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

Example:

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

Power Functions

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

Properties: When n is even, that is $f(x)=ax^{n}{\text{ for }}a>0$ 1. f is an even function 2. Domain = $(-\infty ,\infty )$, while the range = $[0,\infty )$ 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)</math> 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 $(x-r)^{m}$ is a factor of a polynomial function f, but $(x-r)^{m+1}$ 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 $f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots a_{1}x+a_{0}$ will behave like the power function $a_{n}x^{n}$ So if n is even, both ends of the function will point up or down, depending on whether $a_{n}$ 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 $a_{n}$ is positive or negative. If $a_{n}>0$, then the left end will point down, and the right end will point up. The reverse holds if $a_{n}<0$.

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

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