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Properties of Rational Expressions

A ratio of two integers is called a rational number; a ratio of two polynomials is called a rational expression. Rational expressions are as fundamental to algebra as rational numbers are to arithmetic. In this section we look carefully at some of the properties of rational numbers and see how they extend to rational expressions.

 

Definition of Rational Expressions

A rational expression is the ratio of two polynomials with the denominator not equal to zero. For example,

are rational expressions. The rational number is a rational expression because 2 and 3 are monomials and is a ratio of two monomials. If the denominator of a rational expression is 1, it is usually omitted, as in the expression 3a + 5.

 

Domain

The domain of a rational expression is the set of all real numbers that can be used in place of the variable. Because the denominator of a rational expression cannot be zero, the domain of a rational expression consists of the set of real numbers except those that cause the denominator to be zero. The domain of

is the set of all real numbers excluding -5. In set-builder notation this set is written as {x | x -5}, and in interval notation it is written as (-, -5) È (-5, ).

 

Helpful hint

If the domain consists of all real numbers except -5, some people write R - {-5} for the domain. Even though there are several ways to indicate the domain, you should keep practicing interval notation because it is used in algebra, trigonometry, and calculus.

 

Example 1

Domain

Find the domain of each rational expression.

Solution

a) The denominator is zero if x + 9 = 0 or x = -9. The domain is {x | x -9} or

(-, -9) È (-9, ).

b) The denominator is zero if 5y = 0 or y = 0. The domain is {y | y 0} or

(-, 0) È (0, ).

c) The denominator is zero if 2x2 - 2 = 0. Solve this equation.

  2x2 - 2 = 0    
  2(x2 - 1) = 0   Factor out 2.
  2(x + 1)(x - 1) = 0   Factor completely.
x + 1 = 0 or x - 1 = 0   Zero factor property
x = -1 or x = 1    

The domain is the set of all real numbers except -1 and 1. This set is written as {x | x -1 and x 1}, or in interval notation as

(-, 1) È (-1, 1) È (1, ).

Caution

The numbers that you find when you set the denominator equal to zero and solve for x are not in the domain of the rational expression. The solutions to that equation are excluded from the domain.

 

 
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