**Learning Objectives**

After completing this tutorial, you should be able to:

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

** Introduction**

In this tutorial we add on to your factoring repertoire by talking about factoring trinomials. Basically, you will be doing the FOIL method backwards. This is one of those things that just takes practice to master. Make sure that you work through the problems on this page as well as any that you teacher may have assigned you. You never know when your math skills will be put to the test. Now let's get to factoring these trinomials.

** Tutorial**

(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

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 *y* squared as our first term, we will need
the following:

(*y *
)(*y* )

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

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

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

We need to factor out the **GCF,
as shown in Tutorial 27: The GCF and Factoring by Grouping**,
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.

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 negatives here because
the
middle term is negative.**

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.

This is our original polynomial.

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

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 -6.
This would have to be -6 and 1, 6 and -1, 3 and -2, or -3 and 2.
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 .

**First try:**

This is our original polynomial.

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

Sometimes a trinomial does not exactly fit the
form
OR ,
but with
the use of substitution it can be written in that form.

**Step 1: Substitute x for the
non-coefficient
part of the middle term.**

Note that if *x* is
already being used in
the polynomial, you want to use a different variable like *y *to
avoid confusion.

When you substitute, the trinomial should be in the form OR .

If it isn't, then you may have to seek a different method than what we are covering here. All of the ones we are using substitution with on this web page will fit this form.

**Step 2: If the trinomial
is in the form **OR **,
factor it accordingly.**

**Step 3: Substitute back in what you
replaced x with in step 1.**

Note that this trinomial does not have a GCF.

Since *x* is already being
used in this problem,
let's use *y* for our substitution.

Let

***Substitute y in
for x squared**

Now it is in a form that we do know how to
factor.

Now we proceed as we did in examples 3 and 4 above.

**In the second terms of the binomials, we need factors
of -9.
This would have to be -3 and 3, 9 and -1, or -9 and 1. 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 .

**First try:**

This is our original polynomial.

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

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

Note that this trinomial does not have a GCF.

Let

***Substitute x in
for (a + 2)**

Now it is in a form that we do know how to
factor.

Now we proceed as we did in examples 1 and 2 above.

**We need two numbers whose product is 32 and sum is
-12. That
would have to be -8 and -4.**

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

***Combine like terms in
( )**

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

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.**

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 9 and sum is 2.

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 9 is 3 and 3, which does not
add up to be
2.

Another pair of factors are 1 and 9, which also does not
add up
to 2.

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.

** 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 Completely.

** Need Extra Help on these Topics?**

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

This webpage helps you factor trinomials.

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

This webpage helps you factor trinomials.

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

This webpage helps you factor trinomials.

**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.