**Learning Objectives**

After completing this tutorial, you should be able to:

- Find the domain of a rational function.
- Simplify a rational expression.
- 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**

**A rational
expression or function
is
**

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

An example of a rational expression is :

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.**

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.

***"Solve" for x**

***"Solve" for x**

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

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

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**

**AND**

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

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

**Q and S do not equal 0.**

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**

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.

**AND**

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

**Step 2: Multiply
the rational expressions as shown above.**

**AND **

**Step 2: Multiply
the rational expressions as shown above.**

***Factor the num. and den.**

***Div. out the common factors
of **

**( t +
3) and (t - 2)**

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:

***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.

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

** 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.**

Last revised on July 17, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.