**Learning Objectives**

After completing this tutorial, you should be able to:

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

** Introduction**

** Tutorial**

**A rational
expression is one
that **

**can be written in
the form **

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

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 all numbers that must be excluded from
the
domain of .

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

So to find what values we need to exclude, think of what
value(s) of *x*,
if any, would cause the denominator to be 0.

This give us a better look at it.

Since 1 would make the first factor in the denominator
0, then **1
would have to be excluded.**

Since - 4 would make the second factor in the
denominator 0, then **-
4 would also have to be excluded.**

**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, by all means go to **Tutorial
7: Factoring Polynomials.**

** Example
2: **Simplify and find all numbers that must be
excluded
from the domain of the simplified rational expression: .

**AND**

***Divide out the common factor
of ( x + 3)**

***Rational expression simplified**

To find the value(s) needed to be excluded from the
domain, we need
to ask ourselves, what value(s) of *x* would
cause our denominator to be 0?

Looking at the denominator *x* - 9, I would
say it would have to be *x* = 9.
Don’t
you agree?

**9 would be our excluded value.**

** Example
3: **Simplify and find all numbers that must be
excluded
from the domain of the simplified rational expression: .

**AND**

***Factor out a -1 from (5 - x)**

***Divide out the common factor
of ( x - 5)**

***Rational expression simplified**

To find the value(s) needed to be excluded from the
domain, we need
to ask ourselves, what value(s) of *x* would
cause our denominator to be 0?

Looking at the denominator *x* - 5, I would
say it would have to be *x* = 5.
Don’t
you agree?

**5 would be our excluded value.**

** 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 all numbers that must be excluded from the domain of the given rational expression.

Practice Problems 2a - 2b:Simplify and find all numbers that must be excluded from the domain of the simplified rational expression.

** Need Extra Help on these Topics?**

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut32_multrat.htm**

The beginning of this webpage goes through how to simplify a rational
expression.

**http://www.purplemath.com/modules/rtnldefs.htm**

This website helps with simplifying rational expressions.

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

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 14, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.