## Learning Objectives

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

- Find the least common denominator of rational expressions
- Find equivalent rational expressions
- Add rational expressions with different denominators
- Subtract rational expressions with different denominators

## Be Prepared 8.13

Before you get started, take this readiness quiz.

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

Add: $\frac{7}{10}+\frac{8}{15}.$

If you missed this problem, review Example 1.81.

## Be Prepared 8.14

Subtract: $6\left(2x+1\right)-4\left(x-5\right).$

If you missed this problem, review Example 1.139.

## Be Prepared 8.15

Find the Greatest Common Factor of $9{x}^{2}{y}^{3}$ and $12x{y}^{5}$.

If you missed this problem, review Example 7.3.

## Be Prepared 8.16

Factor completely $\mathrm{-48}n-12$.

If you missed this problem, review Example 7.11.

## Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Let’s look at the example $\frac{7}{12}+\frac{5}{18}$ from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.

To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.

$$\frac{\begin{array}{}\\ \\ 12=2\xb72\xb73\hfill \\ 18=2\xb73\xb73\hfill \end{array}}{\begin{array}{}\\ \\ \phantom{\rule{0ex}{0ex}}\text{LCD}=2\xb72\xb73\xb73\hfill \\ \phantom{\rule{0ex}{0ex}}\text{LCD}=36\hfill \end{array}}$$

We do the same thing for rational expressions. However, we leave the LCD in factored form.

## How To

### Find the least common denominator of rational expressions.

- Step 1. Factor each expression completely.
- Step 2. List the factors of each expression. Match factors vertically when possible.
- Step 3. Bring down the columns.
- Step 4. Multiply the factors.

Remember, we always exclude values that would make the denominator zero. What values of *x* should we exclude in this next example?

## Example 8.38

Find the LCD for $\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}$.

### Solution

$\text{Find the LCD for}\phantom{\rule{0ex}{0ex}}\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.$ | |

Factor each expression completely, lining up common factors. Bring down the columns. | $\frac{\begin{array}{c}{x}^{2}-2x-3=(x+1)(x-3)\hfill \\ {x}^{2}+4x+3=(x+1)\phantom{\rule{0ex}{0ex}}(x+3)\hfill \end{array}}{\phantom{\rule{0ex}{0ex}}\text{LCD}=(x+1)(x-3)(x+3)}$ |

Multiply the factors. | $\text{The LCD is}\phantom{\rule{0ex}{0ex}}\left(x+1\right)\left(x-3\right)\left(x+3\right).$ |

## Try It 8.75

Find the LCD for $\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}$.

## Try It 8.76

Find the LCD for $\frac{x}{{x}^{2}+8x+15},\frac{5}{{x}^{2}+9x+18}$.

## Find Equivalent Rational Expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.

We will do the same thing for rational expressions.

## Example 8.39

Rewrite as equivalent rational expressions with denominator $(x+1)(x-3)(x+3)$: $\frac{8}{{x}^{2}-2x-3},\frac{3x}{{x}^{2}+4x+3}.$

### Solution

Factor each denominator. | |

Find the LCD. | |

Multiply each denominator by the 'missing' factor and multiply each numerator by the same factor. | |

Simplify the numerators. |

## Try It 8.77

Rewrite as equivalent rational expressions with denominator $(x+3)(x-4)(x+4)$:

$\frac{2}{{x}^{2}-x-12},\frac{1}{{x}^{2}-16}.$

## Try It 8.78

Rewrite as equivalent rational expressions with denominator $(x+3)(x+5)(x+6)$:

$\frac{x}{{x}^{2}+8x+15},\frac{5}{{x}^{2}+9x+18}.$

## Add Rational Expressions with Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

## Example 8.40

Add: $\frac{7}{12}+\frac{5}{18}.$

### Solution

Find the LCD of 12 and 18. | |

Rewrite each fraction as an equivalent fraction with the LCD. | |

Add the fractions. | |

The fraction cannot be simplified. |

## Try It 8.79

Add: $\frac{11}{30}+\frac{7}{12}.$

## Try It 8.80

Add: $\frac{3}{8}+\frac{9}{20}.$

Now we will add rational expressions whose denominators are monomials.

