Contents

Add and Subtract Rational Expressions with a Common Denominator

Learning Objectives

By the end of this section, you will be able to:

Add rational expressions with a common denominator
Subtract rational expressions with a common denominator
Add and subtract rational expressions whose denominators are opposites

Be Prepared 8.3

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

Add: $\frac{y}{3} + \frac{9}{3} . \frac{y}{3} + \frac{9}{3} .$
If you missed this problem, review Example 1.77.
Subtract: $\frac{10}{x} - \frac{2}{x} . \frac{10}{x} - \frac{2}{x} .$
If you missed this problem, review Example 1.79.
Factor completely: $8 n^{5} - 20 n^{3} . 8 n^{5} - 20 n^{3} .$
If you missed this problem, review Example 7.59.
Factor completely: $45 a^{3} - 5 a b^{2} . 45 a^{3} - 5 a b^{2} .$
If you missed this problem, review Example 7.62.

Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

Rational Expression Addition

If $p, q, and r p, q,$ are polynomials where $r \neq 0 r \neq 0$ , then

\frac{p}{r} + \frac{q}{r} = \frac{p + q}{r} \frac{p}{r} + \frac{q}{r} = \frac{p + q}{r}

To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

We will add two numerical fractions first, to remind us of how this is done.

Example 8.30

Add: $\frac{5}{18} + \frac{7}{18} . \frac{5}{18} + \frac{7}{18} .$

Show solution

Solution

$\begin{matrix} \frac{5}{18} + \frac{7}{18} \\ \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} & \frac{5 + 7}{18} \\ Add in the numerator. & \frac{12}{18} \\ \begin{matrix} Factor the numerator and denominator to \\ show the common factors. \end{matrix} & \frac{6 \cdot 2}{6 \cdot 3} \\ Remove common factors. & \frac{6 \cdot 2}{6 \cdot 3} \\ Simplify. & \frac{2}{3} \end{matrix} \begin{matrix} \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} & Add in the numerator. & \begin{matrix} Factor the numerator and denominator to \\ show the common factors. \end{matrix} & Remove common factors. & Simplify. \end{matrix}$

Try It 8.59

Add: $\frac{7}{16} + \frac{5}{16} . \frac{7}{16} + \frac{5}{16} .$

Try It 8.60

Add: $\frac{3}{10} + \frac{1}{10} . \frac{3}{10} + \frac{1}{10} .$

Remember, we do not allow values that would make the denominator zero. What value of $y y$ should be excluded in the next example?

Example 8.31

Add: $\frac{3 y}{4 y - 3} + \frac{7}{4 y - 3} . \frac{3 y}{4 y - 3} + \frac{7}{4 y - 3} .$

Show solution

Solution

$\begin{matrix} \frac{3 y}{4 y - 3} + \frac{7}{4 y - 3} \\ \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} & \frac{3 y + 7}{4 y - 3} \end{matrix} \begin{matrix} \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} \end{matrix}$

The numerator and denominator cannot be factored. The fraction is simplified.

Try It 8.61

Add: $\frac{5 x}{2 x + 3} + \frac{2}{2 x + 3} . \frac{5 x}{2 x + 3} + \frac{2}{2 x + 3} .$

Try It 8.62

Add: $\frac{x}{x - 2} + \frac{1}{x - 2} . \frac{x}{x - 2} + \frac{1}{x - 2} .$

Example 8.32

Add: $\frac{7 x + 12}{x + 3} + \frac{x^{2}}{x + 3} . \frac{7 x + 12}{x + 3} + \frac{x^{2}}{x + 3} .$

Show solution

Solution

$\begin{matrix} \frac{7 x + 12}{x + 3} + \frac{x^{2}}{x + 3} \\ \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} & \frac{7 x + 12 + x^{2}}{x + 3} \\ Write the degrees in descending order. & \frac{x^{2} + 7 x + 12}{x + 3} \\ Factor the numerator. & \frac{(x + 3) (x + 4)}{x + 3} \\ Simplify by removing common factors. & \frac{(x + 3) (x + 4)}{x + 3} \\ Simplify. & x + 4 \end{matrix} \begin{matrix} \begin{matrix} The fractions have a common \\ denominator, so add the numerators and \\ place the sum over the common \\ denominator. \end{matrix} & Write the degrees in descending order. & Factor the numerator. & Simplify by removing common factors. & Simplify. \end{matrix}$

