**Learning Objectives**

After completing this tutorial, you should be able to:

- Identify a term, coefficient, constant term, and polynomial.
- Tell the difference between a monomial, binomial, and trinomial.
- Find the degree of a term and polynomial.
- Combine like terms.
- Add and subtract polynomials.
- 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 will be looking at the different
components of
polynomials. Then we will move on to adding, subtracting and
multiplying
them. Some of these concepts are based on ideas that were covered
in earlier tutorials. A lot of times in math you use previous
knowledge
to learn new concepts. The trick is to not reinvent the wheel
each
time but recognize what you have done before and draw on that knowledge
to help you work through the problems.

** Tutorial**

Let’s start with defining some words before we get to
our polynomial.

Examples of terms are *,
z.*

Here are the coefficients of the terms listed above:

Examples of constant terms are 4, 100, and -5.

**where n is a non-negative
integer. **

** is
called the leading coefficient.**

** is
a constant.**

In other words, **a polynomial is a finite sum of
terms where the
exponents on the variables are non-negative integers.** Note
that
the terms are separated by +'s and -'s.

An example of a polynomial expression is .

For example, the degree of the term would be 1 + 1 = 2. The exponent on *a *is
1 and on *b* is 1 and the sum of the
exponents
is 2.

The degree of the term would be 3 since the only variable exponent that we have is 3.

Note that the standard form of a polynomial that is
shown above is
written in descending order. **This means that the term that
has
the highest degree is written first, the term with the next highest
degree
is written next, and so forth**.

Also note that a polynomial can be “missing” terms. For example, the polynomial written above starts with a degree of 5, but notice there is not a term that has an exponent of 4. That means the coefficient on it is 0, so we do not write it.

** Example
1**: Find the degree of the polynomial and
indicate
whether the polynomial is a monomial, binomial, trinomial, or none of
these: .

Since the degree of the polynomial is the highest
degree of all the
terms, **it looks like the degree is 3. **

Since there are three terms, **this is a trinomial.**

** Example
2**: Find the degree of the polynomial and
indicate
whether the polynomial is a monomial, binomial, trinomial, or none of
these: .

Since the degree of the polynomial is the highest
degree of all the
terms, **it looks like the degree is 3. **

**Make sure that you don’t fall
into the trap
of thinking it is always the degree of the first term. This
polynomial
is not written in standard form (descending order). So we had to
actually go to the second term to get the highest degree.**

Since there are two terms, **this is a binomial.**

** Example
3**: Find the degree of the polynomial and
indicate
whether the polynomial is a monomial, binomial, trinomial, or none of
these:
-20.

Since the degree of the polynomial is the highest
degree of all the
terms, **it looks like the degree is 0. **

Since there is one term, **this is a monomial.**

Recall that **like terms are terms that have the
exact same variables
raised to the exact same exponents.** One example of like terms
is . Another
example is .

**You can only combine terms that are like terms.**
You can
think of it as the reverse of the distributive property.

It is like counting apples and oranges. **You
just count up how
many variables you have the same and write the number in front of
the common variable part.**

If there is only a + sign in front of ( ), then the
terms inside of
( ) remain the same when you remove the ( ).

If there is a - in front of the ( ), then distribute it by multiplying every term in the ( ) by a -1 (or you can think of it as negating every term in the ( )).

** Example
4**: Perform the indicated operation and
simplify: .

** Example
5**: Perform the indicated operation and
simplify: .

On this page we will look at some of the more common types of polynomials to illustrate this idea.

** Example
6**: Find the product .

** Example
7**: Find the product .

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

This is a fancy way of saying to 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
8**: Find the product .

***Use the FOIL method**

***Combine like terms**

**Special product rule for **

**a binomial 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
9**: Find the product .

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
10**: Find the product .

** Example
11**: Find the product .

***Combine like terms**

Any time you have a binomial cubed you can use this shortcut method to find your product.

** Example
12**: Find the product .

** 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 - 1c:Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these.

1a. -10

(answer/discussion to 1a)

(answer/discussion to 1a)

Practice Problems 2a - 2e:Perform the indicated operation.

** Need Extra Help on these Topics?**

This webpage goes over the basic terminology of polynomials as well as how to add and subtract them.

**http://www.purplemath.com/modules/polydefs.htm**

This webpage helps you with the different parts of a polynomial.

**http://www.purplemath.com/modules/polyadd.htm**

This webpage helps you with adding and subtracting polynomials.

**http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/****int_alg_tut26_multpoly.htm**

This webpage goes over multiplying polynomials.

**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.**

Videos at this site were created and produced by Kim Seward and Virginia Williams Trice.

Last revised on Dec. 13, 2009 by Kim Seward.

All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.