Rational Expressions and Functions
Add and Subtract Rational Expressions
Learning Objectives
By the end of this section, you will be able to:
- Add and subtract rational expressions with a common denominator
- Add and subtract rational expressions whose denominators are opposites
- Find the least common denominator of rational expressions
- Add and subtract rational expressions with unlike denominators
- Add and subtract rational functions
Before you get started, take this readiness quiz.
Add and Subtract 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.
If p, q, and r are polynomials where then
To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result 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.
Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?
Add:
Since the denominator is we must exclude the value
The expression simplifies to but the original expression had a denominator of so
Simplify:
Simplify:
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. Be careful of the signs when you subtract a binomial or trinomial.
Subtract:
Subtract:
Subtract:
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
Let’s see how this works.
| Multiply the second fraction by | |
| The denominators are the same. | |
| Simplify. |
Be careful with the signs as you work with the opposites when the fractions are being subtracted.
Subtract:
| The denominators are opposites, so multiply the
second fraction by |
|
| Simplify the second fraction. | |
| The denominators are the same. Subtract the numerators. | |
| Distribute. | |
| Combine like terms. | |
| Factor the numerator and denominator. | |
| Simplify by removing common factors. | |
| Simplify. |
Subtract:
Subtract:
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 rat