Intermediate Algebra Tutorial 25

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.

Term

A term is a number, variable or the product of a number and variable(s).

Examples of terms are , , , z

Coefficient

A coefficient is the numeric factor of your term.

Here are the coefficients of the terms listed above:

Term Coefficient

2 z 1

Constant Term

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 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 .

Degree of a Term

The degree of a term is the sum of the exponents on the variables contained in the term.

Degree of the Polynomial

The degree of the polynomial is the largest degree of all its terms.

Descending Order

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.

Polynomial Function

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.

Example 7: If

find P(-2).

Plugging -2 into the polynomial function we get:

*Replace x with -2
*Exponent
*Multiplication
*Subtraction

Combining Like Terms

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

Adding Polynomials

Step 1: Remove the ( ) .

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

Subtracting Polynomials

Step 1: 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 ( ).

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.

1a. -3
(answer/discussion to 1a)

1b.

(answer/discussion to 1b)

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

2a.
(answer/discussion to 2a)