8.3 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

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.

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

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

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.

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

Be careful of the signs when you subtract a binomial!

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 −1/−1.

Let’s see how this works.

Section 8.3 Exercises

Practice Makes Perfect

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

In the following exercises, subtract.

Everyday Math


Writing Exercises

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?