Try It 8.63

Add: $\frac{9 x + 14}{x + 7} + \frac{x^{2}}{x + 7} . \frac{9 x + 14}{x + 7} + \frac{x^{2}}{x + 7} .$

Try It 8.64

Add: $\frac{x^{2} + 8 x}{x + 5} + \frac{15}{x + 5} . \frac{x^{2} + 8 x}{x + 5} + \frac{15}{x + 5} .$

Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

Rational Expression Subtraction

If $p, q, and r p, q,$ are polynomials where $r \neq 0 r \neq 0$ , then

\frac{p}{r} - \frac{q}{r} = \frac{p - q}{r} \frac{p}{r} - \frac{q}{r} = \frac{p - q}{r}

To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Example 8.33

Subtract: $\frac{n^{2}}{n - 10} - \frac{100}{n - 10} . \frac{n^{2}}{n - 10} - \frac{100}{n - 10} .$

Show solution

Solution

$\begin{matrix} \frac{n^{2}}{n - 10} - \frac{100}{n - 10} \\ \begin{matrix} The fractions have a common \\ denominator, so subtract the numerators \\ and place the difference over the \\ common denominator. \end{matrix} & \frac{n^{2} - 100}{n - 10} \\ Factor the numerator. & \frac{(n - 10) (n + 10)}{n - 10} \\ Simplify by removing common factors. & \frac{(n - 10) (n + 10)}{n - 10} \\ Simplify. & n + 10 \end{matrix} \begin{matrix} \begin{matrix} The fractions have a common \\ denominator, so subtract the numerators \\ and place the difference over the \\ common denominator. \end{matrix} & Factor the numerator. & Simplify by removing common factors. & Simplify. \end{matrix}$

Try It 8.65

Subtract: $\frac{x^{2}}{x + 3} - \frac{9}{x + 3} . \frac{x^{2}}{x + 3} - \frac{9}{x + 3} .$

Try It 8.66

Subtract: $\frac{4 x^{2}}{2 x - 5} - \frac{25}{2 x - 5} . \frac{4 x^{2}}{2 x - 5} - \frac{25}{2 x - 5} .$

Be careful of the signs when you subtract a binomial!

Example 8.34

Subtract: $\frac{y^{2}}{y - 6} - \frac{2 y + 24}{y - 6} . \frac{y^{2}}{y - 6} - \frac{2 y + 24}{y - 6} .$

Show solution

Solution

$\begin{matrix} \frac{y^{2}}{y - 6} - \frac{2 y + 24}{y - 6} \\ \begin{matrix} The fractions have a common \\ denominator, so subtract the numerators \\ and place the difference over the \\ common denominator. \end{matrix} & \frac{y^{2} - (2 y + 24)}{y - 6} \\ Distribute the sign in the numerator. & \frac{y^{2} - 2 y - 24}{y - 6} \\ Factor the numerator. & \frac{(y - 6) (y + 4)}{y - 6} \\ Remove common factors. & \frac{(y - 6) (y + 4)}{y - 6} \\ Simplify. & y + 4 \end{matrix} \begin{matrix} \begin{matrix} The fractions have a common \\ denominator, so subtract the numerators \\ and place the difference over the \\ common denominator. \end{matrix} & Distribute the sign in the numerator. & Factor the numerator. & Remove common factors. & Simplify. \end{matrix}$

Try It 8.67

Subtract: $\frac{n^{2}}{n - 4} - \frac{n + 12}{n - 4} . \frac{n^{2}}{n - 4} - \frac{n + 12}{n - 4} .$

Try It 8.68

Subtract: $\frac{y^{2}}{y - 1} - \frac{9 y - 8}{y - 1} . \frac{y^{2}}{y - 1} - \frac{9 y - 8}{y - 1} .$

Example 8.35

Subtract: $\frac{5 x^{2} - 7 x + 3}{x^{2} - 3 x - 18} - \frac{4 x^{2} + x - 9}{x^{2} - 3 x - 18} . \frac{5 x^{2} - 7 x + 3}{x^{2} - 3 x - 18} - \frac{4 x^{2} + x - 9}{x^{2} - 3 x - 18} .$

