Introduction to Polynomial Functions
On this page we will introduce you to a couple of things. First, a semi-formal definition of polynomial functions. The notation here is a bit confusing so we’ll take our time exploring what it all means. We will also see what polynomial functions are and what they are not.
Be sure to take notes using the first tab below, watch the video, taking more notes, then try the practice problems. As this is the first section in this unit, it is very important that you fully understand what is discussed here.
Then, when you’re done, pat yourself on the back for a job well done!
Polynomial Functions
In this unit on polynomial equations we will introduce you to the various features, graphs, methods, techniques and concepts associated to polynomial equations. Let’s start with the basics, like some definitions and ideas.
A polynomial equation is always a function, and that can be seen from the definition. Here it is:
Definition
A generic polynomial function is shown below.
This certainly looks confusing, but let’s break it down. Here, n is the degree, the largest exponent. The value of n must be a Whole Number (0, 1, 2, 3, …). The value of a is any Real Number, and it is just a coefficient. For example, say that the degree was 4, then 
is the leading coefficient.
Here the value of n is 4. So, 
because 5 is the leading coefficient. Do you see what 
equals? It is zero, because 0x3 = 0, and there’s no need to write + 0x3.
Let’s take a look at n, which is the exponent. Because n is a Whole Number it cannot be negative or a fraction. If n was a negative number, like – 3, this is what would happen. This is NOT a polynomial function.
Negative exponents are a way of writing repeated division. So, we can rewrite this as follows:
Polynomial functions will never have the input, x, in the denominator. However, you can have a coefficient that is a fraction.
The value of n cannot be a fraction, like 2/3. Here’s what that would look like.
Coefficients can be in radical notation, so long as they’re Real Numbers (not the square root of a negative number for example). But, the input, x, cannot be the radicand. If it were, then the function would NOT be a polynomial function. You will learn about those types of functions in a future unit.
Here are some examples of polynomial functions you probably are familiar with.
Some examples, that are not arbitrary, but specific are below.
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