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==Introduction== Polynomials are one of the simplest collection of functions that we can understand. In this section we discuss the immed..."

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==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, <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + x_0 \text{ for } n\ge 0
\text{ and }a_n, a_{n-1}, \ldots a_1, a_0</math> are numbers called coefficients. For this section we only care about <math>a_n</math>, which is called the leading coefficient and the parity of n, whether n is even or odd.

Example:

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

==Power Functions==

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

Properties:
When n is even, that is <math>f(x) = ax^n \text{ for }a > 0</math>
1. f is an even function
2. Domain = <math>(-\infty, \infty)</math>, while the range = <math>[0, \infty)</math>
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 <math>(x - r)^m</math> is a factor of a polynomial function f, but <math>(x - r)^{m + 1}</math> 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 <math>f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots a_1x + a_0</math> will behave like the power function <math> a_nx^n</math>
So if n is even, both ends of the function will point up or down, depending on whether <math>a_n</math> 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 <math>a_n</math> is positive or negative. If <math> a_n > 0</math>, then the left end will point down,
and the right end will point up. The reverse holds if <math>a_n < 0</math>.

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

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Polynomial Functions - Revision history

MathAdmin: Created page with "__TOC__ ==Introduction== Polynomials are one of the simplest collection of functions that we can understand. In this section we discuss the immed..."

MathAdmin: Created page with "
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==Introduction== Polynomials are one of the simplest collection of functions that we can understand. In this section we discuss the immed..."