- The Logic Behind The Process
- The Process

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

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.

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.

Objectives of this module: understand and be able to use the process of reducing rational expressions.

- The Logic Behind The Process
- The Process

When working with rational expressions, it is often best to write them in the simplest possible form. For example, the rational expression

can be reduced to the simpler expression

From our discussion of equality of fractions in Section (Reference), we know that

The process of removing common factors is commonly called *cancelling*.

Remove the three factors of 1;

Notice that in

Remove the factor of 1;

Notice that in

Remove the factor of 1;

Notice that in

Problems 1, 2, and 3 shown above could all be reduced. The process in each reduction included the following steps:

- Both the numerator and denominator were factored.
- Factors that were common to both the numerator and denominator were noted and removed by dividing them out.

We know that we can divide both sides of an equation by the same nonzero number, but why should we be able to divide both the numerator and denominator of a fraction by the same nonzero number? The reason is that any nonzero number divided by itself is 1, and that if a number is multiplied by 1, it is left unchanged.

Consider the fraction

The answer,

Multiplying or dividing the numerator and denominator by the same nonzero number does not change the value of a fraction.

We can now state a process for reducing a rational expression.

- Factor the numerator and denominator completely.
- Divide the numerator and denominator by all factors they have in common, that is, remove all factors of 1.

- A rational expression is said to be reduced to lowest terms when the numerator and denominator have
*no*factors in common.

Reduce the following rational expressions.

The expression *factors* common to both the numerator and denominator. Although there is an *common term*, not a *common factor*, and therefore cannot be divided out.

CAUTION — This is a common error: *incorrect!*

Since

Sometimes we may reduce a rational expression by using the division rule of exponents.

Reduce each of the following fractions to lowest terms.

−1

For the following problems, reduce each rational expression to lowest terms.

1

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms.

*((Reference))* Write

*((Reference))* Factor

*((Reference))* Factor

*((Reference))* Supply the missing word. An equation expressed in the form

form.

*((Reference))* Find the domain of the rational expression

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