1. Learning objectives

Learning objectives

Multiply rational expressions

Divide rational expressions

2. Multiply Rational Expressions

Multiply 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.

If $p,q,r,s$ are polynomials where $q\ne{}0\,\mathrm{and}\,s\ne{}0$ , then

\frac{p}{q}\cdot\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.

3. Example 8.17

Example 8.17

Multiply: $\frac{10}{28}\cdot\frac{8}{15}.$

Solution

Remember, throughout this chapter, 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 $y\ne{}0$ .

4. Example 8.18

Example 8.18

Mulitply: $\frac{2x}{3{y}^{2}}\cdot\frac{6x{y}^{3}}{{x}^{2}y}.$

Solution

5. Example 8.19

Example 8.19

How to Multiply Rational Expressions

Mulitply: $\frac{2x}{{x}^{2}-7x+12}\cdot\frac{{x}^{2}-9}{6{x}^{2}}.$

Multiply a rational expression.

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

6. Example 8.20

Example 8.20

Multiply: $\frac{{n}^{2}-7n}{{n}^{2}+2n+1}\cdot\frac{n+1}{2n}.$

Solution


	$\frac{{n}^{2}-7n}{{n}^{2}+2n+1}\cdot\frac{n+1}{2n}$
Factor each numerator and denominator.	$\frac{n(n-7)}{(n+1)(n+1)}\cdot\frac{n+1}{2n}$
Multiply the numerators and the denominators.	$\frac{n(n-7)(n+1)}{(n+1)(n+1)2n}$
Remove common factors.	$\frac{n(n-7)(n+1)}{(n+1)(n+1)2n}$
Simplify.	$\frac{n-7}{2(n+1)}$

7. Example 8.21

Example 8.21

Multiply: $\frac{16-4x}{2x-12}\cdot\frac{{x}^{2}-5x-6}{{x}^{2}-16}.$

Solution


	$\frac{16-4x}{2x-12}\cdot\frac{{x}^{2}-5x-6}{{x}^{2}-16}$
Factor each numerator and denominator.	$\frac{4(4-x)}{2(x-6)}\cdot\frac{(x-6)(x+1)}{(x-4)(x+4)}$
Multiply the numerators and the denominators.	$\frac{4(4-x)(x-6)(x+1)}{2(x-6)(x-4)(x+4)}$
Remove common factors.	$(-1)\frac{2\cdot{}2(4-x)(x-6)(x+1)}{2(x-6)(x-4)(x+4)}$
Simplify.	$-\frac{2(x+1)}{(x+4)}$

8. Example 8.22

Example 8.22

Multiply: $\frac{2x-6}{{x}^{2}-8x+15}\cdot\frac{{x}^{2}-25}{2x+10}.$

Solution

9. Divide Rational Expressions

Divide 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.

If $p,q,r,s$ are polynomials where $q\ne{}0,r\ne{}0,s\ne{}0$ , then

\frac{p}{q}\mathrm{÷}\frac{r}{s}=\frac{p}{q}\cdot\frac{s}{r}

To divide rational expressions multiply the first fraction by the reciprocal of the second.

10. Example 8.23

Example 8.23

How to Divide Rational Expressions

Divide: $\frac{x+9}{6-x}\mathrm{÷}\frac{{x}^{2}-81}{x-6}.$

Divide rational expressions.

Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Step 2. Factor the numerators and denominators completely.
Step 3. Multiply the numerators and denominators together.
Step 4. Simplify by dividing out common factors.

11. Example 8.24

Example 8.24

Divide: $\frac{3{n}^{2}}{{n}^{2}-4n}\mathrm{÷}\frac{9{n}^{2}-45n}{{n}^{2}-7n+10}.$

Solution

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.

