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 Intermediate Algebra Tutorial 29: Factoring Special Products

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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

 In this tutorial we help you expand your knowledge of polynomials by looking at multiplying polynomials together.  We will look at using the distributive property, initially shown in Tutorial 5: Properties of Real Numbers, to help us out.  Again, we are using a concept that you have already seen to apply to the new concept.  After going through this tutorial you should have multiplying polynomials down pat.

Tutorial

 Factoring a Perfect Square Trinomial 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 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:

 a.  b.  c.

III. Trinomials:

 a.  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.

Practice Problems 1a - 1e: Factor Completely.

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.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.

Last revised on July 15, 2011 by Kim Seward.