Show solution

Solution

$\begin{matrix} \frac{5 x^{2} - 7 x + 3}{x^{2} - 3 x - 18} - \frac{4 x^{2} + x - 9}{x^{2} - 3 x - 18} \\ \begin{matrix} Subtract the numerators and place the \\ difference over the common \\ denominator. \end{matrix} & \frac{5 x^{2} - 7 x + 3 - (4 x^{2} + x - 9)}{x^{2} - 3 x - 18} \\ Distribute the sign in the numerator. & \frac{5 x^{2} - 7 x + 3 - 4 x^{2} - x + 9}{x^{2} - 3 x - 18} \\ Combine like terms. & \frac{x^{2} - 8 x + 12}{x^{2} - 3 x - 18} \\ \begin{matrix} Factor the numerator and the \\ denominator. \end{matrix} & \frac{(x - 2) (x - 6)}{(x + 3) (x - 6)} \\ Simplify by removing common factors. & \frac{(x - 2) (x - 6)}{(x + 3) (x - 6)} \\ Simplify. & \frac{(x - 2)}{(x + 3)} \end{matrix} \begin{matrix} \begin{matrix} Subtract the numerators and place the \\ difference over the common \\ denominator. \end{matrix} & Distribute the sign in the numerator. & Combine like terms. & \begin{matrix} Factor the numerator and the \\ denominator. \end{matrix} & Simplify by removing common factors. & Simplify. \end{matrix}$

Try It 8.69

Subtract: $\frac{4 x^{2} - 11 x + 8}{x^{2} - 3 x + 2} - \frac{3 x^{2} + x - 3}{x^{2} - 3 x + 2} . \frac{4 x^{2} - 11 x + 8}{x^{2} - 3 x + 2} - \frac{3 x^{2} + x - 3}{x^{2} - 3 x + 2} .$

Try It 8.70

Subtract: $\frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81} . \frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81} .$

Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by $\frac{−1}{−1} \frac{−1}{−1}$ .

Let’s see how this works.


Multiply the second fraction by $\frac{−1}{−1} \frac{−1}{−1}$ .
The denominators are the same.
Simplify.

Example 8.36

Add: $\frac{4 u - 1}{3 u - 1} + \frac{u}{1 - 3 u} . \frac{4 u - 1}{3 u - 1} + \frac{u}{1 - 3 u} .$

Show solution

Solution


The denominators are opposites, so multiply the second fraction by $\frac{−1}{−1} \frac{−1}{−1}$ .
Simplify the second fraction.
The denominators are the same. Add the numerators.
Simplify.
Simplify.

Try It 8.71

Add: $\frac{8 x - 15}{2 x - 5} + \frac{2 x}{5 - 2 x} . \frac{8 x - 15}{2 x - 5} + \frac{2 x}{5 - 2 x} .$

Try It 8.72

Add: $\frac{6 y^{2} + 7 y - 10}{4 y - 7} + \frac{2 y^{2} + 2 y + 11}{7 - 4 y} . \frac{6 y^{2} + 7 y - 10}{4 y - 7} + \frac{2 y^{2} + 2 y + 11}{7 - 4 y} .$

Example 8.37

Subtract: $\frac{m^{2} - 6 m}{m^{2} - 1} - \frac{3 m + 2}{1 - m^{2}} . \frac{m^{2} - 6 m}{m^{2} - 1} - \frac{3 m + 2}{1 - m^{2}} .$

Show solution

Solution


The denominators are opposites, so multiply the second fraction by $\frac{−1}{−1} \frac{−1}{−1}$ .
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.	$\frac{m^{2} - 6 m + 3 m + 2}{m^{2} - 1} \frac{m^{2} - 6 m + 3 m + 2}{m^{2} - 1}$
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Try It 8.73

Subtract: $\frac{y^{2} - 5 y}{y^{2} - 4} - \frac{6 y - 6}{4 - y^{2}} . \frac{y^{2} - 5 y}{y^{2} - 4} - \frac{6 y - 6}{4 - y^{2}} .$

Try It 8.74

Subtract: $\frac{2 n^{2} + 8 n - 1}{n^{2} - 1} - \frac{n^{2} - 7 n - 1}{1 - n^{2}} . \frac{2 n^{2} + 8 n - 1}{n^{2} - 1} - \frac{n^{2} - 7 n - 1}{1 - n^{2}} .$

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

129.

$\frac{2}{15} + \frac{7}{15} \frac{2}{15} + \frac{7}{15}$

130.

$\frac{4}{21} + \frac{3}{21} \frac{4}{21} + \frac{3}{21}$

131.

$\frac{7}{24} + \frac{11}{24} \frac{7}{24} + \frac{11}{24}$

132.

$\frac{7}{36} + \frac{13}{36} \frac{7}{36} + \frac{13}{36}$

133.

