<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.</para><para>The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.</para><para>Objectives of this module: be able to recognize a rational expression, be familiar with the equality and negative properties of fractions.</para>
Overview
- Rational Expressions
- Zero-Factor Property
- The Equality Property of Fractions
- The Negative Property of Fractions
Rational expressions
In arithmetic it is noted that a fraction is a quotient of two whole numbers. The expression
$\frac{a}{b}$ , where
$a$ and
$b$ are any two whole numbers and
$b\ne 0$ , is called a fraction. The top number,
$a$ , is called the numerator, and the bottom number,
$b$ , is called the denominator.
Simple algebraic fraction
We define a simple algebraic fraction in a similar manner. Rather than restricting ourselves only to numbers, we use polynomials for the numerator and denominator. Another term for a simple algebraic fraction is a
rational expression . A rational expression is an expression of the form
$\frac{P}{Q}$ , where
$P$ and
$Q$ are both polynomials and
$Q$ never represents the zero polynomial.
Rational expression
A
rational expression is an algebraic expression that can be written as the quotient of two polynomials.
Examples 1–4 are rational expressions:
$\frac{x+9}{x-7}$ is a rational expression:
$P$ is
$x+9$ and
$Q$ is
$x-7$ .
$\frac{{x}^{3}+5{x}^{2}-12x+1}{{x}^{4}-10}$ is a rational expression:
$P$ is
${x}^{3}+5{x}^{2}-12x+1$ and
$Q$ is
${x}^{4}-10$ .
$\frac{3}{8}$ is a rational expression:
$P$ is 3 and
$Q$ is 8.
$4x-5$ is a rational expression since
$4x-5$ can be written as
$\frac{4x-5}{1}$ :
$P$ is
$4x-5$ and
$Q$ is 1.
$\frac{\sqrt{5{x}^{2}-8}}{2x-1}$ is
not a rational expression since
$\sqrt{5{x}^{2}-8}$ is not a polynomial.
In the rational expression
$\frac{P}{Q}$ ,
$P$ is called the numerator and
$Q$ is called the denominator.
Domain of a rational expression
Since division by zero is not defined, we must be careful to note the values for which the rational expression is valid. The collection of values for which the rational expression is defined is called the
domain of the rational expression. (Recall our study of the domain of an equation in Section
[link] .)
Finding the domain of a rational expression
To find the domain of a rational expression we must ask, "What values, if any, of the variable will make the denominator zero?" To find these values, we set the denominator equal to zero and solve. If any zero-producing values are obtained, they are not included in the domain. All other real numbers are included in the domain (unless some have been excluded for particular situational reasons).