Intermediate Algebra
Tutorial 29: Factoring Special Products

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Learning Objectives
After completing this tutorial, you should be able to:
 Factor a perfect square trinomial.
 Factor a difference of squares.
 Factor a sum or difference of cubes.
 Apply the factoring strategy to factor a polynomial
completely.

Introduction
In this tutorial we help you expand your knowledge of
polynomials by
looking at multiplying polynomials together. We will look at
using
the distributive property,
initially
shown in Tutorial 5: Properties of Real Numbers, to help us
out.
Again, we are using a concept that you have already seen to apply to
the
new concept. After going through this tutorial you should have
multiplying
polynomials down pat.

Tutorial
Factoring
a Perfect Square
Trinomial
OR

Example
1: Factor . 

*Fits the form of a perfect sq.
trinomial
*Factor as the sum of bases squared

Note that if we would multiply this out, we would get
the original
polynomial. 
Example
2: Factor . 

*Fits the form of a perfect sq.
trinomial
*Factor as the diff. of bases
squared

Note that if we would multiply this out, we would get
the original
polynomial. 
Factoring
a Difference of
Two Squares

Example
3: Factor . 
First note that there is no GCF to factor out of this
polynomial.
This fits the form of a the difference
of two squares.
So we will factor using that rule: 

*Fits the form of a diff. of two
squares
*Factor as the prod. of sum and
diff. of bases

Note that if we would multiply this out, we would get
the original
polynomial. 
Example
4: Factor . 
First note that there is no GCF to factor out of this
polynomial.
This fits the form of the difference
of two squares.
So we will factor using that rule: 

*Fits the form of a diff. of two
squares
*Factor as the prod. of sum and
diff. of bases
*Combine like terms in ( )

Note that if we would multiply this out and the
original expression
out we would get the same polynomial. 
Factoring
a Sum of Two Cubes

The sum of two cubes has to be exactly in this
form to use this
rule. When you have the sum of two cubes, you have a product
of
a binomial and a trinomial. The binomial is the sum of the bases
that are being cubed. The trinomial is the first base squared,
the
second term is the opposite of the product of the two bases found, and
the third term is the second base squared. 
Example
5: Factor . 
First note that there is no GCF to factor out of this
polynomial.
This fits the form of the
sum of cubes.
So we will factor using that rule: 

*Fits the form of a sum of two
cubes
*Binomial is sum of bases
*Trinomial is 1st base squared,
minus prod.
of bases, plus 2nd base squared 
Note that if we would multiply this out, we would get
the original
polynomial. 
Factoring
a Difference of
Two Cubes

This is factored in a similar fashion to the sum of two
cubes.
Note the only difference is that the sign in the binomial is a  which
matches the original sign, and the sign in front of ax is positive,
which
is the opposite sign. The difference of two cubes has to be exactly in this
form to use this
rule. When you have the difference of two cubes, you have a
product
of a binomial and a trinomial. The binomial is the difference of
the bases that are being cubed. The trinomial is the first base
squared,
the second term is the opposite of the product of the two bases found,
and the third term is the second base squared. 
Example
6: Factor . 
First note that there is no GCF to factor out of this
polynomial.
This fits the form of the
difference of
cubes. So we will factor using that rule: 

*Fits the form of a diff. of
two cubes
*Binomial is diff. of bases
*Trinomial is 1st base squared,
plus prod.
of bases, plus 2nd base squared 
Note that if we would multiply this out, we would get
the original
polynomial. 
Now that you have a list of different factoring rules,
let’s put it
all together. The following is a checklist of the factoring rules
that we have covered in our tutorials.
When you need to factor, you
ALWAYS look for
the GCF first. Whether you have a GCF or not, then
you continue looking to see if you have anything else that
factors.
Below is a checklist to make sure you do not miss
anything. Always
factor until you can not factor any further. 
Factoring Strategy
I. GCF:
Always check for the GCF first, no matter what. 
II. Binomials:
III. Trinomials:
a.
b. Trial and error:
c. Perfect square trinomial:

IV. Polynomials with four terms:

Example
7: Factor completely. 

*Factor a 4 out of every term

Next, we assess to see if there is anything else that
we can factor.
We have a trinomial inside the ( ). It fits the form
of a perfect square trinomial, so we will
factor
it accordingly: 

*Fits the form of a perfect sq.
trinomial
*Factor as the sum of bases squared

There is no more factoring that we can do in this
problem.
Note that if we would multiply this out, we would get
the original polynomial. 
Example
8: Factor completely. 
The first thing that we always check when we are
factoring is WHAT?
The GCF. In this case, there is not one.
So we assess what we have. It fits the form of a difference
of two squares, so we will factor it accordingly: 

*Fits the form of a diff. of two
squares
*Factor as the prod. of sum and
diff. of bases

Next we assess to see if there is anything else that we
can factor.
Note how the second binomial is another difference of
two squares. That means we have to continue factoring this
problem. 

*Fits the form of a diff. of two
squares
*Factor as the prod. of sum and
diff. of bases

There is no more factoring that we can do in this
problem.
Note that if we would multiply this out, we would get
the original polynomial. 
Example
9: Factor completely. 
The first thing that we always check when we are
factoring is WHAT?
The GCF. In this case, there is not one.
So we assess what we have. It fits the form of a sum
of two cubes, so we will factor it accordingly: 

*Fits the form of a sum of two
cubes
*Binomial is sum of bases
*Trinomial is 1st base squared,
minus prod.
of bases, plus 2nd base squared

There is no more factoring that we can do in this
problem.
Note that if we would multiply this out, we would get
the original polynomial. 
Example
10: Factor completely. 
The first thing that we always check when we are
factoring is WHAT?
The GCF. In this case, there is not one.
So we assess what we have. This is a trinomial that does
not fit the
form of a perfect square trinomial. Looks like we will have to
use trial
and error as shown in Tutorial 28: Factoring Trinomials: 
=

*Factor by trial and error

There is no more factoring that we can do in this
problem.
Note that if we would multiply this out, we would get
the original polynomial. 
Example
11: Factor completely. 

*Group in two's
*Factor out the GCF out of each
separate (
)
*Factor out the GCF of (x + 5b)

There is no more factoring that we can do in this
problem.
Note that if we would multiply this out, we would get
the original polynomial. 
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  1e: Factor Completely.
Need Extra Help on these Topics?
Last revised on July 15, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.