12. Example 8.25

Example 8.25

Divide: $\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\mathrm{÷}\frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}.$

Solution


	$\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\mathrm{÷}\frac{2{x}^{2}-13x+15}{{x}^{2}-8x+16}$
Rewrite the division as multiplication ofthe first expression by the reciprocal of the second.	$\frac{2{x}^{2}+5x-12}{{x}^{2}-16}\cdot\frac{{x}^{2}-8x+16}{2{x}^{2}-13x+15}$
Factor the numerators and denominators and then multiply.	$\frac{(2x-3)(x+4)(x-4)(x-4)}{(x-4)(x+4)(2x-3)(x-5)}$
Simplify by dividing out common factors.	$\frac{(2x-3)(x+4)(x-4)(x-4)}{(x-4)(x+4)(2x-3)(x-5)}$
Simplify.	$\frac{x-4}{x-5}$

13. Example 8.26

Example 8.26

Divide: $\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}\mathrm{÷}\frac{{p}^{2}-{q}^{2}}{6}.$

Solution


	$\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}\mathrm{÷}\frac{{p}^{2}-{q}^{2}}{6}$
Rewrite the division as a multiplicationof the first expression times thereciprocal of the second.	$\frac{{p}^{3}+{q}^{3}}{2{p}^{2}+2pq+2{q}^{2}}\cdot\frac{6}{{p}^{2}-{q}^{2}}$
Factor the numerators and denominators and then multiply.	$\frac{(p+q)({p}^{2}-pq+{q}^{2})6}{2({p}^{2}+pq+{q}^{2})(p-q)(p+q)}$
Simplify by dividing out common factors.	$\frac{(p+q)({p}^{2}-pq+{q}^{2}){6}^{3}}{2({p}^{2}+pq+{q}^{2})(p-q)(p+q)}$
Simplify.	$\frac{3({p}^{2}-pq+{q}^{2})}{(p-q)({p}^{2}+pq+{q}^{2})}$

Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide $\frac{3}{5}\mathrm{÷}4$ , we first write 4 as a fraction so that we can find its reciprocal.

\begin{array}{l} \frac{3}{5}\mathrm{÷}4 \\ \frac{3}{5}\mathrm{÷}\frac{4}{1} \\ \frac{3}{5}\cdot\frac{1}{4} \\ \end{array}

We do the same thing when we divide rational expressions.

14. Example 8.27

Example 8.27

Divide: $\frac{{a}^{2}-{b}^{2}}{3ab}\mathrm{÷}({a}^{2}+2ab+{b}^{2}).$

Solution


	$\frac{{a}^{2}-{b}^{2}}{3ab}\mathrm{÷}({a}^{2}+2ab+{b}^{2})$
Write the second expression as a fraction.	$\frac{{a}^{2}-{b}^{2}}{3ab}\mathrm{÷}\frac{{a}^{2}+2ab+{b}^{2}}{1}$
Rewrite the division as the firstexpression times the reciprocal of thesecond expression.	$\frac{{a}^{2}-{b}^{2}}{3ab}\cdot\frac{1}{{a}^{2}+2ab+{b}^{2}}$
Factor the numerators and thedenominators, and then multiply.	$\frac{(a-b)(a+b)\cdot{}1}{3ab\cdot(a+b)(a+b)}$
Simplify by dividing out common factors.	$\frac{(a-b)(a+b)}{3ab\cdot(a+b)(a+b)}$
Simplify.	$\frac{(a-b)}{3ab(a+b)}$

Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

15. Example 8.28

Example 8.28

Divide: $\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}.$

Solution


	$\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}}$
Rewrite with a division sign.	$\frac{6{x}^{2}-7x+2}{4x-8}\mathrm{÷}\frac{2{x}^{2}-7x+3}{{x}^{2}-5x+6}$
Rewrite as product of first timesreciprocal of second.	$\frac{6{x}^{2}-7x+2}{4x-8}\cdot\frac{{x}^{2}-5x+6}{2{x}^{2}-7x+3}$
Factor the numerators and thedenominators, and then multiply.	$\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}$
Simplify by dividing out common factors.	$\frac{(2x-1)(3x-2)(x-2)(x-3)}{4(x-2)(2x-1)(x-3)}$
Simplify.	$\frac{3x-2}{4}$

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.

16. Example 8.29

Example 8.29

Divide: $\frac{3x-6}{4x-4}\cdot\frac{{x}^{2}+2x-3}{{x}^{2}-3x-10}\mathrm{÷}\frac{2x+12}{8x+16}.$

Solution

17. Section 8.2 Exercises

Section 8.2 Exercises

Multiply Rational Expressions

In the following exercises, multiply.

