Intermediate Algebra Tutorial 26


Intermediate Algebra
Tutorial 26: Multiplying Polynomials


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deskLearning Objectives


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




desk 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.  

 

 

desk 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.

 
 
 
notebook Example 1:   Multiply example 1a.

 
example 1b
*Mult. like bases add exp.

 
 
  (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.

 
 
 
notebook Example 2:   Multiply example 2a.

 
example 2b

*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.


 
 
 
notebook Example 3:   Multiply example 3a.

 
example 3b

 
 

*Use the FOIL method

*Combine like terms
 


 

Binomial Squared

binomial squared
 

Special Product Rule for 
a Binomial Squared: 

binomial squared

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 binomial squared  , you will already be familiar with the product it came from.


 
 
 
notebook Example 4:   Multiply example 4a.

 
example 4b
*binomial squared

 
 
 
notebook Example 5:   Multiply example 5a.

 
example 5b
*binomial squared

*binomial squared
 

*Write in desc. order
 


 


 

Product of the sum and difference 
of two terms

difference of squares


 
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.

 
 
 
notebook Example 6:   Multiply example 6a.

 
example 6b
*difference of squares

 
 
notebook Example 7:   Multiply  using a special product example 7a.

 
example 7b
*difference of squares
*binomial squared

*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.

 
 
 
notebook Example 8:   Multiply example 8a.

 
example 8b
*Use Dist. Prop. twice

*Combine like terms

 


 

 
desk 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.

 

pencil Practice Problems 1a - 1e: Multiply.

 

1a. problem 1a
(answer/discussion to 1a)
1b. problem 1b
(answer/discussion to 1b)

1c. problem 1c
(answer/discussion to 1c)

1d. problem 1d
(answer/discussion to 1d)

1e. problem 1e
(answer/discussion to 1e)

 

 

 

desk 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.


 


WTAMU > Virtual Math Lab > Intermediate Algebra


Last revised on July 14, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.