Learning Objectives
Introduction
Tutorial
Rational Expression
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 .
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.
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.
Fundamental Principle of
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.
Simplifying (or reducing) a
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
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
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.