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Intermediate Algebra
Tutorial 32: 
Multiplying and Dividing Rational Functions


Learning Objectives

 
After completing this tutorial, you should be able to:
  1. Find the domain of a rational function.
  2. Simplify a rational expression.
  3. Multiply and divide rational expressions.

 
 
 
Introduction

 
Do you ever feel like running and hiding when you see a fraction? If so, you are not alone.  But don't fear!  Help is here!  Hey, that rhymes.  Anyway, over the next several tutorials we will be showing you several aspects of rational expressions (fractions).   In this section we will be simplifying them.  Again, we will be putting your knowledge of factoring to the test.  Factoring plays a big part of simplifying these rational expressions. We will also look at multiplying and dividing them.  I think you are ready to tackle these rational expressions.

 
 
Tutorial

 
 
Rational Expression or Function

A rational expression or function is one that can be written in the form 

where P and Q are polynomials and Q does not equal 0.


 
An example of a rational expression is :


 
 
 
Domain of a Rational Function

 
Recall from Tutorial 13: Introduction to Functions that the domain of a function is the set of all input values to which the rule applies.

With rational functions, we need to watch out for values that cause our denominator to be 0.  If our denominator is 0, then we have an undefined value. 

So, when looking for the domain of a given rational function, we use a back door approach.  We find the values that we cannot use, which would be values that make the denominator 0.


 
 
 
Example 1:   Find the domain of the function .

 
Our restriction is that the denominator of a fraction can never be equal to 0. 

So to find our domain, we want to set the denominator “not equal” to 0 to restrict those values.


 
*Restrict values that cause den. to be 0
 
 

*"Solve" for x
 
 
 

*"Solve" for x

 


 
Our domain is all real numbers except 5 and 1 because they both make the denominator equal to 0, which would not give us a real number answer for our function.

 
 
Fundamental Principle of 
Rational Expressions

For any rational expression , and any polynomial R, where ,, then 


 
In other words, if you multiply the EXACT SAME thing to the numerator and denominator, then you have an equivalent rational expression.

This will come in handy when we simplify rational expressions, which is coming up next.


 

Simplifying (or reducing) a 
Rational Expression

 
Step 1: Factor the numerator and the denominator.

 
If you need a review on factoring, you can go to any or all of the following tutorials:


 
Step 2: Divide out all common factors that the numerator and the denominator have.

 
 
 
Example 2:  Write the rational expression in lowest terms .

 
Step 1: Factor the numerator and the denominator

AND


 
Step 2: Divide out all common factors that the numerator and the denominator have.

 
*Factor the num. and den.

*Divide out the common factor of (x + 10)

 


 

Multiplying Rational Expressions

Q and S do not equal 0.


 
Step 1: Factor both the numerator and the denominator.

 
If you need a review on factoring, you can go to any or all of the following tutorials:


 
 
Step 2: Write as one fraction.

 
Write it as a product of the factors of the numerators over the product of the factors of the denominators.  DO NOT multiply anything out at this point.

 
Step 3: Simplify the rational expression.

 
 
 
Example 3:   Multiply .

 
 
Step 1: Factor both the numerator and the denominator

AND


 
Step 2: Write as one fraction.

 
*Factor the num. and den.

 


 
Step 3: Simplify the rational expression.

 
*Divide out the common factors of y and (y - 1)

 


 
 
 
Dividing Rational Expressions

where Q, S, and R do not equal 0.


 
Step 1: Write as multiplication of the reciprocal.

Step 2: Multiply the rational expressions as shown above.


 
 
 
 
Example 4:   Divide .

 
 
Step 1: Write as multiplication of the reciprocal

AND 

Step 2: Multiply the rational expressions as shown above.


 
 
*Rewrite as mult. of reciprocal
 

*Factor the num. and den.

*Div. out the common factors of 
(t + 3) and (t - 2)


 
 
 
 
Example 5:   Multiply and divide .

 
Since we have a division, let’s go ahead and rewrite that part of it as multiplication of the reciprocal and proceed with multiplying the whole expression:

 
*Rewrite the division as mult. of recip.

*Factor the num. and den.
*All factors divide out
 


 
Be careful.  0 is not our answer here.  When everything divides out like this, it doesn’t mean nothing is left, there is still a 1 there.

 
 
 
Practice Problems

 
These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.


 

Practice Problem 1a:

Find the domain of the rational function.


 
1a. 
(answer/discussion to 1a)


 

Practice Problems 2a - 2c:

Multiply or divide.


 
 
2a. 
(answer/discussion to 2a)

 
2b. 
(answer/discussion to 2b)

 
2c. 
(answer/discussion to 2c)

 
 
 
Need Extra Help on These Topics?

 
 
No appropriate web pages could be found to help you with the topics on this page.
 

for some more suggestions.


 




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Jan. 8, 2002