Add and Subtract Rational Expressions with a Common Denominator

Lynn Marecek; MaryAnne Anthony-Smith

Rational Expressions and Equations

64 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

Before you get started, take this readiness quiz.

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

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

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 $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , then

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

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.

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{5}{18}+\frac{7}{18}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5+7}{18}\hfill \\ \\ \\ \text{Add in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{12}{18}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator to}\hfill \\ \text{show the common factors.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{6·2}{6·3}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)6}·2}{\overline{)6}·3}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{2}{3}\hfill \end{array}$

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

$\frac{3}{4}$

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

$\frac{2}{5}$

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

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{3y}{4y-3}+\frac{7}{4y-3}\hfill \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{3y+7}{4y-3}\hfill \end{array}$

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

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

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

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

$\frac{x+1}{x-2}$

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{7x+12+{x}^{2}}{x+3}\hfill \\ \\ \\ \text{Write the degrees in descending order.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}+7x+12}{x+3}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x+3\right)\left(x+4\right)}{x+3}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(x+3\right)}\left(x+4\right)}{\overline{)x+3}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}x+4\hfill \end{array}$

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

$x+2$

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

$x+3$

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 $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , then

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

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.

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so subtract the numerators}\hfill \\ \text{and place the difference over the}\hfill \\ \text{common denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{n}^{2}-100}{n-10}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(n-10\right)\left(n+10\right)}{n-10}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(n-10\right)}\left(n+10\right)}{\overline{)n-10}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}n+10\hfill \end{array}$

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

$x-3$

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

$2x+5$

Be careful of the signs when you subtract a binomial!

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so subtract the numerators}\hfill \\ \text{and place the difference over the}\hfill \\ \text{common denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-\left(2y+24\right)}{y-6}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{y}^{2}-2y-24}{y-6}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(y-6\right)\left(y+4\right)}{y-6}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)\left(y-6\right)}\left(y+4\right)}{\overline{)y-6}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}y+4\hfill \end{array}$

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

$n+3$

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

$y-8$

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\hfill \\ \\ \\ \begin{array}{c}\text{Subtract the numerators and place the}\hfill \\ \text{difference over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-\left(4{x}^{2}+x-9\right)}{{x}^{2}-3x-18}\hfill \\ \\ \\ \text{Distribute the sign in the numerator.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\hfill \\ \\ \\ \text{Combine like terms.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and the}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\left(x-6\right)}{\left(x+3\right)\left(x-6\right)}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)\overline{)\left(x-6\right)}}{\left(x+3\right)\overline{)\left(x-6\right)}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\left(x-2\right)}{\left(x+3\right)}\hfill \end{array}$

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

$\frac{x-11}{x-2}$

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

$\frac{x-3}{x+9}$

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 $\frac{-1}{-1}$ .

Let’s see how this works.


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

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

Solution


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

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

$3$

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

$y+3$

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

Solution


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

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

$\frac{y+3}{y+2}$

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

$\frac{3n-2}{n-1}$

Key Concepts

Rational Expression Addition
- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , then
  
  $\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$
- To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
Rational Expression Subtraction
- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$ , then
  
  $\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}$
- To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

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

$\frac{3}{5}$

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

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

$\frac{3}{4}$

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

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

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

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

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

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

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

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

$p+8$

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

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

$r+8$

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

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

$8t$

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

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

$\frac{2w}{w-4}$

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

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

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

$y-8$

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

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

$3a+7$

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

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

$c+2$

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

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

$\frac{m-2}{3}$

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

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

$\frac{p+3}{p+5}$

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

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

$\frac{r+9}{r+7}$

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

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

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

$4$

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

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

$x+2$

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

In the following exercises, subtract.

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

$\frac{z+4}{z-5}$

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

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

$\frac{4b-3}{b-7}$

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

Everyday Math

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

$\frac{32r+32}{r\left(r+4\right)}$

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