8.2 Multiply and Divide 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.

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

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, $x not equal to 0$ and $y not equal to 0$ .

Example 5

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

	$\frac{16 - 4 x}{2 x - 12} \cdot \frac{x^{2} - 5 x - 6}{x^{2} - 16}$
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.	$- \frac{2 (x + 1)}{(x + 4)}$

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.

Example 8

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

	$A mathematical expression showing the division of two algebraic fractions. The first fraction is 3n^2 over n^2 - 4n, and the second is 9n^2 - 45n over n^2 - 7n + 10.$
Rewrite the division as the product of the first rational expression and the reciprocal of the second.
Factor the numerators and denominators and then multiply.	$A mathematical expression showing a fraction. The numerator is 3 * n * n * (n - 5)(n - 2). The denominator is n(n - 4) * 3 * 3 * n * (n - 5).$
Simplify by dividing out common factors.	$An algebraic fraction undergoing simplification, with identical terms 'B', 'n', and 'n(n-5)' struck through in red in both the numerator and denominator.$
	$A mathematical expression showing the fraction (n-2) divided by 3(n-4).$

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.

Example 9

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

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

Example 10

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

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

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

$\begin{matrix} \frac{3}{5} \div 4 \\ \frac{3}{5} \div \frac{4}{1} \\ \frac{3}{5} \cdot \frac{1}{4} \end{matrix}$

We do the same thing when we divide rational expressions.

Example 11

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

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

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

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.

Example 13

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


Rewrite the division as multiplication by the reciprocal.	$A mathematical expression showing the multiplication of three rational algebraic fractions: (3x-6)/(4x-4), (x^2+2x-3)/(x^2-3x-10), and (8x+16)/(2x+12), with the last fraction in red.$
Factor the numerators and the denominators, and then multiply.	$A complex algebraic fraction displays a numerator of 3 * 8 * (x-2)(x+3)(x-1)(x+2) and a denominator of 4 * 2 * (x-1)(x+2)(x-5)(x+6), set up for simplification.$
Simplify by dividing out common factors.
Simplify.	$A mathematical expression showing a fraction. The numerator is 3(x-2)(x+3) and the denominator is (x-5)(x+6).$

Key Concepts

Multiplication of Rational Expressions
- If $p, q, r, s$ are polynomials where $q not equal to 0, s not equal to 0$ , then $\frac{p}{q} \cdot \frac{r}{s} = \frac{p r}{q s}$ .
- To multiply rational expressions, multiply the numerators and multiply the denominators
Multiply a Rational Expression
1. Factor each numerator and denominator completely.
2. Multiply the numerators and denominators.
3. Simplify by dividing out common factors.
Division of Rational Expressions
- If $p, q, r, s$ are polynomials where $q not equal to 0, r not equal to 0, s not equal to 0$ , then $\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \cdot \frac{s}{r}$ .
- To divide rational expressions multiply the first fraction by the reciprocal of the second.
Divide Rational Expressions
1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Factor the numerators and denominators completely.
3. Multiply the numerators and denominators together.
4. Simplify by dividing out common factors.

Practice Makes Perfect

Multiply Rational Expressions

In the following exercises, multiply.

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

$\frac{3}{10}$

$\frac{32}{5} \cdot \frac{16}{24}$

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

$\frac{6}{25}$

$\frac{21}{36} \cdot \frac{45}{24}$

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

$\frac{x^{3}}{8 y}$

$\frac{8 w^{3} y}{9 y^{2}} \cdot \frac{3 y}{4 w^{4}}$

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

$\frac{8 a^{4}}{3 b^{2}}$

$\frac{4 m n^{2}}{5 n^{3}} \cdot \frac{m n^{3}}{8 m^{2} n^{2}}$

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

$\frac{p (p - 4)}{2 (p - 9)}$

$\frac{3 q^{2}}{q^{2} + q - 6} \cdot \frac{q^{2} - 9}{9 q}$

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

$\frac{r + 5}{2 r (r + 2)}$

$\frac{s}{s^{2} - 9 s + 14} \cdot \frac{s^{2} - 49}{7 s^{2}}$

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

$\frac{x - 7}{4 (x + 3)}$

$\frac{2 y^{2} - 10 y}{y^{2} + 10 y + 25} \cdot \frac{y + 5}{6 y}$

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

$\frac{z + 3}{z (z + 1)}$

$\frac{2 a^{2} + 8 a}{a^{2} - 9 a + 20} \cdot \frac{a - 5}{a^{2}}$

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

$- \frac{4 (b + 9)}{3 (b + 7)}$

$\frac{18 c - 2 c^{2}}{6 c + 30} \cdot \frac{c^{2} + 7 c + 10}{c^{2} - 81}$

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

$−7$

$\frac{72 m - 12 m^{2}}{8 m + 32} \cdot \frac{m^{2} + 10 m + 24}{m^{2} - 36}$

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

$1$

$\frac{6 p^{2} - 6 p}{p^{2} + 7 p - 18} \cdot \frac{p^{2} - 81}{3 p^{2} - 27 p}$

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

$1$

$\frac{2 r^{2} - 2 r}{r^{2} + 4 r - 5} \cdot \frac{r^{2} - 25}{2 r^{2} - 10 r}$

Divide Rational Expressions

In the following exercises, divide.

