# 8.4 Add and Subtract Rational Expressions with Unlike Denominators

### 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

### 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 7/12+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.

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?

### 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.

### 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.

Now we will add rational expressions whose denominators are monomials.

Now we are ready to tackle polynomial denominators.

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

### How To

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.

### 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.

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.

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

### 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.

### Section 8.4 Exercises

#### Practice Makes Perfect

In the following exercises, find the LCD.

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

In the following exercises, subtract.

In the following exercises, add and subtract.

In the following exercises, simplify.

Everyday Math

Writing Exercises

#### 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?