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.
A polynomial function is a function given by a polynomial, are numbers called coefficients. For this section we only care about , which is called the leading coefficient and the parity of n, whether n is even or odd.
Two examles of polynomials are:
Before we talk about polynomial we will discuss a simpler version, power functions. These are monomials of the form where a 0 and n is a positive integer.
Properties: When n is even, that is 1. f is an even function 2. Domain = , while the range = 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
Sometimes whena a polynomial is completely factored a factor of the form (x - r) may occur multiple times.
If is a factor of a polynomial function f, but 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.
For large positive or negative values of x, the polynomial will behave like the power function So if n is even, both ends of the function will point up or down, depending on whether 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 is positive or negative. If , then the left end will point down, and the right end will point up. The reverse holds if .
The way I keep them straight is by comparing odd power functions to what does. For even power functions, I look at