## Example 8.41

Add: $\frac{5}{12{x}^{2}y}+\frac{4}{21x{y}^{2}}.$

### Solution

Find the LCD of $12{x}^{2}y$ and $21x{y}^{2}$. | ||

Rewrite each rational expression as an equivalent fraction with the LCD. | ||

Simplify. | ||

Add the rational expressions. | ||

There are no factors common to the numerator and denominator. The fraction cannot be simplified. |

## Try It 8.81

Add: $\frac{2}{15{a}^{2}b}+\frac{5}{6a{b}^{2}}.$

## Try It 8.82

Add: $\frac{5}{16c}+\frac{3}{8c{d}^{2}}.$

Now we are ready to tackle polynomial denominators.

## Example 8.42

### How to Add Rational Expressions with Different Denominators

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

### Solution

## Try It 8.83

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

## Try It 8.84

Add: $\frac{4}{m+3}+\frac{3}{m+4}.$

The steps to use to add rational expressions are summarized in the following procedure box.

## How To

### Add rational expressions.

- Step 1. Determine if the expressions have a common denominator.
**Yes**– go to step 2.**No**– Rewrite each rational expression with the LCD.

Find the LCD.

Rewrite each rational expression as an equivalent rational expression with the LCD. - Step 2. Add the rational expressions.
- Step 3. Simplify, if possible.

## Example 8.43

Add: $\frac{2a}{2ab+{b}^{2}}+\frac{3a}{4{a}^{2}-{b}^{2}}.$

### Solution

Do the expressions have a common denominator? No. Rewrite each expression with the LCD. | |

Find the LCD. | |

Rewrite each rational expression as an equivalent rational expression with the LCD. | |

Simplify the numerators. | |

Add the rational expressions. | |

Simplify the numerator. | |

Factor the numerator. | |

There are no factors common to the numerator and denominator. The fraction cannot be simplified. |

## Try It 8.85

Add: $\frac{5x}{xy-{y}^{2}}+\frac{2x}{{x}^{2}-{y}^{2}}.$

## Try It 8.86

Add: $\frac{7}{2m+6}+\frac{4}{{m}^{2}+4m+3}.$

Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as $\frac{2a\left(2a-b\right)}{b\left(2a+b\right)\left(2a-b\right)}$ to be able to add it to $\frac{3a\xb7b}{\left(2a+b\right)\left(2a-b\right)\xb7b}$. Simplify only after you have combined the numerators.

## Example 8.44

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

### Solution

Do the expressions have a common denominator? No. Rewrite each expression with the LCD. | |

Find the LCD. | |

Rewrite each rational expression as an equivalent fraction with the LCD. | |

Simplify the numerators. | |

Add the rational expressions. | |

Simplify the numerator. | |

The numerator is prime, so there are no common factors. |

## Try It 8.87

Add: $\frac{1}{{m}^{2}-m-2}+\frac{5m}{{m}^{2}+3m+2}.$

## Try It 8.88

Add: $\frac{2n}{{n}^{2}-3n-10}+\frac{6}{{n}^{2}+5n+6}.$

## Subtract Rational Expressions with Different Denominators

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

## Example 8.45

### How to Subtract Rational Expressions with Different Denominators

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

### Solution

## Try It 8.89

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

## Try It 8.90

Subtract: $\frac{z+3}{z+2}-\frac{z}{z+3}.$

The steps to take to subtract rational expressions are listed below.

## How To

### Subtract rational expressions.

- Step 1. Determine if they have a common denominator.
**Yes**– go to step 2.**No**– Rewrite each rational expression with the LCD.

Find the LCD.

Rewrite each rational expression as an equivalent rational expression with the LCD. - Step 2. Subtract the rational expressions.
- Step 3. Simplify, if possible.

## Example 8.46

Subtract: $\frac{8y}{{y}^{2}-16}-\frac{4}{y-4}.$

### Solution

Do the expressions have a common denominator? No. Rewrite each expression with the LCD. | |

Find the LCD. | |

Rewrite each rational expression as an equivalent rational expression with the LCD. | |

Simplify the numerators. | |

Subtract the rational expressions. | |

Simplify the numerators. | |

Factor the numerator to look for common factors. | |

Remove common factors. | |

Simplify. |

## Try It 8.91

Subtract: $\frac{2x}{{x}^{2}-4}-\frac{1}{x+2}.$

## Try It 8.92

Subtract: $\frac{3}{z+3}-\frac{6z}{{z}^{2}-9}.$

There are lots of negative signs in the next example. Be extra careful!

