Polynomial Functions of Higher Degree

When you are done with this section, you will be able to do the following


We have already looked at two types of polynomial functions:  Linear Functions and Quadratic Functions.

Let's first define a polynomial.  A polynomial in x is an algebraic expression of the form where n is a nonnegative integer, , and .

There are several parts to the polynomial that we will want to know the names of:
n is the degree of the polynomial.
is the leading coefficient.
is the constant term.

What is end behavior of a polynomial?
We now want to look at the graphs of several polynomials to see what we can discover.
Pull out your graphing calculator and graph the following functions in the same window.
What do you notice about these graphs?

Let's start with the way the graphs look.  They all look more or less like a U.  They only differ in being fatter or skinnier.  If this were a live class, you would see me with both my arms up in the air much like this guy . 

Now let's look at the functions themselves.  What is the degree of the polynomial f(x)?  What is the degree of the polynomial g(x)?  What is the degree of the polynomial h(x)?

If you said 2, 4 and 6 respectively, you were correct.  What can you say that the numbers 2, 4, and 6 have in common?  What type of numbers are they?  Hopefully, you are thinking even.  That is the critical information that we need.

Now let's look at  the leading coefficients of the functions.  What is the leading coefficient of the polynomial f(x)?  What is the leading coefficient of the polynomial g(x)?  What is the leading coefficient of the polynomial h(x)?  Yeah, yeah, they are all 1.

Try graphing a few more graphs like f(x), g(x), and h(x) on you calculator changing the leading coefficient to numbers like 2, 5, 100, , .  Does changing the leading coefficient to a number like one of these change the basic look of the graph from this .  Now what do all of these numbers have in common.  Hopefully, you are thinking, they are all positive numbers. 

Now if we put this information together, we can say that a polynomial that has a positive leading coefficient and an even degree basically looks like this .  The end behavior is for the y-values to increase toward infinity.

You may still be skeptical.  You should try graphing several polynomial functions on your calculator that fit this criteria.  You can add terms on to the polynomial and you will still see that the basic look is .  Since we are only looking at end behaviors, what happens under the yellow part of the smiley face is not relevant.  The graph may run into the x-axis several times under the smiley face.  On the ends, it will still look like .

We are going to start a table to put together all the information about end behaviors.

Degree
 Leading Coefficient
 Left-hand Behavior
()
 Right-hand Behavior
()
 Picture
Even
 Positive

Now graphs the following functions on your graphing calculator.
What do you notice about these graphs?

Let's start with the way the graphs look.  They all look more or less like an upside down U.  They only differ in being fatter or skinnier.  If this were a live class, you would see me with both my arms down bent at the elbows much like this guy . 

Now let's look at the functions themselves.  What is the degree of the polynomial f(x)?  What is the degree of the polynomial g(x)?  What is the degree of the polynomial h(x)?

If you said 2, 4 and 6 respectively, you were correct.  What can you say that the numbers 2, 4, and 6 have in common?  What type of numbers are they?  Hopefully, you are still thinking even.  That is the critical information that we need.

Now let's look at  the leading coefficients of the functions.  What is the leading coefficient of the polynomial f(x)?  What is the leading coefficient of the polynomial g(x)?  What is the leading coefficient of the polynomial h(x)?  Okay, they are all -1.

Try graphing a few more graphs like f(x), g(x), and h(x) on you calculator changing the leading coefficient to numbers like -2, -5, -100, , .  Does changing the leading coefficient to a number like one of these change the basic look of the graph from this .  Now what do all of these numbers have in common.  Hopefully, you are thinking, they are all negative numbers. 

Now if we put this information together, we can say that a polynomial that has a negative leading coefficient and an even degree basically looks like this .  The end behavior is for the y-values to decrease toward negative infinity.

You may still be skeptical.  You should try graphing several polynomial functions on your calculator that fit this criteria.  You can add terms on to the polynomial and you will still see that the basic look is .  Since we are only looking at end behaviors, what happens under the yellow part of the smiley face is not relevant.  The graph may run into the x-axis several times under the smiley face.  On the ends, it will still look like .

Let's continue our table where we are putting together all the information about end behaviors.

Degree
 Leading Coefficient
 Left-hand Behavior
()
 Right-hand Behavior
()
 Picture
Even
 Positive
Even
 Negative

Now graphs the following functions on your graphing calculator.
What do you notice about these graphs?

Let's start with the way the graphs look.  They all look more or less like snake slithering up from left to right.  They only differ in being straighter or wigglier.  If this were a live class, you would see me with my left arm pointing down and my right arm pointing up much like this guy . 

Now let's look at the functions themselves.  What is the degree of the polynomial f(x)?  What is the degree of the polynomial g(x)?  What is the degree of the polynomial h(x)?

If you said 1, 3 and 5 respectively, you were correct.  What can you say that the numbers 1, 3, and 5 have in common?  What type of numbers are they?  Hopefully, you are thinking odd.  That is the critical information that we need.

Now let's look at  the leading coefficients of the functions.  What is the leading coefficient of the polynomial f(x)?  What is the leading coefficient of the polynomial g(x)?  What is the leading coefficient of the polynomial h(x)?  Okay, they are all 1.

