Print This Page
**Common Factors**: At a Glance

- Topics At a Glance
- Polynomial Division
- Common Factors
- Factoring
- Long Division
**More Polynomial Division****Common Factors**- Factoring
- Long Division
- A Clever Trick
- Rational Expressions
- Evaluating Rational Expressions
- Simplifying Rational Expressions
- Multiplying and Dividing Rational Expressions
- Adding and Subtracting Rational Expressions
- Simplifying Complex Rational Expressions
- Equations Involving Rational Expressions
- Word Problems
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

The first thing to do when dividing polynomials is to look for and remove common factors. In terms of rational expressions, "looking for common factors" means that we're looking for those that go into every term in the numerator *and* into every term in the denominator. These are the factors we can pull out and cancel. At least it's way easier than canceling your LA Fitness membership. Those people are relentless.

The difference between what we did with common factors earlier and what we're doing now is that now we may cancel a common factor and still have a rational expression instead of a polynomial. That's fine. We can live with that, right? We've had more than our fill of polynomials lately anyway.

The rational expression has a common factor of *x*^{2} in the numerator and denominator. There are *x*'s all over the place, but this is the most we can grab out of every single term. We pull this factor out of both the numerator and denominator to find

and then we can cancel *x*^{2} from the top and bottom to get

.

Example 1

Remove the common factors from the expression . |

Exercise 1

Simplify the following expression by canceling common factors, if possible:

Exercise 2

Simplify the following expression by canceling common factors, if possible:

Exercise 3

Simplify the following expression by canceling common factors, if possible:

Exercise 4

A student was asked to remove all possible common factors from the following rational expression: . The student wrote:. Identify the mistake, if any.

Exercise 5

A student was asked to remove all possible common factors from the following rational expression: . The student wrote: . Identify the mistake, if any.

Exercise 6

A student was asked to remove all possible common factors from the following rational expression: . The student wrote:. Identify the mistake, if any.