Intermediate Algebra
Tutorial 28: Factoring Trinomials

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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

Factoring Trinomials of the Form

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

Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials that when you multiply them you get the given trinomial.

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 positive, your factors are going to both have the same sign depending on b’s sign.

If c is negative, your factors are going to have opposite signs depending on b’s sign.

Example 1: Factor the trinomial

Note that this trinomial does not have a GCF.

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

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.

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

(y )(y )

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

We need two numbers whose product is 6 and sum is -5. That would have to be -2 and -3.

Putting that into our factors we get:

*-2 and -3 are two numbers whose prod. is 6 and sum is -5

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

Example 2: Factor the trinomial

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

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.

*Factor out the GCF of 2y

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.

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

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

Step 2 (trinomial): Find the factors that go in the first positions.

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

2y(x )(x )

Step 3 (trinomial): Find the factors that go in the last positions.

We need two numbers whose product is -20 and sum is 1. That would have to be 5 and -4.

Putting that into our factors we get:

*5 and -4 are two numbers whose prod. is -20 and sum is 1

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

Factoring Trinomials of the Form

(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 factors of a will go in the first terms of the binomials and the factors of c will go in the last terms of the binomials.

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 3: Factor the trinomial

Note that this trinomial does not have a GCF.

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

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.

In the first terms of the binomials, we need factors of 3 x squared. This would have to be 3x and x.

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 .

Possible Factors

Check using the FOIL method Tutorial 26: Multiplying Polynomials

First try:

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.

This process takes some practice. After a while you will get used to it and be able to come up with the right factor on the first try.

Example 4: Factor the trinomial

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form

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.

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

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 .

Possible Factors

Check using the FOIL method Tutorial 26: Multiplying Polynomials

First try:

This is our original polynomial.
So this is the correct combination of factors for this polynomial.

Factor a Trinomial by Substitution

Sometimes a trinomial does not exactly fit the form

, 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.

Example 5: Factor the trinomial

Note that this trinomial does not have a GCF.

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

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

Let	*Substitute y in for x squared

Step 2: If the trinomial is in the form OR

, factor it accordingly

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 first terms of the binomials, we need factors of 2 y squared. This would have to be 2y and y.

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 .

Possible Factors	Check using the FOIL method Tutorial 26: Multiplying Polynomials
First try:	This is our original polynomial. So this is the correct combination of factors for this polynomial.

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

*Substitute x squared back in for y

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

Example 6: Factor the trinomial

Note that this trinomial does not have a GCF.

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

Let	*Substitute x in for (a + 2)

Step 2: If the trinomial is in the form OR

, factor it accordingly.

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:

*-8 and -4 are two numbers whose prod. is 32and sum is -12

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

*Substitute (a + 2) back in for x

*Combine like terms in ( )

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

Prime Polynomials

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 7: Factor the trinomial

Note that this trinomial does not have a GCF.

So we go right into factoring the trinomial of the form

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.

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

(x )(x )

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

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.

1a.

(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

1c.
(answer/discussion to 1c)

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/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.

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