Intermediate Algebra
Tutorial 27: The GCF and Factoring by Grouping
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Learning Objectives
After completing this tutorial, you should be able to:
- Find the Greatest Common Factor (GCF) of a polynomial.
- Factor out the GCF of a polynomial.
- Factor a polynomial with four terms by grouping.
Introduction
Factoring is to write an expression as a product of
factors.
For example, we can write 10 as (5)(2), where 5 and 2 are called
factors
of 10. We can also do this with polynomial expressions. In
this tutorial we are going to look at two ways to factor polynomial
expressions,
factoring out the greatest common factor and factoring by
grouping.
In the next two tutorials we will add on other types of
factoring.
Something to look forward to! By the time I'm are through with
you,
you will be a factoring machine. Others will be asking
you
for help with factoring.
Basically, when we factor, we reverse the process of multiplying
the polynomial which was covered in Tutorial 26: Multiplying
Polynomials.
Tutorial
Greatest Common Factor (GCF)
The GCF for a polynomial is the largest monomial
that divides (is
a factor of) each term of the polynomial.
Example
1: Find the GCF of the list of monomials:
We need to figure out what the largest monomial that we
can divide
out of each of these terms would be. Note how they all have an x,
so it look like x will be involved.
The exponents on the x’s
are 8, 7, and 6.
We have to decide which exponent we are going to use. If we use
the
exponent 8, we are in trouble. We cannot divide 
or 
by 
, we don’t have enough x’s to do
that.
But, if we use ,
we would have a monomial that we could divide out of ALL the terms.
Hence our GCF is .
Note that if all terms have the
same variable,
the GCF for the variable part is that variable raised to the lowest
exponent
that is listed.
Example
2: Find the GCF of the list of monomials:
We need to figure out what the largest monomial that we
can divide
out of each of these terms would be. What do you think?
Let’s first look at the numerical part. We have a
3, 9, and 18.
The largest number that can be divided out of those numbers is 3.
So our numerical GCF is 3.
Now onto the variable part. It looks like each
term has an x and a y. In both cases the lowest
exponent
is 1.
So the GCF of our variable part is xy.
Putting this together we have a GCF of 3xy.
Factoring out the GCF
Step 1: Identify the GCF of the polynomial.
Step 2: Divide the GCF
out of every term
of the polynomial.
Example
3: Factor out the GCF:
Step 1: Identify the GCF of the polynomial.
The largest monomial that we can factor out of each
term is 2x.
Step 2: Divide the GCF out of
every term of
the polynomial.
*Divide 2x out
of every term of the poly.
Be careful. If a term
of the polynomial
is exactly the same as the GCF, when you divide it by the GCF you are
left
with 1, NOT 0. Don’t think, 'oh I have nothing left',
there is actually a 1. As shown above when we divide 2x by 2x we get 1, so we need a 1 as the
third
term inside of the ( ).
Note that if we multiply our answer out, we should get
the original
polynomial. In this case, it does check out. Factoring
gives
you another way to write the expression so it will be equivalent to the
original problem.
Example
4: Factor out the GCF:
Step 1: Identify the GCF of the polynomial.
The largest monomial that we can factor out of each
term is
.
Step 2: Divide the GCF out of
every term of
the polynomial .
*Divide out of every term of the poly.
Note that if we multiply our answer out, we do get the
original polynomial.
Example
5: Factor out the GCF:
This problem looks a little different, because now our
GCF is a binomial.
That is ok, we treat it in the same manner that we do when we have a
monomial
GCF.
Note that this is not in factored form because of the
minus sign we
have before the 7 in the problem. To be in factored form, it must
be written as a product of factors.
Step 1: Identify the GCF of the polynomial.
This time it isn't a monomial but a binomial that we
have in common.
Our GCF is (x + 5).
Step 2: Divide the GCF out of
every term of
the polynomial .
*Divide (x + 5) out of both parts
When we divide out the (x + 5) out of the first term,
we are left with x squared.
When we divide it out of the second term, we are left with -7.
That is how we get the 
for our second ( ).
Factoring a Polynomial with
Four Terms by Grouping
In some cases there is not a GCF for ALL the terms
in a polynomial.
If you have four terms with no GCF, then try factoring by grouping.
Step 1: Group the first two terms
together and then
the last two terms together.
Step 2: Factor out a GCF from each
separate binomial.
Step 3: Factor out the common
binomial.
Example
6: Factor by grouping:
Note how there is not a GCF for ALL the terms. So
let’s go ahead
and factor this by grouping.
Step 1: Group the first two terms
together and then
the last two terms together.
*Two groups of two terms
Step 2: Factor out a GCF from each
separate binomial.
*Factor out an x squared
from the 1st ( )
*Factor out a 2 from the 2nd
( )
Step 3: Factor out the common
binomial.
*Divide (x + 7) out of both parts
Note that if we multiply our answer out, we do get the
original polynomial.
Example
7: Factor by grouping:
Note how there is not a GCF for ALL the terms. So
let’s go ahead
and factor this by grouping.
Step 1: Group the first two terms
together and then
the last two terms together.
*Two groups of two terms
Be careful. When the first term of the second
group of two has
a minus sign in front of it, you want to put the minus in front of the
second ( ). When you do this you need to change the
sign
of both terms of the second ( ) as shown above.
Step 2: Factor out a GCF from each
separate binomial.
*Factor out an x from
the 1st ( )
*Factor out a 4 from the 2nd
( )
Step 3: Factor out the common
binomial.
*Divide (3x + y)
out of both parts
Note that if we multiply our answer out that we do 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 - 1d: Factor.
1d. 
(answer/discussion
to 1d)
Need Extra Help on these Topics?
The following are webpages
that can assist
you in the topics that were covered on this page:
http://www.purplemath.com/modules/simpfact.htm
This webpage helps you with factoring out the GCF.
http://www.mathpower.com/tut111.htm
This webpage will help you with factoring out the GCF.
Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.
Last revised on July 15, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.
