Intermediate Algebra Tutorial 32


Intermediate Algebra
Tutorial 32: Multiplying and Dividing Rational Expressions


WTAMU > Virtual Math Lab > Intermediate Algebra  



deskLearning 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.




desk 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.

 

 

desk Tutorial


 
 

Rational Expression or Function

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

rational function

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


 
An example of a rational expression is :

rational function


 
 
  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.


 
 
 
notebook Example 1:   Find the domain of the function example 1a.

 
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.


 
example 1b
*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 rational function, and
any polynomial R, where ,rational function, then 

rational function


 
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:

Tutorial 27: The GCF and Factoring by Grouping

Tutorial 28: Factoring Trinomials

Tutorial 29: Factoring Special Products


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

 
 
 
notebook Example 2:  Write the rational expression in lowest terms example 2a.

 
Step 1: Factor the numerator and the denominator

AND

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


  
example 2b
*Factor the num. and den.

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

 


 

Multiplying Rational Expressions

multiply

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:

Tutorial 27: The GCF and Factoring by Grouping

Tutorial 28: Factoring Trinomials

Tutorial 29: Factoring Special Products


 
 
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.

 
 
 
notebook Example 3:   Multiply example 3a.

 
 
Step 1: Factor both the numerator and the denominator

AND


 
Step 2: Write as one fraction.

 
example 3b
*Factor the num. and den.

 


 
Step 3: Simplify the rational expression.

 
example 3c
*Divide out the common factors of y and (y - 1)

 


 
 
  Dividing Rational Expressions

division

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.


 
 
 
 
notebook Example 4:   Divide example 4a.

 
 
Step 1: Write as multiplication of the reciprocal

AND 

Step 2: Multiply the rational expressions as shown above.


 
 
example 4b
*Rewrite as mult. of reciprocal
 

*Factor the num. and den.

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


 
 
 
 
notebook Example 5:   Multiply and divide example 5a.

 
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:

  example 5b
*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.

 

 
desk 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.

 

pencil Practice Problem 1a: Find the domain of the rational function.

 

1a. problem 1a
(answer/discussion to 1a)
 

 

pencil Practice Problems 2a - 2c: Multiply or divide.

 

2a. problem 2a
(answer/discussion to 2a)

2b. problem 2b
(answer/discussion to 2b)

 

2c. problem 2c
(answer/discussion to 2c)

 

 


desk Need Extra Help on these Topics?



 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 


WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 17, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.