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

$\frac{(6 - t) (t + 3)}{t - 5}$

$\frac{v - 5}{11 - v} \div \frac{v^{2} - 25}{v - 11}$

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

$- \frac{1}{10 - w}$

$\frac{7 + x}{x - 6} \div {\frac{49 - x}{x + 6}}^{2}$

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

$\frac{3 y^{2} (y + 6) (y + 7)}{(y - 7) (y^{2} + 6)}$

$\frac{24 z^{2}}{2 z - 8} \div \frac{4 z - 28}{z^{2} - 11 z + 28}$

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

$\frac{a (a - 5)}{a - 6}$

$\frac{24 b^{2}}{2 b - 4} \div \frac{12 b^{2} + 36 b}{b^{2} - 11 b + 18}$

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

$\frac{(c + 2) (c + 2)}{(c - 2) (c - 3)}$

$\frac{2 d^{2} + d - 3}{d^{2} - 16} \div \frac{2 d^{2} - 9 d - 18}{d^{2} - 8 d + 16}$

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

$- \frac{(m - 2) (m - 3)}{(3 + m) (m + 4)}$

$\frac{2 n^{2} - 3 n - 14}{25 - n^{2}} \div \frac{2 n^{2} - 13 n + 21}{n^{2} - 10 n + 25}$

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

$\frac{3 s}{s + 4}$

$\frac{r^{2} - 9}{15} \div \frac{r^{3} - 27}{5 r^{2} + 15 r + 45}$

$\frac{p^{3} + q^{3}}{3 p^{2} + 3 p q + 3 q^{2}} \div \frac{p^{2} - q^{2}}{12}$

$\frac{4 (p^{2} - p q + q^{2})}{(p - q) (p^{2} + p q + q^{2})}$

$\frac{v^{3} - 8 w^{3}}{2 v^{2} + 4 v w + 8 w^{2}} \div \frac{v^{2} - 4 w^{2}}{4}$

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

$\frac{t + 3}{2 t (t - 3)}$

$\frac{x^{2} + 3 x - 10}{4 x} \div (2 x^{2} + 20 x + 50)$

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

$\frac{y + 3 z}{2 y (2 y - 1)}$

$\frac{2 m^{2} - 98 n^{2}}{2 m + 6} \div (m^{2} - 7 m n)$

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

$\frac{2 a - 7}{5}$

$\frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}}$

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

$3 open parenthesis 3 c minus 5 close parenthesis$

$\frac{\frac{4 d^{2} + 7 d - 2}{35 d + 10}}{\frac{d^{2} - 4}{7 d^{2} - 12 d - 4}}$

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

$\frac{4 (m + 8) (m + 7)}{3 (m - 4) (m + 2)}$

$\frac{4 n^{2} + 32 n}{3 n + 2} \cdot \frac{3 n^{2} - n - 2}{n^{2} + n - 30}$
$\div \frac{108 n^{2} - 24 n}{n + 6}$

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

$\frac{(4 p + 1) (p - 4)}{3 p (p + 9) (p - 1)}$

$\frac{6 q + 3}{9 q^{2} - 9 q} \div \frac{q^{2} + 14 q + 33}{q^{2} + 4 q - 5}$
$\cdot \frac{4 q^{2} + 12 q}{12 q + 6}$

Everyday Math

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 $w equals 5$ and $m equals 10$ .

ⓐ $\frac{w (w - 1)}{(w + m) (w + m - 1)}$
ⓑ $\frac{2}{21}$

Area of a triangle The area of a triangle with base b and height h is $\frac{b h}{2} .$ If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is $\frac{9 b h}{2} .$ Calculate how the area of the new triangle compares to the area of the original triangle by dividing $\frac{9 b h}{2}$ by $\frac{b h}{2}$ .

Writing Exercises

ⓐ 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.

Answers will vary.

ⓐ Divide $\frac{24}{5} \div 6$ and explain all your steps.
ⓑ Divide $\frac{x^{2} - 1}{x} \div (x + 1)$ and explain all your steps.

Self Check

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

The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.

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

	$The image displays the multiplication of two fractions: 10/28 multiplied by 8/15, represented as (10/28) (8/15).$
Multiply the numerators and denominators.	$A mathematical expression displaying the fraction (10 multiplied by 8) divided by (28 multiplied by 15).$
Look for common factors, and then remove them.	$A mathematical fraction is shown with terms being cancelled out. In the numerator, 5 and 4 are crossed out, leaving 2 multiplied by 2. In the denominator, 4 and 5 are crossed out, leaving 7 multiplied by 3.$
Simplify.	$The fraction 4/21 is displayed on a white background, with 4 as the numerator and 21 as the denominator, separated by a horizontal line.$

	$A mathematical expression showing the product of two fractions: (2x / 3y^2) multiplied by (6xy^3 / x^2y).$
Multiply.	$An algebraic fraction is displayed with the numerator '2x multiplied by 6xy^3' and the denominator '3y^2 multiplied by x^2y'.$
Factor the numerator and denominator completely, and then remove common factors.	$A mathematical fraction showing simplification. Common factors like 'x', 'B', and 'y' are struck through in red from both the numerator and denominator, illustrating term cancellation.$
Simplify.

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

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

	$A mathematical expression showing the multiplication of two algebraic fractions: (2x-6)/(x^2-8x+15) * (x^2-25)/(2x+10).$
Factor each numerator and denominator.	$An algebraic expression illustrating the multiplication of two rational fractions: 2(x-3) over (x-3)(x-5) multiplied by (x-5)(x+5) over 2(x+5).$
Multiply the numerators and denominators.
Remove common factors.	$A fraction displays identical algebraic expressions in the numerator and denominator, with Z, (x-3), (x-5), and (x+5) terms all struck through, demonstrating cancellation.$
Simplify.