## Example 8.47

Subtract: $\frac{\mathrm{-3}n-9}{{n}^{2}+n-6}-\frac{n+3}{2-n}.$

### Solution

Factor the denominator. | |

Since $n-2$ and $2-n$ are opposites, we will mutliply the second rational expression by$\frac{\mathrm{-1}}{\mathrm{-1}}$. | |

Simplify. | |

Do the expressions have a common denominator? No. | |

Find the LCD. | |

Rewrite each rational expression as an equivalent rational expression with the LCD. | |

Simplify the numerators. | |

Simplify the rational expressions. | |

Simplify the numerator. | |

Factor the numerator to look for common factors. | |

Simplify. |

## Try It 8.93

Subtract: $\frac{3x-1}{{x}^{2}-5x-6}-\frac{2}{6-x}.$

## Try It 8.94

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

When one expression is not in fraction form, we can write it as a fraction with denominator 1.

## Example 8.48

Subtract: $\frac{5c+4}{c-2}-3.$

### Solution

Write $3$ as $\frac{3}{1}$ to have 2 rational expressions. | |

Do the rational expressions have a common denominator? No. | |

Find the LCD of $c-2$ and $1.$ LCD = $c-2.$ | |

Rewrite $\frac{3}{1}$ as an equivalent rational expression with the LCD. | |

Simplify. | |

Subtract the rational expressions. | |

Simplify. | |

Factor to check for common factors. | |

There are no common factors; the rational expression is simplified. |

## Try It 8.95

Subtract: $\frac{2x+1}{x-7}-3.$

## Try It 8.96

Subtract: $\frac{4y+3}{2y-1}-5.$

## How To

### Add or subtract rational expressions.

- Step 1. Determine if the expressions have a common denominator.
**Yes**– go to step 2.**No**– Rewrite each rational expression with the LCD.

Find the LCD.

Rewrite each rational expression as an equivalent rational expression with the LCD. - Step 2. Add or subtract the rational expressions.
- Step 3. Simplify, if possible.

We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.

## Example 8.49

Simplify: $\frac{2u}{u-1}+\frac{1}{u}-\frac{2u-1}{{u}^{2}-u}.$

### Solution

Do the rational expressions have a common denominator? No. | |

Find the LCD. | |

Rewrite each rational expression as an equivalent rational expression with the LCD. | |

Write as one rational expression. | |

Simplify. | |

Factor the numerator, and remove common factors. | |

Simplify. |

## Try It 8.97

Simplify: $\frac{v}{v+1}+\frac{3}{v-1}-\frac{6}{{v}^{2}-1}.$

## Try It 8.98

Simplify: $\frac{3w}{w+2}+\frac{2}{w+7}-\frac{17w+4}{{w}^{2}+9w+14}.$

## Section 8.4 Exercises

### Practice Makes Perfect

In the following exercises, find the LCD.

169.

$\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}$

170.

$\frac{8}{{y}^{2}+12y+35},\frac{3y}{{y}^{2}+y-42}$

171.

$\frac{9}{{z}^{2}+2z-8},\frac{4z}{{z}^{2}-4}$

172.

$\frac{6}{{a}^{2}+14a+45},\frac{5a}{{a}^{2}-81}$

173.

$\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}$

174.

$\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-10c+16}$

175.

$\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}$

176.

$\frac{3}{5{m}^{2}-3m-2},\frac{6m}{5{m}^{2}+17m+6}$

In the following exercises, write as equivalent rational expressions with the given LCD.

177.

$\frac{5}{{x}^{2}-2x-8},\frac{2x}{{x}^{2}-x-12}$

LCD $(x-4)(x+2)(x+3)$

178.

$\frac{8}{{y}^{2}+12y+35},\frac{3y}{{y}^{2}+y-42}$

LCD $(y+7)(y+5)(y-6)$

179.

$\frac{9}{{z}^{2}+2z-8},\frac{4z}{{z}^{2}-4}$

LCD $(z-2)(z+4)(z+2)$

180.

$\frac{6}{{a}^{2}+14a+45},\frac{5a}{{a}^{2}-81}$

LCD $(a+9)(a+5)(a-9)$

181.