Try graphing a few more graphs like f(x), g(x), and h(x) on you calculator changing the leading coefficient to numbers like 2, 5, 100, , .  Does changing the leading coefficient to a number like one of these change the basic look of the graph from this .  Now what do all of these numbers have in common.  Hopefully, you are thinking, they are all positive numbers. 

Now if we put this information together, we can say that a polynomial that has a positive leading coefficient and an odd degree basically looks like this .  The end behavior is for the y-values to the left of the x-axis decrease toward negative infinity and the y-values to the right of the x-axis increase toward infinity.

You may still be skeptical.  You should try graphing several polynomial functions on your calculator that fit this criteria.  You can add terms on to the polynomial and you will still see that the basic look is .  Since we are only looking at end behaviors, what happens under the yellow part of the smiley face is not relevant.  The graph may run into the x-axis several times under the smiley face.  On the ends, it will still look like .
Let's continue our table where we are putting together all the information about end behaviors.

Degree
 Leading Coefficient
 Left-hand Behavior
()
 Right-hand Behavior
()
 Picture
Even
 Positive
Even
 Negative
Odd
 Positive

Now graphs the following functions on your graphing calculator.
What do you notice about these graphs?

Let's start with the way the graphs look.  They all look more or less like snake slithering down from left to right.  They only differ in being straighter or wigglier.  If this were a live class, you would see me with my left arm pointing up and my right arm pointing down much like this guy . 

Now let's look at the functions themselves.  What is the degree of the polynomial f(x)?  What is the degree of the polynomial g(x)?  What is the degree of the polynomial h(x)?

If you said 1, 3 and 5 respectively, you were correct.  What can you say that the numbers 1, 3, and 5 have in common?  What type of numbers are they?  Hopefully, you are thinking odd.  That is the critical information that we need.

Now let's look at  the leading coefficients of the functions.  What is the leading coefficient of the polynomial f(x)?  What is the leading coefficient of the polynomial g(x)?  What is the leading coefficient of the polynomial h(x)?  Okay, they are all -1.

Try graphing a few more graphs like f(x), g(x), and h(x) on you calculator changing the leading coefficient to numbers like -2, -5, -100, , .  Does changing the leading coefficient to a number like one of these change the basic look of the graph from this .  Now what do all of these numbers have in common.  Hopefully, you are thinking, they are all negative numbers. 

Now if we put this information together, we can say that a polynomial that has a negative leading coefficient and an odd degree basically looks like this .  The end behavior is for the y-values to the left of the x-axis increase toward infinity and the y-values to the right of the x-axis decrease toward negative infinity.

You may still be skeptical.  You should try graphing several polynomial functions on your calculator that fit this criteria.  You can add terms on to the polynomial and you will still see that the basic look is .  Since we are only looking at end behaviors, what happens under the yellow part of the smiley face is not relevant.  The graph may run into the x-axis several times under the smiley face.  On the ends, it will still look like .
Let's finish our table where we are putting together all the information about end behaviors.

Degree
 Leading Coefficient
 Left-hand Behavior
()
 Right-hand Behavior
()
 Picture
Even
 Positive
Even
 Negative
Odd
 Positive
Odd
 Negative

The table above summarizes the information about end behaviors.  I fully expect to see all of you doing the little "end-behavior dance" as you are taking your midterm and final.
  
Additional On-line Resources:
What are roots of a polynomial?
The roots of a polynomial are all numbers that make the polynomial equal to 0.  In this section, we will only have polynomials that we can solve by factoring.  In the next section, we will learn about polynomial division and will be able to find zeros that can be found by methods other than factoring but are still real.  Finally, we will learn about complex number and be able to find all the zeros (both real and complex) of a polynomial functions.

Example 1:
Find all the real zeros of the polynomial function algebraically.
Solution:
Let's start with what do we mean by real zeros or roots.  Real zeros are the x-coordinates of the x-intercepts of the graph of the polynomial.

For this polynomial, we can factor.  We will start by setting the function equal to 0 and then factoring out .  This will give us

Now we can factor the quadratic factor of to get

Now just for clarity, we are going to break the polynomial up into linear factors (factors with a degree of 1).  This will give us


We will now set each factor to zero and solve for x.   (Please note that this only works if the polynomial is equal to zero and does not work if the polynomial is equal to any other number.)

This gives us the real zeros of the polynomial as 0, 5, and -4.  (Please note, for many WeBWorK problems, you will need to put the 0 in twice since it is produced as a root by two factors.)

 Additional On-line Resources:


Polynomial Functions of Higher Degree Homework:
Polynomial Functions of Higher Degree Homework contains 7 problems.
Things to remember:
  • Intercepts are points and must be written as ordered pairs (x-intercepts look like (x, 0) and y-intercepts look like (0, y)).
  • The point on the graph that is not a root (zero) is used to determine the leading coefficient.
  • When find the equation of a graph, first determine the zeros of the graph and then follow the same method as if just the zeros were given instead of the graph.
    •
Homework examples:
 also-recommended Homework examples:
Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Polynomial Functions of Higher Degree Homework.  There are many different ways to ask the same questions.  This will allow you to see additional problems that are related to this topic.

© 2005 Elizabeth E. K. Jones and the ASU Department of Mathematics and Statistics - All rights reserved.