**Learning Objectives**

After completing this tutorial, you should be able to:

- Find the least common denominator of rational expressions.
- Add and subtract rational expressions.

**Introduction**

** Tutorial**

**Adding or Subtracting Rational Expressions **

**with Common Denominators**

**Step 2: Put the sum or
difference found in step 1 over the common denominator.**

**Step 3: Reduce to lowest
terms as shown in Tutorial 8: Simplifying
Rational Expressions.**

**Why do we have to have a common denominator
when
we add or subtract rational expressions?????**

Good question. The denominator indicates what type of fraction
that you have and the numerator is counting up how many of that type you
have. You can only directly combine fractions that are of the same
type (have the same denominator). For example if 2 was my denominator,
I would be counting up how many halves I had. If 3 was my denominator,
I would be counting up how many thirds I had. But I would not be
able to add a fraction with a denominator of 2 directly with a fraction
that had a denominator of 3 because they are not the same type of fraction.
I would have to find a common denominator first, which we will cover after
the next two examples.

** Example
1**: Add .

Since the two denominators are the same, we can go right into adding
these two rational expressions.

**AND**

**Step 2: Put the sum or
difference found in step 1 over the common denominator.**

***Combine the numerators**

***Write over common denominator**

***Excluded values of the original den.**

Note that neither the numerator nor the denominator will factor.
The rational expression is as simplified as it gets.

Also note that **the value that would be excluded from the domain is
2/5.** This is the **value that makes the original denominator
equal to 0**.

** Example
2**: Subtract .

Since the two denominators are the same, we can go right into subtracting
these two rational expressions.

**AND**

**Step 2: Put the sum or
difference found in step 1 over the common denominator.**

***Combine the numerators**

***Write over common denominator**

***Factor the num.**

***Simplify by div. out the common factor of ( y - 1)**

***Excluded values of the original den.**

Note that **the value that would be excluded from the domain is 1.**
This is the **value that makes the original denominator equal to 0**.

**Least Common Denominator (LCD)**

If you need a review on factoring, feel free to go back to **Tutorial
7: Factoring Polynomials.**

**Step 2: The LCD is the
list of all the DIFFERENT factors in the denominators raised to the highest
power that there is of each factor.**

If we multiply the numerator and denominator by the exact same expression
it is the same as multiplying it by the number 1. If that is the
case, we will have equivalent expressions when we do this.

**Now the question is WHAT do we multiply top and bottom by to get
what we want? ** We need to have the LCD, so you look to see what
factor(s) are missing from the original denominator that is in the LCD.
If there are any missing factors then that is what you need to multiply
the numerator AND denominator by.

** Example
3**: Add .

The **first denominator** has the following two factors:

The **second denominator** has the following factor:

Since the **first rational expression** already has the LCD, we
do not need to change this fraction.

***Rewriting denominator in factored form**

Rewriting the **second expression** with the LCD:

***Mult. top and bottom by ( y - 4)**

***Combine the numerators**

***Write over common denominator**

***Simplify by div. out the common factor of y**

***Excluded values of the original den.**

Note that **the values that would be excluded from the domain are
0 and 4.** These are the **values that make the original denominator
equal to 0**.

** Example
4**: Add .

The **first denominator** has the following factor:

The **second denominator** has the following two factors:

Rewriting the **first expression** with the LCD:

***Missing the factor of ( x +
1) in the den.**

Since the **second rational expression** already has the LCD, we
do not need to change this fraction.

***Excluded values of the original den.**

This rational expression cannot be simplified down any farther.

Also note that **the values that would be excluded from the domain
are -1 and 1.** These are the **values that make the original
denominator equal to 0**.

** Example
5**: Subtract .

The **first denominator** has the following two factors:

The **second denominator** has the following two factors:

Rewriting the **first expression** with the LCD:

***Missing the factor of ( x -
8) in the den.**

Rewriting the **second expression** with the LCD:

***Missing the factor of ( x +
5) in the den.**

***Distribute the minus sign through the (
)**

***No common factors to divide out**

***Excluded values of the original den.**

Note that **the values that would be excluded from the domain are
-5, -1 and 8.** These are the **values that make the original
denominator equal to 0**.

** 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 Problems 1a - 1b:Perform the indicated operation.

** Need Extra Help on these Topics?**

This webpage will help you with adding and subtracting rational expressions.

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

This webpage goes over finding the least common denominator and combining
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. 15, 2009 by Kim Seward.

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