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.
![](./articles_imgs/1032/pic2.gif)
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 ![](./articles_imgs/1032/pic3.gif)
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.
![](./articles_imgs/1032/pic7.gif)
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.
![](./articles_imgs/1032/pic9.gif)
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 · 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.
![](./articles_imgs/1032/pic11.gif)
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