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
On this page we will look at some of the more common
types of polynomials
to illustrate this idea. 
In this case, there is only one term in each
polynomial. You
simply multiply the two terms together. 
Example
1: Multiply . 
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 . 
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

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



