# Polynomial Functions

## Introduction

Polynomials are one of the simplest collection of functions that we can understand. In this section we discuss the immediate consequences of the information given in a problem.

## Definition

A polynomial function is a function given by a polynomial, ${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+x_{0}{\text{ for }}n\geq 0{\text{ and }}a_{n},a_{n-1},\ldots a_{1},a_{0}}$ are numbers called coefficients. For this section we only care about ${\displaystyle a_{n}}$, which is called the leading coefficient and the parity of n, whether n is even or odd.

Example:

Two examles of polynomials are: ${\displaystyle x^{3}+-2x^{2}+x-5{\text{ and }}x^{4}-3x^{2}+9}$

## Power Functions

Before we talk about polynomial we will discuss a simpler version, power functions. These are monomials of the form ${\displaystyle ax^{n}}$ where a ${\displaystyle \neq }$ 0 and n is a positive integer.

Properties: When n is even, that is ${\displaystyle f(x)=ax^{n}{\text{ for }}a>0}$ 1. f is an even function 2. Domain = ${\displaystyle (-\infty ,\infty )}$, while the range = ${\displaystyle [0,\infty )}$ 3. (-1, a), (0, 0), and (1, a) are always points on the graph of f.

When n is odd

1. f is an odd function 2. Domain = Range = <mat>(-\infty, \infty)[/itex] 3. The points (-1, -a), (0, 0), and (1, a) are always on the graph of f.

## Zeros of a Polynomial Function

Definition: If f is a function, it does not have to be a polynomial, and r is a real number such that f(r) = 0, then r is called a real zero of f.

The following three statements are equivalent for all functions, and the fourth is equivalent to the first three when f is a polynomial function:

1) r is a real zero of f(x) 2)r is an x-intercept 3)r is a solution of f(x)= 0 4) (x - r) is a factor of f

## Multiplicities

Sometimes whena a polynomial is completely factored a factor of the form (x - r) may occur multiple times.

Definition:

If ${\displaystyle (x-r)^{m}}$ is a factor of a polynomial function f, but ${\displaystyle (x-r)^{m+1}}$ is not a factor, then r is called a zero of multiplicity m of f.

Note: Other sources may say r is a root of multiplicity m.

Since a root corresponds to an x-intercept, the multiplicity of the root gives us information about the behavior around the zero.

If r is a zero of even multiplicity: The sign of f(x) does not change from one side of r to the other. So if f(x) is positive(negative) to the left of r it is also positive(negative) to the right of r. From a more geometric(graphical) standpoint, the function bounces off of the x-axis.

If r is a zero of odd multiplicity: The sign of f(x) changes sign from side of r to the other. So, if f(x) is positive the the left of r it is negative to the right of r, and vice versa. Once again, from the geometric standpoint, f(x) will cross form below the x-axis to above the x-axis or vice versa.

## End Behavior

For large positive or negative values of x, the polynomial ${\displaystyle f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots a_{1}x+a_{0}}$ will behave like the power function ${\displaystyle a_{n}x^{n}}$ So if n is even, both ends of the function will point up or down, depending on whether ${\displaystyle a_{n}}$ is positive or negative, respectively.

If n is odd, one end will point up and the other will point down. Once again, this depends on whether ${\displaystyle a_{n}}$ is positive or negative. If ${\displaystyle a_{n}>0}$, then the left end will point down, and the right end will point up. The reverse holds if ${\displaystyle a_{n}<0}$.

The way I keep them straight is by comparing odd power functions to what ${\displaystyle f(x)=x,{\text{ or }}f(x)=-x}$ does. For even power functions, I look at ${\displaystyle f(x)=x^{2},{\text{ or }}f(x)=-x^{2}}$