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Polynomial Function

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Lesson Summary:

A polynomial is a function like $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ where $n$ is a positive integer.

Lesson Inputs:

Lesson Outputs:

Lesson:

A polynomial function is a function that is of the form <Sign in to see all the formulas> where $n$ is a positive integer. The variables $a_0$, $a_1$, <Sign in to see all the formulas>, $a_n$ are numbers.

Here are some concrete examples of polynomials:

And here are some functions that are not polynomials:

There are several terms associated with polynomials that are important to know:

The <Sign in to see all the formulas> and $a_0$ are called coefficients.
$a_n$ is called the leading coefficient.
The degree of the polynomial is the highest integer for which $a_n$ is not zero.

Given our examples above let's find out what the degree and leading coefficient is:

Function	Leading Coefficient	Degree	Graph
<Sign in to see all the formulas>	$m$	1	Enlarge
<Sign in to see all the formulas>	4	2	Enlarge
<Sign in to see all the formulas>	200	7	Enlarge
<Sign in to see all the formulas>	10	2	Enlarge

For the last function we needed to expand it out to <Sign in to see all the formulas> to determine the leading coefficient and degree.

Finally, we call a function linear if it is of the form

quadratic if it is of the form

and cubic if it is of the form

Note that each of these is degree $1$, $2$, and $3$, respectively. By setting the quadratic form to zero we get a Quadratic Equation.