Multiplying and Dividing Rational Expressions

 

Learning Objective(s)

·         Multiply rational expressions and simplify.

·         Divide rational expressions and simplify.

 

Introduction

 

Just as you can multiply and divide fractions, you can multiply and divide rational expressions. In fact, you use the same processes for multiplying and dividing rational expressions as you use for multiplying and dividing numeric fractions. The process is the same even though the expressions look different!

 

Multiplication of Rational Expressions

 

Remember that there are two ways to multiply numeric fractions.

 

One way is to multiply the numerators and the denominators and then simplify the product, as shown here.

 

 

A second way is to factor and simplify the fractions before performing the multiplication.

 

 

Notice that both methods result in the same product. In some cases you may find it easier to multiply and then simplify, while in others it may make more sense to simplify fractions before multiplying.

 

The same two approaches can be applied to rational expressions. In the following examples, both techniques are shown. First, let’s multiply and then simplify.

 

 

Example

Problem

Multiply. State the product in simplest form.

 

 

Multiply the numerators, and then multiply the denominators.

 

Simplify by finding common factors in the numerator and denominator.

 

 

Use the common factors to rewrite as multiplication by 1.

 

 

Simplify.

Answer

 

 

 

 

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

 

 

Example

Problem

Multiply. State the product in simplest form.

 

Factor the numerators and denominators. Look for the greatest common factors.

 

 

  

Regroup the fractions to express common factors as multiplication by 1, and then multiply.

 

 

Answer

 

 

 

Both methods produced the same answer.

 

Also, remember that when working with rational expressions, you should get into the habit of identifying any values for the variables that would result in division by 0. These excluded values must be eliminated from the domain, the set of all possible values of the variable. In the example above, , the domain is all real numbers where a is not equal to 0. When a = 0, the denominator of the fraction  equals 0, which will make the fraction undefined.

 

Some rational expressions contain quadratic expressions and other multi-term polynomials. To multiply these rational expressions, the best approach is to first factor the polynomials and then look for common factors. (Multiplying the terms before factoring will often create complicated polynomials…and then you will have to factor these polynomials anyway! For this reason, it is easier to factor, simplify, and then multiply.) Just take it step by step, like in the examples below.

 

 

Example

Problem

Multiply. State the product in simplest form.

 

Factor the numerators and denominators.

 

 

 

 

Regroup to express rational expressions equivalent to 1.

 

 

Multiply simplified rational expressions. This expression can be left with the numerator in factored form or multiplied out.

Answer

 

 

 

Example

Problem

Multiply. State the product in simplest form.

 

 

Factor the numerators and denominators.

 

Regroup to express rational expressions equivalent to 1.

 

Multiply simplified rational expressions. This expression can be left with the denominator in factored form or multiplied out.

Answer

 

 

 

Note that in the answer above, you cannot simplify the rational expression any further. It may be tempting to express the 5’s in the numerator and denominator as the fraction , but these 5’s are terms because they are being added or subtracted. Remember that only common factors, not terms, can be regrouped to form factors of 1!

 

 

Multiply, and express the product as a simplified rational expression.

 

, y ≠ −8, −2, 0, 2,

 

A)

 

B)

 

C)  

 

D)

 

Show/Hide Answer

A)

Incorrect. You simplify by factoring the numerators and denominators. The fraction  can be written as , and the fraction  can be written as . 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 = .

 

 

 

Dividing Rational Expressions

 

You've seen that you multiply rational expressions as you multiply numeric fractions. It should come as no surprise that you also divide rational expressions the same way you divide numeric fractions. Specifically, to divide rational expressions, keep the first rational expression, change the division sign to multiplication, and then take the reciprocal of the second rational expression.

 

Let’s begin by recalling division of numerical fractions.

 

 

Use the same process to divide rational expressions. You can think of division as multiplication by the reciprocal, and then use what you know about multiplication to simplify.

 

You do still need to think about the domain, specifically the variable values that would make either denominator equal zero. But there's a new consideration this time—because you divide by multiplying by the reciprocal of one of the rational expressions, you also need to find the values that would make the numerator of that expression equal zero. Have a look.

 

 

Example

Problem

Identify the domain of the expression.

 

 

 

 

15x3 = 0

 

x = 0 is an excluded value.

Find excluded values. 9 and 27 can never equal 0.

 

Because 15x3 becomes the denominator in the reciprocal of , you must find the values of x that would make 15x3 equal 0.

Answer

The domain is all real numbers except 0.

 

 

Knowing how to find the domain may seem unimportant here, but it will help you when you learn how to solve rational equations. To divide, multiply by the reciprocal.

 

 

Example

Problem

Divide. State the quotient in simplest form.

 

Rewrite division as multiplication by the reciprocal.

 

Factor the numerators and denominators.

 

 

 

Reorder factors and express as multiplication by 1.

 

 

Simplify.

Answer

 

 

 

Example

Problem

Divide. State the quotient in simplest form, and express the domain of the expression.

 

(x +2) = 0

x = −2

 

x = −3 or −2

 

6x4 = 0

x = 0

 

Domain is all real numbers except 0, −2, and −3.

Determine the excluded values that make the denominators and the numerator of the divisor equal to 0.

 

Rewrite division as multiplication by the reciprocal.

 

Factor the numerators and denominators.

 

 

Reorder factors and express as multiplication by 1.

 

Simplify.

Answer

.

 

The domain is all real numbers except 0, −2, and −3.

 

 

Notice that once you rewrite the division as multiplication by a reciprocal, you follow the same process you used to multiply rational expressions.

 

 

Find the quotient and express as a simplified rational expression.

 

, domain is all real numbers except −2 and −1.

 

A)

 

B)

 

C)

 

D)

 

Show/Hide Answer

A)

Incorrect. This operation is division, not multiplication. The first step in division is to rewrite it as multiplication by the reciprocal. The correct answer is.

 

B)

Correct. Division is the same as multiplication by the reciprocal, so this problem can be written: .

 

C)  

Incorrect. You appear to have taken the reciprocal, factored correctly, and simplified your common factors to 1, getting .  But you need to multiply the two numerators together . The correct answer is .

 

D)

Incorrect.

This is a division problem. The first step in division is to rewrite it as multiplication by the reciprocal. You also appear to have simplified incorrectly. The correct answer is .

 

 

 

Summary

 

Rational expressions are multiplied and divided the same way as numeric fractions. 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.