### Learning Outcomes

- Add and subtract rational expressions that share no common factors
- Add and subtract more than two rational expressions

So far all the rational expressions you’ve added and subtracted have shared some factors. What happens when they don’t have factors in common?

## Add and Subtract Rational Expressions with No Common Factor

In the next example, we show how to find a common denominator when there are no common factors in the expressions.

### Example

Subtract [latex]\displaystyle \frac{3y}{2y-1}-\frac{4}{y-5}[/latex], and give the domain.

State the difference in simplest form.

In the video that follows, we present an example of adding two rational expression whose denominators are binomials with no common factors.

### Try It

## Add and Subtract More Than Two Rational Expressions

You may need to combine more than two rational expressions. While this may seem pretty straightforward if they all have the same denominator, what happens if they do not?

In the example below, notice how a common denominator is found for three rational expressions. Once that is done, the addition and subtraction of the terms looks the same as earlier, when you were only dealing with two terms.

### Example

Simplify[latex]\frac{2{{x}^{2}}}{{{x}^{2}}-4}+\frac{x}{x-2}-\frac{1}{x+2}[/latex], and give the domain.

State the result in simplest form.

In the video that follows we present an example of subtracting [latex]3[/latex] rational expressions with unlike denominators. One of the terms being subtracted is a number, so the denominator is [latex]1[/latex].

### Example

Simplify[latex]\frac{{{y}^{2}}}{3y}-\frac{2}{x}-\frac{15}{9}[/latex], and give the domain.

State the result in simplest form.

In this last video, we present another example of adding and subtracting three rational expressions with unlike denominators.

### Try It

## Summary

The methods shown here will help you when you are solving rational equations later on. To add and subtract rational expressions that share common factors, you first identify which factors are missing from each expression, and build the LCD with them. To add and subtract rational expressions with no common factors, the LCD will be the product of all the factors of the denominators.

### Candela Citations

- Image: No common factors..
**Provided by**: Lumen Learning.**License**:*CC BY: Attribution* - Screenshot: Add and subtract.
**Provided by**: Lumen Learning.**License**:*CC BY: Attribution* - Revision and Adaptation.
**Provided by**: Lumen Learning.**License**:*CC BY: Attribution* - Subtract Rational Expressions with UnLike Denominators - 3 Expressions.
**Authored by**: James Sousa (Mathispower4u.com) for Lumen Learning.**Located at**: https://youtu.be/c-8xQyU0ch0.**License**:*CC BY: Attribution* - Add and Subtract Rational Expressions with UnLike Denominators - 3 Expressions.
**Authored by**: James Sousa (Mathispower4u.com) for Lumen Learning.**Located at**: https://youtu.be/43xPStLm39A.**License**:*CC BY: Attribution*

- Unit 15: Rational Expressions, from Developmental Math: An Open Program.
**Provided by**: Monterey Institute of Technology and Education.**Located at**: http://nrocnetwork.org/resources/downloads/nroc-math-open-textbook-units-1-12-pdf-and-word-formats/.**License**:*CC BY: Attribution* - Ex: Add Rational Expressions with Unlike Denominators.
**Authored by**: James Sousa (Mathispower4u.com) .**Located at**: https://www.youtube.com/watch?v=CKGpiTE5vIg&feature=youtu.be.**License**:*CC BY: Attribution*