$\frac{3 a}{a - b} + \frac{1}{a - b} \frac{3 a}{a - b} + \frac{1}{a - b}$

134.

$\frac{3 c}{4 c - 5} + \frac{5}{4 c - 5} \frac{3 c}{4 c - 5} + \frac{5}{4 c - 5}$

135.

$\frac{d}{d + 8} + \frac{5}{d + 8} \frac{d}{d + 8} + \frac{5}{d + 8}$

136.

$\frac{7 m}{2 m + n} + \frac{4}{2 m + n} \frac{7 m}{2 m + n} + \frac{4}{2 m + n}$

137.

$\frac{p^{2} + 10 p}{p + 2} + \frac{16}{p + 2} \frac{p^{2} + 10 p}{p + 2} + \frac{16}{p + 2}$

138.

$\frac{q^{2} + 12 q}{q + 3} + \frac{27}{q + 3} \frac{q^{2} + 12 q}{q + 3} + \frac{27}{q + 3}$

139.

$\frac{2 r^{2}}{2 r - 1} + \frac{15 r - 8}{2 r - 1} \frac{2 r^{2}}{2 r - 1} + \frac{15 r - 8}{2 r - 1}$

140.

$\frac{3 s^{2}}{3 s - 2} + \frac{13 s - 10}{3 s - 2} \frac{3 s^{2}}{3 s - 2} + \frac{13 s - 10}{3 s - 2}$

141.

$\frac{8 t^{2}}{t + 4} + \frac{32 t}{t + 4} \frac{8 t^{2}}{t + 4} + \frac{32 t}{t + 4}$

142.

$\frac{6 v^{2}}{v + 5} + \frac{30 v}{v + 5} \frac{6 v^{2}}{v + 5} + \frac{30 v}{v + 5}$

143.

$\frac{2 w^{2}}{w^{2} - 16} + \frac{8 w}{w^{2} - 16} \frac{2 w^{2}}{w^{2} - 16} + \frac{8 w}{w^{2} - 16}$

144.

$\frac{7 x^{2}}{x^{2} - 9} + \frac{21 x}{x^{2} - 9} \frac{7 x^{2}}{x^{2} - 9} + \frac{21 x}{x^{2} - 9}$

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

145.

$\frac{y^{2}}{y + 8} - \frac{64}{y + 8} \frac{y^{2}}{y + 8} - \frac{64}{y + 8}$

146.

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

147.

$\frac{9 a^{2}}{3 a - 7} - \frac{49}{3 a - 7} \frac{9 a^{2}}{3 a - 7} - \frac{49}{3 a - 7}$

148.

$\frac{25 b^{2}}{5 b - 6} - \frac{36}{5 b - 6} \frac{25 b^{2}}{5 b - 6} - \frac{36}{5 b - 6}$

149.

$\frac{c^{2}}{c - 8} - \frac{6 c + 16}{c - 8} \frac{c^{2}}{c - 8} - \frac{6 c + 16}{c - 8}$

150.

$\frac{d^{2}}{d - 9} - \frac{6 d + 27}{d - 9} \frac{d^{2}}{d - 9} - \frac{6 d + 27}{d - 9}$

151.

$\frac{3 m^{2}}{6 m - 30} - \frac{21 m - 30}{6 m - 30} \frac{3 m^{2}}{6 m - 30} - \frac{21 m - 30}{6 m - 30}$

152.

$\frac{2 n^{2}}{4 n - 32} - \frac{30 n - 16}{4 n - 32} \frac{2 n^{2}}{4 n - 32} - \frac{30 n - 16}{4 n - 32}$

153.

$\frac{6 p^{2} + 3 p + 4}{p^{2} + 4 p - 5} - \frac{5 p^{2} + p + 7}{p^{2} + 4 p - 5} \frac{6 p^{2} + 3 p + 4}{p^{2} + 4 p - 5} - \frac{5 p^{2} + p + 7}{p^{2} + 4 p - 5}$

154.

$\frac{5 q^{2} + 3 q - 9}{q^{2} + 6 q + 8} - \frac{4 q^{2} + 9 q + 7}{q^{2} + 6 q + 8} \frac{5 q^{2} + 3 q - 9}{q^{2} + 6 q + 8} - \frac{4 q^{2} + 9 q + 7}{q^{2} + 6 q + 8}$

155.

