College Algebra Tutorial 10: Adding and Subtracting Rational
Expressions
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
Do you ever feel dazed and confused when working with fractions?
If so, you are not alone. This is your lucky day! We have a
whole other tutorial devoted to rational expressions (fractions).
In this tutorial we will be looking at adding and subtracting them.
If you need a review on simplifying rational expressions, feel free to
go . It is time to get started with this
tutorial.
Tutorial
Adding or Subtracting Rational Expressions with Common Denominators
Step 1: Combine the numerators
together.
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.
Step 1: Combine the numerators
together
AND
Step 2: Put the sum or
difference found in step 1 over the common denominator.
*Common denominator of 5x
- 2
*Combine the numerators *Write over common denominator
*Excluded values of the original den.
Step 3: Reduce
to lowest terms.
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.
Step 1: Combine the numerators
together
AND
Step 2: Put the sum or
difference found in step 1 over the common denominator.
*Common denominator of y
- 1
*Combine the numerators *Write over common denominator
Step 3: Reduce
to lowest terms.
*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)
Step 1: Factor all the
denominators
If you need a review on factoring, feel free to go
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.
Adding and Subtracting Rational Expressions Without a Common Denominator
Step 1: Find
the LCD as shown above if needed.
Step 2: Write equivalent
fractions using the LCD if needed.
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.
Step 3: Combine
the rational expressions as shown above.
Step 4: Reduce to lowest
terms as shown in Tutorial 8: Simplifying
Rational Expressions.
Example
3: Add .
Step 1: Find
the LCD as shown above if needed.
The first denominator has the following two factors:
*Factor the GCF
The second denominator has the following factor:
Putting all the different factors together and using the highest
exponent, we get the following LCD:
Step 2: Write equivalent
fractions using the LCD if needed.
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:
*Missing the factor of (y
- 4) in the den.
*Mult. top and bottom by (y
- 4)
Step 3: Combine
the rational expressions as shown above.
*Combine the numerators *Write over common denominator
Step 4: Reduce
to lowest terms.
*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 .
Step 1: Find
the LCD as shown above if needed.
The first denominator has the following factor:
The second denominator has the following two factors:
*Factor the difference
of squares
Putting all the different factors together and using the highest
exponent, we get the following LCD:
Step 2: Write equivalent
fractions using the LCD if needed.
Rewriting the first expression with the LCD:
*Missing the factor of (x
+
1) in the den. *Mult. top and bottom by (x
+
1)
Since the second rational expression already has the LCD, we
do not need to change this fraction.
*Rewriting denominator in factored form
Step 3: Combine
the rational expressions as shown above.
*Combine the numerators *Write over common denominator
*Excluded values of the original den.
Step 4: Reduce
to lowest terms.
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 .
Step 1: Find
the LCD as shown above if needed.
The first denominator has the following two factors:
*Factor the trinomial
The second denominator has the following two factors:
*Factor the trinomial
Putting all the different factors together and using the highest
exponent, we get the following LCD:
Step 2: Write equivalent
fractions using the LCD if needed.
Rewriting the first expression with the LCD:
*Missing the factor of (x
-
8) in the den. *Mult. top and bottom by (x
-
8)
Rewriting the second expression with the LCD:
*Missing the factor of (x
+
5) in the den. *Mult. top and bottom by (x
+
5)
Step 3: Combine
the rational expressions as shown above.
*Combine the numerators *Write over common denominator
*Distribute the minus sign through the (
)
Step 4: Reduce
to lowest terms.
*Factor the num.
*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.
1a. (answer/discussion
to 1a)
1b. (answer/discussion
to 1b)
Need Extra Help on These Topics?
The following are webpages that can assist
you in the topics that were covered on this page:
int_alg_tut33_addrat.htm This webpage will help you with adding and subtracting rational expressions.
commondenomratexpr.htm This webpage goes over finding the least common denominator and combining
rational expressions.
Learning Objectives
Example
1: Add 
.
.
.
.
.
Practice
Problems 1a - 1b:
