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 Intermediate Algebra Tutorial 26: Multiplying Polynomials

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Learning Objectives

 After completing this tutorial, you should be able to: Multiply any polynomial times any other polynomial. Use the FOIL method to multiply a binomial times a binomial. Use special product rules to multiply a binomial squared and a product of a sum and difference of two terms

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

 Multiplying Polynomials

 In general, when multiplying two polynomials together, use the distributive property, as shown in Tutorial 5: Properties of Real Numbers,  until every term of one polynomial is multiplied times every term of the other polynomial.  Make sure that you simplify your answer by combining any like terms. On this page we will look at some of the more common types of polynomials to illustrate this idea.

 (Monomial)(Monomial)

 In this case, there is only one term in each polynomial.  You simply multiply the two terms together.

 Example 1:   Multiply .

 (Monomial)(Polynomial)

 In this case, there is only one term in one polynomial and more than one term in the other.  You need to distribute the monomial to EVERY term of the other polynomial.

 Example 2:   Multiply .

 *Dist. -2a *Mult. like bases add exp.

 (Binomial)(Binomial)

In this case, both polynomials have two terms.  You need to distribute both terms of one polynomial times both terms of the other polynomial.

One way to keep track of your distributive property is to
Use the FOIL method.   Note that this method only works on (Binomial)(Binomial).

 F First terms O Outside terms I Inside terms L Last terms

This is a fancy way of saying take every term of the first binomial times every term of the second binomial.  In other words, do the distributive  property for every term in the first binomial.

 Example 3:   Multiply .

 *Use the FOIL method *Combine like terms

 Binomial Squared   Special Product Rule for  a Binomial Squared:

 In other words, when you have a binomial squared, you end up with the first term squared plus (or minus) twice the product of the two terms plus the last term squared. Any time you have a binomial squared you can use this shortcut method to find your product. This is a special products rule.  It would be perfectly ok to use the foil method on this to find the product.  The reason we are showing you this form is that when you get to factoring, you will have to reverse your steps.  So when you see   , you will already be familiar with the product it came from.

 Example 4:   Multiply .

 *

 Example 5:   Multiply .

 * *   *Write in desc. order

 Product of the sum and difference  of two terms

 This is another special products rule.  It would be perfectly ok to use the foil method on this to find the product.  The reason we are showing you this form is that when you get to factoring, you will have to reverse your steps.  So when you see a difference of two squares, you will already be familiar with the product it came from.

 Example 6:   Multiply .

 *

 Example 7:   Multiply  using a special product .

 * * *Write in desc. order

 (Polynomial)(Polynomial)

 As mentioned above, use the distributive property until every term of one polynomial is multiplied times every term of the other polynomial.  Make sure that you simplify your answer by combining any like terms.

 Example 8:   Multiply .

 *Use Dist. Prop. twice *Combine like terms

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: Multiply.

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.algebrahelp.com/lessons/simplifying/distribution/ This website helps with the distributive property. http://www.algebrahelp.com/lessons/simplifying/foilmethod/ This website helps with the FOIL method and (polynomial)(polynomial). http://www.purplemath.com/modules/polymult.htm This webpage helps with multiplying polynomials.

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 14, 2011 by Kim Seward.