$\frac{4}{{b}^{2}+6b+9},\frac{2b}{{b}^{2}-2b-15}$

LCD $(b+3)(b+3)(b-5)$

182.

$\frac{5}{{c}^{2}-4c+4},\frac{3c}{{c}^{2}-10c+16}$

LCD $(c-2)(c-2)(c-8)$

183.

$\frac{2}{3{d}^{2}+14d-5},\frac{5d}{3{d}^{2}-19d+6}$

LCD $(3d-1)(d+5)(d-6)$

184.

$\frac{3}{5{m}^{2}-3m-2},\frac{6m}{5{m}^{2}+17m+6}$

LCD $(5m+2)(m-1)(m+3)$

In the following exercises, add.

185.

$\frac{5}{24}+\frac{11}{36}$

186.

$\frac{7}{30}+\frac{13}{45}$

187.

$\frac{9}{20}+\frac{11}{30}$

188.

$\frac{8}{27}+\frac{7}{18}$

189.

$\frac{7}{10{x}^{2}y}+\frac{4}{15x{y}^{2}}$

190.

$\frac{1}{12{a}^{3}{b}^{2}}+\frac{5}{9{a}^{2}{b}^{3}}$

191.

$\frac{1}{2m}+\frac{7}{8{m}^{2}n}$

192.

$\frac{5}{6{p}^{2}q}+\frac{1}{4p}$

193.

$\frac{3}{r+4}+\frac{2}{r-5}$

194.

$\frac{4}{s-7}+\frac{5}{s+3}$

195.

$\frac{8}{t+5}+\frac{6}{t-5}$

196.

$\frac{7}{v+5}+\frac{9}{v-5}$

197.

$\frac{5}{3w-2}+\frac{2}{w+1}$

198.

$\frac{4}{2x+5}+\frac{2}{x-1}$

199.

$\frac{2y}{y+3}+\frac{3}{y-1}$

200.

$\frac{3z}{z-2}+\frac{1}{z+5}$

201.

$\frac{5b}{{a}^{2}b-2{a}^{2}}+\frac{2b}{{b}^{2}-4}$

202.

$\frac{4}{cd+3c}+\frac{1}{{d}^{2}-9}$

203.

$\frac{2m}{3m-3}+\frac{5m}{{m}^{2}+3m-4}$

204.

$\frac{3}{4n+4}+\frac{6}{{n}^{2}-n-2}$

205.

$\frac{3}{{n}^{2}+3n-18}+\frac{4n}{{n}^{2}+8n+12}$

206.

$\frac{6}{{q}^{2}-3q-10}+\frac{5q}{{q}^{2}-8q+15}$

207.

$\frac{3r}{{r}^{2}+7r+6}+\frac{9}{{r}^{2}+4r+3}$

208.

$\frac{2s}{{s}^{2}+2s-8}+\frac{4}{{s}^{2}+3s-10}$

In the following exercises, subtract.

209.

$\frac{t}{t-6}-\frac{t-2}{t+6}$

210.

$\frac{v}{v-3}-\frac{v-6}{v+1}$

211.

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

212.

$\frac{x-3}{x+6}-\frac{x}{x+3}$

213.

$\frac{y-4}{y+1}-\frac{1}{y+7}$

214.

$\frac{z+8}{z-3}-\frac{z}{z-2}$

215.

$\frac{5a}{a+3}-\frac{a+2}{a+6}$

216.

$\frac{3b}{b-2}-\frac{b-6}{b-8}$

217.

$\frac{6c}{{c}^{2}-25}-\frac{3}{c+5}$

218.

$\frac{4d}{{d}^{2}-81}-\frac{2}{d+9}$

219.

$\frac{6}{m+6}-\frac{12m}{{m}^{2}-36}$

220.

$\frac{4}{n+4}-\frac{8n}{{n}^{2}-16}$

221.

$\frac{\mathrm{-9}p-17}{{p}^{2}-4p-21}-\frac{p+1}{7-p}$

222.

$\frac{-13q-8}{{q}^{2}+2q-24}-\frac{q+2}{4-q}$

223.

$\frac{\mathrm{-2}r-16}{{r}^{2}+6r-16}-\frac{5}{2-r}$

224.

$\frac{2t-30}{{t}^{2}+6t-27}-\frac{2}{3-t}$

225.

$\frac{5v-2}{v+3}-4$

226.

