div {\frac{x^{2}-4}{x^{2}-x-20}}={}}\)

Multiplying and Dividing Rational Expressions with More Than One Variable
29.

Simplify this expression.

\(\displaystyle{{\frac{8\!\left(r+y\right)}{r-y}} \cdot {\frac{r-y}{2\!\left(2r+y\right)}} =}\)

30.

Simplify this expression.

\(\displaystyle{{\frac{20\!\left(r+y\right)}{r-y}} \cdot {\frac{r-y}{5\!\left(4r+y\right)}} =}\)

31.

Simplify this expression.

\(\displaystyle{{\frac{4t^{5}x}{5t^{2}}} \cdot {\frac{15t^{3}x^{2}}{8x^{5}}} =}\)

32.

Simplify this expression.

\(\displaystyle{{\frac{2t^{4}r}{3t^{3}}} \cdot {\frac{15tr}{8r^{4}}} =}\)

33.

Simplify this expression.

\(\displaystyle{{\frac{x^{2}-5xy+4y^{2}}{x-5y}} \cdot {\frac{4x-20y}{x-y}} =}\)

34.

Simplify this expression.

\(\displaystyle{{\frac{x^{2}+7xt+12t^{2}}{x-5t}} \cdot {\frac{3x-15t}{x+3t}} =}\)

35.

Simplify this expression.

\(\displaystyle{{\frac{y^{4}r^{9}}{6}} \div {\frac{y^{4}r^{5}}{18}} =}\)

36.

Simplify this expression.

\(\displaystyle{{\frac{y^{3}x^{10}}{2}} \div {\frac{y^{3}x^{5}}{10}} =}\)

37.

Simplify this expression.

\(\displaystyle{({r^{5}+4r^{4}t+4r^{3}t^{2}}) \div ({r^{7}+2r^{6}t}) =}\)

38.

Simplify this expression.

\(\displaystyle{({r^{4}+4r^{3}x+4r^{2}x^{2}}) \div ({r^{5}+2r^{4}x}) =}\)

39.

Simplify this expression.

\(\displaystyle{{\frac{1}{r^{2}-rx-6x^{2}}} \div {\frac{r^{5}}{r^{2}-3rx}} =}\)

40.

Simplify this expression.

\(\displaystyle{{\frac{1}{t^{2}-3tr-4r^{2}}} \div {\frac{t^{2}}{t^{2}-4tr}} =}\)

41.

Simplify this expression.

\(\displaystyle{{\frac{t^{3}}{t^{2}r-2t}} \div {\frac{1}{t^{2}r^{2}+4tr-12}} =}\)

42.

Simplify this expression.

\(\displaystyle{{\frac{x^{3}}{x^{2}r-x}} \div {\frac{1}{x^{2}r^{2}+xr-2}} =}\)

43.

Simplify this expression.

\(\displaystyle{{\frac{4x^{5}t^{5}}{x+2t}} \div {\frac{2x^{7}t}{x^{2}-4t^{2}}} =}\)

44.

Simplify this expression.

\(\displaystyle{{\frac{24y^{3}r^{4}}{y-r}} \div {\frac{6y^{5}r}{y^{2}-r^{2}}} =}\)

45.

Simplify this expression.

\(\displaystyle{\frac{{\frac{a}{b}}}{{\frac{3a}{2b^{2}}}}=}\)

46.

Simplify this expression.

\(\displaystyle{\frac{{\frac{s}{t}}}{{\frac{6s}{5t^{2}}}}=}\)

47.

Simplify this expression.

\(\displaystyle{\frac{{\frac{pq^{2}}{6r}}}{{\frac{p}{9qr}}}=}\)

48.

Simplify this expression.

\(\displaystyle{\frac{{\frac{pq^{2}}{3r}}}{{\frac{p}{4qr}}}=}\)

Challenge
49.

Simplify the following: \(\frac{1}{x+1} \div \frac{x+2}{x+1} \div \frac{x+3}{x+2} \div \frac{x+4}{x+3} \div \dots \frac{x+85}{x+84} \text{.}\) For this exercise, you do not have to write the restricted domain of the simplified expression.