**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.
- Factor a trinomial of the form .
- Factor a trinomial of the form .
- Indicate if a polynomial is a prime polynomial.
- 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**

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 several ways to factor polynomial
expressions. By the time I'm through with you, you will be a
factoring machine.

Basically, when we factor, we reverse the process of **multiplying
the polynomial which was covered in Tutorial 6: Polynomials.**

** Tutorial**

**Step 2:
Divide
the GCF out of every term of the polynomial. **

This process is basically the reverse of the
distributive property.

** Example
1**: Factor out the GCF: .

The largest monomial that we can factor out of each
term is 2*y*.

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 2*y* by 2*y *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
2**: 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 plus sign we have before the 5 in the problem. To be in factored form, it must be written as a product of factors.

This time it isn't a monomial but a binomial that we
have in common.

**Our GCF is (3 x -1).**

When we divide out the (3*x* - 1) out of
the first term, we are left with *x*.
When
we divide it out of the second term, we are left with 5.

That is how we get the (*x *+
5) for our second
( ).

**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
3**: Factor by grouping: .

Note how there is not a GCF for ALL the terms. So
let’s go ahead
and factor this by grouping.

Note that if we multiply our answer out, we do get the
original polynomial.

** Example
4**: Factor by grouping: .

Note how there is not a GCF for ALL the terms. So
let’s go ahead
and factor this by grouping.

Note that if we multiply our answer out that we do get
the original
polynomial.

(Where the number in front of *x* squared is 1)

**Step 1: Set up a product
of two ( ) where each will hold two terms.**

It will look like this:
( )(
).

**Step 2: Find the factors
that go in the first positions.**

To get the *x* squared (which is the F in
FOIL), we would have to have an *x *in
the first
positions in each ( ).

So it would look like this: (*x *
)(*x* ).

**Step 3: Find the
factors that go in the last positions.**

The factors that would go in the last position
would have to be two
expressions such that **their product equals ***c* (the constant) and at the same time their sum equals *b* (number in front of *x *term).

As you are finding these factors, you have to
consider the sign of the
expressions:

**If c is negative**,
your factors are going to have opposite signs depending on

** Example
5**: Factor the trinomial: .

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form .

It will look like
this:
(
)( )

Since we have *a *squared
as our first term,
we will need the following:

(*a *
)(*a * )

**Putting that into our factors we get:**

Note that if we would multiply this out, we would get
the original
trinomial.

** Example
6**: Factor the trinomial: .

Note that this trinomial does have a GCF of 2*y*.

We need to factor out the **GCF**
before
we tackle the trinomial part of this.

We are not finished, we can still factor the
trinomial. It is
of the form .

Anytime you are factoring, you need to make sure that you factor everything that is factorable. Sometimes you end up having to do several steps of factoring before you are done.

It will look like this: 2*y*(
)( )

Since we have x squared as our first term, we will need
the following:

2*y*(*x *
)(*x* )

**Putting that into our factors we get:**

Note that if we would multiply this out, we would get
the original
trinomial.

(where *a* does not
equal 1)

Again, **this is the reverse of the FOIL method.**

The difference between this trinomial and the one
discussed above, is
there is a number other than 1 in front of the *x* squared. **This means, that not only do
you
need to find factors of c, but also a.**

**Step 1:
Set up a
product of two ( ) where each will hold two terms.**

It will look like this
( )(
)

**Step 2: Use trial and
error to find the factors needed.**

The trick is to get the right combination of these factors. You can check this by applying the FOIL method. If your product comes out to be the trinomial you started with, you have the right combination of factors. If the product does not come out to be the given trinomial, then you need to try again.

** Example
7**: Factor the trinomial .

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form .

It will look like this:
(
)( )

**In the second terms of the binomials, we need factors
of 2.
This would have to be 2 and 1. I used positives here because the
middle term is positive.**

Also, we need to make sure that we get the right combination of these factors so that when we multiply them out we get .

**This is not our original polynomial. **

**So we need to try again.**

**Second try:**

**This is our original polynomial.**

**So this is the correct combination of factors for
this polynomial.**

** Example
8**: Factor the trinomial .

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form

It will look like this:
(
)( )

In the first terms of the binomials, we need factors
of 5 *x *squared.
This would have to be 5*x* and *x.*

In the second terms of the binomials, we need factors of -8. This would have to be -8 and 1, 8 and -1, 2 and -4, or -2 and 4. Since the product of these factors has to be a negative number, we need one positive factor and one negative factor.

Also we need to make sure that we get the right combination of these factors so that when we multiply them out we get .

