Intermediate Algebra
Tutorial 25: Polynomials and Polynomial Functions

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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. 
A term is a number, variable or the product of a
number and variable(s).
Examples of terms are , , , z 
A coefficient is the numeric factor of your
term.
Here are the coefficients of the terms listed above:
Term

Coefficient


3


5


2

z

1


A constant term is a term that contains only a number.
In other
words, there is no variable in a constant term.
Examples of constant terms are 4, 100, and 5. 
Standard Form of a
Polynomial
where n is a nonnegative
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 nonnegative integers. Note
that
the terms are separated by +’s and ‘s. An example of a polynomial expression is . 
The degree of a term is the sum of the exponents on
the variables
contained in the term. 
The degree of the polynomial is the largest degree
of all its terms. 
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. 
Some Types of Polynomials

Type

Definition

Example

Monomial

A polynomial with one term

5x

Binomial

A polynomial with two terms

5x  10

Trinomial

A polynomial with three terms


Let’s go through some examples that illustrate these
different definitions.
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. 
Example
2: Find the degree of the term 8. 
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. 
Example
3: Find the degree of the term . 
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. 
Example
4: 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 2.
Since there are three terms, this is a trinomial. 
Example
5: 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 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. 
Example
6: 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. 
Example
7: If find P(2). 
Plugging 2 into the polynomial function we get: 

*Replace x with
2
*Exponent
*Multiplication
*Subtraction

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. 
Example
8: Simplify by combining like terms: . 
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: 

*Combine the x squared terms
together
and then the x terms together 
If there is only a + sign in front of ( ), then the
terms inside of
( ) remain the same when you remove the ( ). 
Step 2: Combine like terms. 
Example
9: Perform the indicated operation and
simplify: 

*Remove the ( )
*Add like terms together

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 ( ). 
Step 2: Combine like terms. 
Example
10: Perform the indicated operation and
simplify: 

*Dist. the  through second ( )
*Combine like terms

Example
11: Perform the indicated operation and
simplify: 

*Dist. the  through second ( )
*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  1b: Find the degree of the term.
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?
Last revised on July 13, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.