Divide Rational Expressions

In the following exercises, divide.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

18. Practice 1

Mulitply: $\frac{6}{10}\cdot\frac{15}{12}.$

Mulitply: $\frac{20}{15}\cdot\frac{6}{8}.$

Mulitply: $\frac{3pq}{{q}^{2}}\cdot\frac{5{p}^{2}q}{6pq}.$

Mulitply: $\frac{6{x}^{3}y}{7{x}^{2}}\cdot\frac{2x{y}^{3}}{{x}^{2}y}.$

19. Practice 2

Mulitply: $\frac{5x}{{x}^{2}+5x+6}\cdot\frac{{x}^{2}-4}{10x}.$

Mulitply: $\frac{9{x}^{2}}{{x}^{2}+11x+30}\cdot\frac{{x}^{2}-36}{3{x}^{2}}.$

Multiply: $\frac{{x}^{2}-25}{{x}^{2}-3x-10}\cdot\frac{x+2}{x}.$

Multiply: $\frac{{x}^{2}-4x}{{x}^{2}+5x+6}\cdot\frac{x+2}{x}.$

20. Practice 3

Multiply: $\frac{12x-6{x}^{2}}{{x}^{2}+8x}\cdot\frac{{x}^{2}+11x+24}{{x}^{2}-4}.$

Multiply: $\frac{9v-3{v}^{2}}{9v+36}\cdot\frac{{v}^{2}+7v+12}{{v}^{2}-9}.$

Multiply: $\frac{3a-21}{{a}^{2}-9a+14}\cdot\frac{{a}^{2}-4}{3a+6}.$

Divide: $\frac{c+3}{5-c}\mathrm{÷}\frac{{c}^{2}-9}{c-5}.$

21. Practice 4

Divide: $\frac{2-d}{d-4}\mathrm{÷}\frac{4-{d}^{2}}{4-d}.$

Divide: $\frac{2{m}^{2}}{{m}^{2}-8m}\mathrm{÷}\frac{8{m}^{2}+24m}{{m}^{2}+m-6}.$

Divide: $\frac{15{n}^{2}}{3{n}^{2}+33n}\mathrm{÷}\frac{5n-5}{{n}^{2}+9n-22}.$

Divide: $\frac{3{a}^{2}-8a-3}{{a}^{2}-25}\mathrm{÷}\frac{3{a}^{2}-14a-5}{{a}^{2}+10a+25}.$

22. Practice 5

Divide: $\frac{4{b}^{2}+7b-2}{1-{b}^{2}}\mathrm{÷}\frac{4{b}^{2}+15b-4}{{b}^{2}-2b+1}.$

Divide: $\frac{{x}^{3}-8}{3{x}^{2}-6x+12}\mathrm{÷}\frac{{x}^{2}-4}{6}.$

Divide: $\frac{2{z}^{2}}{{z}^{2}-1}\mathrm{÷}\frac{{z}^{3}-{z}^{2}+z}{{z}^{3}-1}.$

Divide: $\frac{2{x}^{2}-14x-16}{4}\mathrm{÷}({x}^{2}+2x+1).$

23. Practice 6

Divide: $\frac{{y}^{2}-6y+8}{{y}^{2}-4y}\mathrm{÷}(3{y}^{2}-12y).$

Divide: $\frac{\frac{3{x}^{2}+7x+2}{4x+24}}{\frac{3{x}^{2}-14x-5}{{x}^{2}+x-30}}.$

Divide: $\frac{\frac{{y}^{2}-36}{2{y}^{2}+11y-6}}{\frac{2{y}^{2}-2y-60}{8y-4}}.$

Divide: $\frac{4m+4}{3m-15}\cdot\frac{{m}^{2}-3m-10}{{m}^{2}-4m-32}\mathrm{÷}\frac{12m-36}{6m-48}.$

24. Practice 7

Divide: $\frac{2{n}^{2}+10n}{n-1}\mathrm{÷}\frac{{n}^{2}+10n+24}{{n}^{2}+8n-9}\cdot\frac{n+4}{8{n}^{2}+12n}.$