$\frac{5 r^{2} + 7 r - 33}{r^{2} - 49} - \frac{4 r^{2} - 5 r - 30}{r^{2} - 49} \frac{5 r^{2} + 7 r - 33}{r^{2} - 49} - \frac{4 r^{2} - 5 r - 30}{r^{2} - 49}$

156.

$\frac{7 t^{2} - t - 4}{t^{2} - 25} - \frac{6 t^{2} + 2 t - 1}{t^{2} - 25} \frac{7 t^{2} - t - 4}{t^{2} - 25} - \frac{6 t^{2} + 2 t - 1}{t^{2} - 25}$

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

157.

$\frac{10 v}{2 v - 1} + \frac{2 v + 4}{1 - 2 v} \frac{10 v}{2 v - 1} + \frac{2 v + 4}{1 - 2 v}$

158.

$\frac{20 w}{5 w - 2} + \frac{5 w + 6}{2 - 5 w} \frac{20 w}{5 w - 2} + \frac{5 w + 6}{2 - 5 w}$

159.

$\frac{10 x^{2} + 16 x - 7}{8 x - 3} + \frac{2 x^{2} + 3 x - 1}{3 - 8 x} \frac{10 x^{2} + 16 x - 7}{8 x - 3} + \frac{2 x^{2} + 3 x - 1}{3 - 8 x}$

160.

$\frac{6 y^{2} + 2 y - 11}{3 y - 7} + \frac{3 y^{2} - 3 y + 17}{7 - 3 y} \frac{6 y^{2} + 2 y - 11}{3 y - 7} + \frac{3 y^{2} - 3 y + 17}{7 - 3 y}$

In the following exercises, subtract.

161.

$\frac{z^{2} + 6 z}{z^{2} - 25} - \frac{3 z + 20}{25 - z^{2}} \frac{z^{2} + 6 z}{z^{2} - 25} - \frac{3 z + 20}{25 - z^{2}}$

162.

$\frac{a^{2} + 3 a}{a^{2} - 9} - \frac{3 a - 27}{9 - a^{2}} \frac{a^{2} + 3 a}{a^{2} - 9} - \frac{3 a - 27}{9 - a^{2}}$

163.

$\frac{2 b^{2} + 30 b - 13}{b^{2} - 49} - \frac{2 b^{2} - 5 b - 8}{49 - b^{2}} \frac{2 b^{2} + 30 b - 13}{b^{2} - 49} - \frac{2 b^{2} - 5 b - 8}{49 - b^{2}}$

164.

$\frac{c^{2} + 5 c - 10}{c^{2} - 16} - \frac{c^{2} - 8 c - 10}{16 - c^{2}} \frac{c^{2} + 5 c - 10}{c^{2} - 16} - \frac{c^{2} - 8 c - 10}{16 - c^{2}}$

Everyday Math

165.

Sarah ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. If $r r$ represents Sarah’s speed when she ran, then her running time is modeled by the expression $\frac{8}{r} \frac{8}{r}$ and her biking time is modeled by the expression $\frac{24}{r + 4} . \frac{24}{r + 4} .$ Add the rational expressions $\frac{8}{r} + \frac{24}{r + 4} \frac{8}{r} + \frac{24}{r + 4}$ to get an expression for the total amount of time Sarah ran and biked.

166.

If Pete can paint a wall in $p p$ hours, then in one hour he can paint $\frac{1}{p} \frac{1}{p}$ of the wall. It would take Penelope 3 hours longer than Pete to paint the wall, so in one hour she can paint $\frac{1}{p + 3} \frac{1}{p + 3}$ of the wall. Add the rational expressions $\frac{1}{p} + \frac{1}{p + 3} \frac{1}{p} + \frac{1}{p + 3}$ to get an expression for the part of the wall Pete and Penelope would paint in one hour if they worked together.

Writing Exercises

167.

Donald thinks that $\frac{3}{x} + \frac{4}{x} \frac{3}{x} + \frac{4}{x}$ is $\frac{7}{2 x} . \frac{7}{2 x} .$ Is Donald correct? Explain.

168.

Explain how you find the Least Common Denominator of $x^{2} + 5 x + 4 x^{2} + 5 x + 4$ and $x^{2} - 16 . x^{2} - 16 .$

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 “add rational expressions with a common denominator.”, the next row reads “subtract rational expressions with a common denominator.”, the next row reads, “add and subtract rational expressions whose denominators are opposites.”, the last row reads “What does this checklist tell you about your mastery of this section? What steps will you take to improve?” The remaining columns are blank.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?