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# Simplifying Rational Expressions

Objective Learn how to find excluded values of a rational expression, and to simplify rational expressions.

## Excluded Values

Definition of Rational Expression A rational expression is an algebraic fraction whose numerator and denominator are polynomials.

, where p(x) and q(x) are polynomials

The term rational comes from the word ratio. A rational expression can be described as a ratio of polynomials. A rational function is also a ratio of polynomials, as shown below.

Notice that f ( x ) cannot be defined for values of x when the denominator q ( x ) equals 0, since division by 0 is undefined. So these values are excluded from the domain of f ( x ).

Example 1

Find the excluded values of

Solution

The excluded values are those values of x where the denominator x 2 + 2x - 3 equals 0. Since these are the roots of the denominator, solve x 2 + 2x - 3 = 0.

The easiest way to solve this quadratic equation is by factoring and then using the Zero Product Property.

 x 2 + 2x - 3 = 0 (x - 1)(x + 3) = 0 Factor. x - 1 = 0 or x + 3 = 0 Zero Product Property x = 1 or x = -3

So, the excluded values are x = 1 and x = - 3. Said another way, the domain of the rational function consists of all values of x except x = 1 and x = -3.

Example 2

Find the excluded values of .

Solution

The excluded values are the zeros of denominator, z( y + 1)x 3 . Use the Zero Product Property to find these values.

z( y + 1)x 3 = 0

z = 0 or y + 1 + 0 or x 3 = 0

y = -1, or x = 0

So, the excluded values are z = 0, y = -1, and x = 0.

## Simplifying Rational Expressions

Remember that to simplify a rational number such as , you should look for the greatest common factor (GCF) of the numerator and the denominator, and cancel this common factor.

Simplifying rational expressions can be done the same way. Factor the numerator and the denominator, and cancel the greatest common factor.

Example 3

Simplify .

Solution

First, find the GCF of the numerator and the denominator. The numerator x + 2 is completely factored. Factor the denominator.

x 2 - 4 = ( x - 2)( x + 2)

So, the GCF of the numerator and the denominator is x + 2. Now simplify the expression by canceling this GCF.

Example 4

Simplify .

Solution

First, find the GCF of the numerator and the denominator. Notice that the denominator 2xy 3( x + 1) is completely factored. However, the numerator can be factored further.

4x 2y( x 3 + 2 x + 1) = 2 Â· 2x 3y( x + 1) 3

Comparing the factors of the numerator and the denominator, we see that the greatest common factor of the numerator and denominator is

2xy( x + 1).

Therefore, simplify the expression by canceling this GCF.