Intermediate Algebra Tutorial on Factoring Special Products

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Intermediate Algebra
Tutorial 29:
Factoring Special Products

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

We complete our trilogy of factoring by showing you some factoring by special products. Just to recap what we have done with factoring so far: in Tutorial 27 we looked at the greatest common factor and factoring by grouping, and in Tutorial 28 we looked at factoring trinomials.

At the end of this tutorial, I will give you a checklist to help you strategize when you are having to factor. As mentioned in the last tutorial, factoring is one of those things that takes practice to master. So again, make sure that you are doing the problems your teacher assigns you as well as the ones on this page. You never know when your math skills will be put to the test. After we are through with you, you are going to be a factoring genius. Now let's get to factoring these special products.

Tutorial

Factoring a Perfect Square Trinomial

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 26: Multiplying Polynomials. Recall that factoring is the reverse of multiplication.

Example 1: Factor

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 as shown in Tutorial 28: Factoring Trinomials. But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time.

*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

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 as shown in Tutorial 28 (Factoring Trinomials). But if you can recognize that it fits the form of a perfect square trinomial, you can save yourself some time.

*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

Note that the sum of two squares DOES NOT factor.

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 26: Multiplying Polynomials. Recall that factoring is the reverse of multiplication.

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:

b. Trial and error:

c. Perfect square trinomial:

IV. Polynomials with four terms:

Factor by grouping

Example 7: 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 4 as was shown in Tutorial 27: The GCF and Factoring by Grouping, we get:

*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.

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 as shown in Tutorial 27: The Greatest Common Factor and Factoring by Grouping:

*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.