$\frac{6w+5}{w-1}+2$

227.

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

228.

$\frac{8y-4}{5y+2}-6$

In the following exercises, add and subtract.

229.

$\frac{5a}{a-2}+\frac{9}{a}-\frac{2a+18}{{a}^{2}-2a}$

230.

$\frac{2b}{b-5}+\frac{3}{2b}-\frac{2b-15}{2{b}^{2}-10b}$

231.

$\frac{c}{c+2}+\frac{5}{c-2}-\frac{10c}{{c}^{2}-4}$

232.

$\frac{6d}{d-5}+\frac{1}{d+4}-\frac{7d-5}{{d}^{2}-d-20}$

In the following exercises, simplify.

233.

$\frac{6a}{3ab+{b}^{2}}+\frac{3a}{9{a}^{2}-{b}^{2}}$

234.

$\frac{2c}{2c+10}+\frac{7c}{{c}^{2}+9c+20}$

235.

$\frac{6d}{{d}^{2}-64}-\frac{3}{d-8}$

236.

$\frac{5}{n+7}-\frac{10n}{{n}^{2}-49}$

237.

$\frac{4m}{{m}^{2}+6m-7}+\frac{2}{{m}^{2}+10m+21}$

238.

$\frac{3p}{{p}^{2}+4p-12}+\frac{1}{{p}^{2}+p-30}$

239.

$\frac{\mathrm{-5}n-5}{{n}^{2}+n-6}+\frac{n+1}{2-n}$

240.

$\frac{\mathrm{-4}b-24}{{b}^{2}+b-30}+\frac{b+7}{5-b}$

241.

$\frac{7}{15p}+\frac{5}{18pq}$

242.

$\frac{3}{20{a}^{2}}+\frac{11}{12a{b}^{2}}$

243.

$\frac{4}{x-2}+\frac{3}{x+5}$

244.

$\frac{6}{m+4}+\frac{9}{m-8}$

245.

$\frac{2q+7}{q+4}-2$

246.

$\frac{3y-1}{y+4}-2$

247.

$\frac{z+2}{z-5}-\frac{z}{z+1}$

248.

$\frac{t}{t-5}-\frac{t-1}{t+5}$

249.

$\frac{3d}{d+2}+\frac{4}{d}-\frac{d+8}{{d}^{2}+2d}$

250.

$\frac{2q}{q+5}+\frac{3}{q-3}-\frac{13q+15}{{q}^{2}+2q-15}$

### Everyday Math

251.

**Decorating cupcakes** Victoria can decorate an order of cupcakes for a wedding in $t$ hours, so in 1 hour she can decorate $\frac{1}{t}$ of the cupcakes. It would take her sister 3 hours longer to decorate the same order of cupcakes, so in 1 hour she can decorate $\frac{1}{t+3}$ of the cupcakes.

- ⓐ Find the fraction of the decorating job that Victoria and her sister, working together, would complete in one hour by adding the rational expressions $\frac{1}{t}+\frac{1}{t+3}.$
- ⓑ Evaluate your answer to part (a) when $t=5.$

252.

**Kayaking** When Trina kayaks upriver, it takes her $\frac{5}{3-c}$ hours to go 5 miles, where $c$ is the speed of the river current. It takes her $\frac{5}{3+c}$ hours to kayak 5 miles down the river.

- ⓐ Find an expression for the number of hours it would take Trina to kayak 5 miles up the river and then return by adding $\frac{5}{3-c}+\frac{5}{3+c}.$
- ⓑ Evaluate your answer to part (a) when $c=1$ to find the number of hours it would take Trina if the speed of the river current is 1 mile per hour.

### Writing Exercises

253.

Felipe thinks $\frac{1}{x}+\frac{1}{y}$ is $\frac{2}{x+y}.$

- ⓐ Choose numerical values for
*x*and*y*and evaluate $\frac{1}{x}+\frac{1}{y}.$ - ⓑ Evaluate $\frac{2}{x+y}$ for the same values of
*x*and*y*you used in part (a). - ⓒ Explain why Felipe is wrong.
- ⓓ Find the correct expression for $\frac{1}{x}+\frac{1}{y}.$

254.

Simplify the expression $\frac{4}{{n}^{2}+6n+9}-\frac{1}{{n}^{2}-9}$ and explain all your steps.

### Self Check

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

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?