**This is our original polynomial.**

**So this is the correct combination of factors for
this polynomial.**

Not every polynomial is factorable. Just like not
every number
has a factor other than 1 or itself. A prime number is a number
that
has exactly two factors, 1 and itself. 2, 3, and 5 are examples
of
prime numbers.

The same thing can occur with polynomials. **If
a polynomial
is not factorable we say that it is a prime polynomial.**

Sometimes you will not know it is prime until you start
looking for
factors of it. Once you have exhausted all possibilities, then
you
can call it prime. **Be careful. Do not think because you
could not factor it on the first try that it is prime. You must
go
through ALL possibilities first before declaring it prime.**

** Example
9**: Factor the trinomial .

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form .

It will look like
this:
(
)( )

Since we have *x* squared
as our first term,
we will need the following:

(*x *
)(*x* )

We need two numbers whose product is 12 and sum is 5.

Can you think of any????

Since the product is a positive number and the sum is a positive number, we only need to consider pairs of numbers where both signs are positive.

One pair of factors of 12 is 3 and 4, which does not
add up to be 5.

Another pair of factors are 2 and 6, which also does not
add up
to 5.

Another pair of factors are 1 and 12, which also does
not add up
to 5.

Since we have looked at ALL the possible factors, and none of them worked, we can say that this polynomial is prime. In other words, it does not factor.

OR

It has to be exactly in this form to use this
rule. **When you
have a base being squared plus or minus twice the product of the two
bases
plus another base squared, it factors as the sum (or difference) of the
bases being squared. **

This is the **reverse
of the
binomial squared found in Tutorial 6: Polynomials**. Recall
that factoring is the reverse of multiplication.

** Example
10**: Factor the perfect square trinomial: .

First note that there is no GCF to factor out of this
polynomial.

Since it is a trinomial, you can try factoring this by** trial and error shown above**. But if you can recognize
that
it fits the form of a **perfect square trinomial**,
you can save yourself some time.

Note that if we would multiply this out, we would get
the original
polynomial.

** Example
11**: Factor the perfect square trinomial:.

First note that there is no GCF to factor out of this
polynomial.

Since it is a trinomial, you can try factoring this by** trial and error shown above**. But if you can recognize
that
it fits the form of a **perfect square trinomial**,
you can save yourself some time.

Note that if we would multiply this out, we would get
the original
polynomial.

Just like the perfect square trinomial, the difference
of two squares
has to be exactly in this form to use this rule. **When
you
have the difference of two bases being squared, it factors as the
product
of the sum and difference of the bases that are being squared.**

This is the **reverse
of the
product of the sum and difference of two terms found in Tutorial 6:
Polynomials**.
Recall that factoring is the reverse of multiplication.

** Example
12**: Factor the difference of two squares: .

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:

Note that if we would multiply this out, we would get
the original
polynomial.

** Example
13**: Factor the difference of two squares: .

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:

Note that if we would multiply this out and the
original expression
out we would get the same polynomial.

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
14**: Factor the sum of cubes: .

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:

Note that if we would multiply this out, we would get
the original
polynomial.

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
15**: Factor the difference of cubes: .

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:

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.**

I. **GCF:**

Always check for the GCF first, no matter what.

II. **Binomials:**

a.

b.

c.

III. **Trinomials:**

a.

b. **Trial and error:**

c. **Perfect square trinomial:**

IV. **Polynomials with four terms:**

Factor by grouping

** Example
16**: Factor completely.

The first thing that we always check when we are
factoring is WHAT?

The GCF. In this case, there is one.

**Factoring out the GCF of 3 we get:**

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:

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
17**: 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:

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.

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
18**: 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:

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
19**: 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:**

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
20**: 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 polynomial with
four terms.
Looks like we will have to try **factoring it by
grouping:**

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 - 1f:Factor completely.

** Need Extra Help on these Topics?**

This webpage goes over how to factor out a GCF and how to factor by grouping.

**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.

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut28_facttri.htm**

This webpage goes over how to factor trinomials.

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

This webpage helps you factor trinomials.** **

**http://www.mathpower.com/tut47.htm**

This website helps you factor trinomials.

**http://www.mathpower.com/tut31.htm**

This website helps you factor trinomials.

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/**
**int_alg_tut29_specfact.htm**

This webpage goes over how to factor perfect square trinomial,
difference
of squares, and sum or difference of cubes.

**http://www.sosmath.com/algebra/factor/fac05/fac05.html**

This webpage helps you with the factoring by special products discussed
in this tutorial.

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

This webpage helps you with the factoring by special products discussed
in this tutorial.

**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. 13, 2009 by Kim Seward.

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