Multiplying and Dividing Rational Expressions

 

Learning Objective(s)

·         Multiply and divide rational expressions and simplify.

 

 

Just as we can multiply and divide fractions, we can multiply and divide rational expressions, or fractions that include polynomials. In fact, we use the same processes for multiplying and dividing rational expressions as we use for multiplying and dividing numeric fractions.

 

 

There are two ways to go about multiplying fractions:

 

We can multiply the numerators and the denominators and then simplify the product:

 

 =

 

Or we can factor and simplify the fractions before performing the multiplication:

 

 =  =  =  =

 

The same two approaches can be applied to rational expressions. In the following examples, we'll try both techniques:  multiply, then simplify; and simplify, then multiply. An important difference between fractions and rational expressions, though, is that we must identify any values for the variables that would result in division by 0 since this is undefined. These excluded values must be eliminated from the domain, the set of all possible values of the variable.

 

First, we'll multiply and then simplify:

 

 

Okay, that worked. But this time we'll simplify first, then multiply. When using this method, it helps to look for the greatest common factor. We can factor out any common factors, but finding the greatest one will take fewer steps.

 

 

Both methods produced the same answer.

 

Some rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, factor the polynomials and then look for common factors. Just take it step by step, like in the example below.

 

 

Note that in the answer above, we cannot simplify the rational expression any further. It may be tempting to express the 5s in the numerator and denominator as the fraction , but these 5s are terms that are part of a factor, not factors by themselves. They cannot be pulled out of their expressions.

 

Perform the indicated operation and express the answer as a simplified rational expression:   , y ≠ 0, 2, -8, -2   A)   B) C) D)     Show/Hide Answer A) Incorrect. You simplify by factoring like terms in the numerator and denominator. You can only make equivalent fractions to 1 using factors, not terms. The correct answer is .   B) Incorrect. This expression is equivalent, but it can be further simplified since there is a common factor of y in the numerator and denominator. The correct answer is .   C) Incorrect. This expression is equivalent, but it can be further simplified since there are common factors in the numerator and denominator. .The correct answer is .   D) Correct. Factoring the numerators and denominators, you get .  Regrouping, you get =

 

 

 

We've seen that we multiply rational expression like we multiply fractions. It should come as no surprise that we also divide rational expressions the same way we divide numerical fractions. Specifically, to divide rational expressions, multiply the rational expression numerator by the reciprocal of the rational expression denominator.

 

Let’s begin by recalling division of numerical fractions:

 

=  =

 

To divide rational expressions, the process is the same. But remember, we need to find the excluded values, the variable values that would make either denominator equal zero.  But there's an new wrinkle this time—because we divide by multiplying by the reciprocal of one of the rational expressions, we also need to find the values that would make the numerator of that expression equal zero. Check it out:

 

 

 

 

Notice that once we rewrite the division as multiplication by a reciprocal, we follow the same process we used to multiply rational expressions.  We can't forget to mention excluded values—all values of the variable that result in a denominator of 0 at any step in the process must be left out of the domain.

 

Find the quotient and express as a simplified rational expression:   , y ≠ -2 or -1   A)     B)   C)   D)   Show/Hide Answer A) Incorrect. This operation is division, not multiplication. The first step in division is to rewrite as multiplication by the reciprocal. The correct answer is .   B) Correct.  = .   C) Incorrect. You can only express common factors, not terms, as fractions equivalent to one. The simplifying is incorrect here. The correct answer is .   D) Incorrect. This is a division problem. The first step in division is to rewrite as multiplication by the reciprocal. You can only express common factors, not terms, as fractions equivalent to one. The simplifying is incorrect here. The correct answer is .

 

 

Rational expressions are multiplied and divided the same way numeric fractions are. To multiply, first find the greatest common factors of the numerator and denominator. Next, regroup the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication.

 

When expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.