**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.
- Evaluate a polynomial function.
- Combine like terms.
- Add and subtract polynomials

** Introduction**

In this tutorial we will be looking at the different components of polynomials. Then we will move on to evaluating polynomial functions as well as adding and subtracting them. Some of these concepts are based on ideas that were covered in earlier tutorials. A lot of times in math you are using 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 .

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 term .

What do you think?

Since the degree is the sum of the variable exponents
and 5 is the only
exponent, **the degree would have to be 5.**

What do you think?

This one is a little bit tricky. Where is the
variable? **When
you have a constant term, it’s degree is always 0, because there is no
variable there. **

**Since this is a constant term, it’s degree is 0.**

What do you think?

Since the degree is the sum of the variable exponents
and it looks like
we have a 1 and a 3 as our exponents, **the degree would have to be 1
+ 3 = 4.**

Since the degree of the polynomial is the highest
degree of all the
terms, **it looks like the degree is 2. **

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

Since the degree of the polynomial is the highest
degree of all the
terms, **it looks like the degree is 6. **

**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.**

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.**

Since a polynomial does fit the **definition
of a function, which can be found in Tutorial 13: Introduction to
Functions**,
we can write a polynomial using function notation.

Evaluating a polynomial function is exactly the same
concept as **evaluating
any function, which can be found in Tutorial 13: Introduction to
Functions.**

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 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.**

First we need to identify the like terms.

**Let’s rewrite this so that we have the like terms
next to each other.**

It looks like we have two terms that have an *x* squared that we can combine and we have two terms that have an *x *that
we can combine. The poor 5 does not have anything it can combine
with so it will have to stay 5.

**Adding like terms we get:**

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 ( ).

** 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 - 1b: Find the degree of the term.

1a. -3

(answer/discussion to 1a)

(answer/discussion to 1a)

Practice Problems 2a - 2c:Find the degree of the polynomial and indicate whether the polynomial is a monomial, binomial, trinomial, or none of these.

Practice Problem 3a: Evaluate the polynomial function.

Practice Problems 4a - 4b: Perform the indicated operation and simplify.

** Need Extra Help on these Topics?**

**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 with adding and subtracting 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 13, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.