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By the end of this section, you will be able to:
  • Determine the values for which a rational expression is undefined
  • Evaluate rational expressions
  • Simplify rational expressions
  • Simplify rational expressions with opposite factors

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

  1. Simplify: 90 y 15 y 2 .
    If you missed this problem, review [link] .
  2. Factor: 6 x 2 7 x + 2 .
    If you missed this problem, review [link] .
  3. Factor: n 3 + 8 .
    If you missed this problem, review [link] .

In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.

In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.

Rational expression

A rational expression    is an expression of the form p ( x ) q ( x ) , where p and q are polynomials and q 0 .

Remember, division by 0 is undefined.

Here are some examples of rational expressions:

13 42 7 y 8 z 5 x + 2 x 2 7 4 x 2 + 3 x 1 2 x 8

Notice that the first rational expression listed above, 13 42 , is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.

We will perform same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.

Determine the values for which a rational expression is undefined

When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

Determine the values for which a rational expression is undefined.

  1. Set the denominator equal to zero.
  2. Solve the equation in the set of reals, if possible.

Determine the values for which the rational expression is undefined:

9 y x 4 b 3 2 b + 5 x + 4 x 2 + 5 x + 6

Solution

The expression will be undefined when the denominator is zero.


  1. 9 y x Set the denominator equal to zero. Solve for the variable. x = 0 9 y x is undefined for x = 0 .


  2. 4 b 3 2 b + 5 Set the denominator equal to zero. Solve for the variable. 2 b + 5 = 0 2 b = −5 b = 5 2 4 b 3 2 b + 5 is undefined for b = 5 2 .


  3. x + 4 x 2 + 5 x + 6 Set the denominator equal to zero. Solve for the variable. x 2 + 5 x + 6 = 0 ( x + 2 ) ( x + 3 ) = 0 x + 2 = 0 or x + 3 = 0 x = −2 or x = −3 x + 4 x 2 + 5 x + 6 is undefined for x = −2 or x = −3 .

Saying that the rational expression x + 4 x 2 + 5 x + 6 is undefined for x = −2 or x = −3 is similar to writing the phrase “void where prohibited” in contest rules.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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