$\frac{12}{16}\cdot\frac{4}{10}$

$\frac{18}{10}\cdot\frac{4}{30}$

$\frac{5{x}^{2}{y}^{4}}{12x{y}^{3}}\cdot\frac{6{x}^{2}}{20{y}^{2}}$

25. Practice 8

$\frac{12{a}^{3}b}{{b}^{2}}\cdot\frac{2a{b}^{2}}{9{b}^{3}}$

$\frac{5{p}^{2}}{{p}^{2}-5p-36}\cdot\frac{{p}^{2}-16}{10p}$

$\frac{4r}{{r}^{2}-3r-10}\cdot\frac{{r}^{2}-25}{8{r}^{2}}$

$\frac{{x}^{2}-7x}{{x}^{2}+6x+9}\cdot\frac{x+3}{4x}$

26. Practice 9

$\frac{{z}^{2}+3z}{{z}^{2}-3z-4}\cdot\frac{z-4}{{z}^{2}}$

$\frac{28-4b}{3b-3}\cdot\frac{{b}^{2}+8b-9}{{b}^{2}-49}$

$\frac{35d-7{d}^{2}}{{d}^{2}+7d}\cdot\frac{{d}^{2}+12d+35}{{d}^{2}-25}$

$\frac{4n+20}{{n}^{2}+n-20}\cdot\frac{{n}^{2}-16}{4n+16}$

27. Practice 10

$\frac{{q}^{2}-2q}{{q}^{2}+6q-16}\cdot\frac{{q}^{2}-64}{{q}^{2}-8q}$

$\frac{t-6}{3-t}\mathrm{÷}\frac{t-5}{{t}^{2}-9}$

$\frac{10+w}{w-8}\mathrm{÷}\frac{100-{w}^{2}}{8-w}$

$\frac{27{y}^{2}}{3y-21}\mathrm{÷}\frac{3{y}^{2}+18}{{y}^{2}+13y+42}$

28. Practice 11

$\frac{16{a}^{2}}{4a+36}\mathrm{÷}\frac{4{a}^{2}-24a}{{a}^{2}+4a-45}$

$\frac{5{c}^{2}+9c-2}{{c}^{2}-4}\mathrm{÷}\frac{5{c}^{2}-16c+3}{{c}^{2}+4c+4}$

$\frac{6{m}^{2}-11m-2}{9-{m}^{2}}\mathrm{÷}\frac{6{m}^{2}+25m+4}{{m}^{2}-6m+9}$

$\frac{3{s}^{2}}{{s}^{2}-16}\mathrm{÷}\frac{{s}^{3}+4{s}^{2}+16s}{{s}^{3}-64}$

29. Practice 12

$\frac{{p}^{3}+{q}^{3}}{3{p}^{2}+3pq+3{q}^{2}}\mathrm{÷}\frac{{p}^{2}-{q}^{2}}{12}$

$\frac{{t}^{2}-9}{2t}\mathrm{÷}({t}^{2}-6t+9)$

$\frac{2{y}^{2}-10yz-48{z}^{2}}{2y-1}\mathrm{÷}(4{y}^{2}-32yz)$

$\frac{\frac{2{a}^{2}-a-21}{5a+20}}{\frac{{a}^{2}+7a+12}{{a}^{2}+8a+16}}$

30. Practice 13

$\frac{\frac{12{c}^{2}-12}{2{c}^{2}-3c+1}}{\frac{4c+4}{6{c}^{2}-13c+5}}$

$\frac{10{m}^{2}+80m}{3m-9}\cdot\frac{{m}^{2}+4m-21}{{m}^{2}-9m+20}$ $\,\mathrm{÷}\frac{5{m}^{2}+10m}{2m-10}$

$\frac{12{p}^{2}+3p}{p+3}\mathrm{÷}\frac{{p}^{2}+2p-63}{{p}^{2}-p-12}$ $\,\cdot\frac{p-7}{9{p}^{3}-9{p}^{2}}$

Probability The director of large company is interviewing applicants for two identical jobs. If $w=$ the number of women applicants and $m=$ the number of men applicants, then the probability that two women are selected for the jobs is $\frac{w}{w+m}\cdot\frac{w-1}{w+m-1}.$

ⓐ Simplify the probability by multiplying the two rational expressions.

ⓑ Find the probability that two women are selected when $w=5$ and $m=10$ .

31. Practice 14

1. ⓐ Multiply $\frac{7}{4}\cdot\frac{9}{10}$ and explain all your steps.

ⓑ Multiply $\frac{n}{n-3}\cdot\frac{9}{n+3}$ and explain all your steps.