Rational Expressions and Functions
Simplify Complex Rational Expressions
Learning Objectives
By the end of this section, you will be able to:
- Simplify a complex rational expression by writing it as division
- Simplify a complex rational expression by using the LCD
Before you get started, take this readiness quiz.
Simplify a Complex Rational Expression by Writing it as Division
Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:
In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.
A complex rational expression is a rational expression in which the numerator and/or the denominator contains a rational expression.
Here are a few complex rational expressions:
Remember, we always exclude values that would make any denominator zero.
We will use two methods to simplify complex rational expressions.
We have already seen this complex rational expression earlier in this chapter.
We noted that fraction bars tell us to divide, so rewrote it as the division problem:
Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.
This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.
Simplify the complex rational expression by writing it as division:
Are there any value(s) of x that should not be allowed? The original complex rational expression had denominators of and This expression would be undefined if or
Simplify the complex rational expression by writing it as division:
Simplify the complex rational expression by writing it as division:
Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.
Simplify the complex rational expression by writing it as division:
| Simplify the numerator and denominator.
Find the LCD and add the fractions in the numerator. Find the LCD and subtract the fractions in the denominator. |
|
| Simplify the numerator and denominator. | |
| Rewrite the complex rational expression as a division
problem. |
|
| Multiply the first by the reciprocal of the second. | |
| Simplify. | 3 |
Simplify the complex rational expression by writing it as division:
Simplify the complex rational expression by writing it as division:
We follow the same procedure when the complex rational expression contains variables.
Simplify the complex rational expression by writing it as division:
Simplify the complex rational expression by writing it as division:
Simplify the complex rational expression by writing it as division: .
We summarize the steps here.
- Simplify the numerator and denominator.
- Rewrite the complex rational expres