Polynomial Functions and Models

Module 12

A polynomial of degree n is a function of the form

P(x) = anxn + an-1xn-1 + ... + a1x + a0

Where an 0. The numbers a0, a1, a2, . . . , an are

called the coefficients of the polynomial.

The a0 is the constant coefficient or constant term.

The number an, the coefficient of the highest

power, is the leading coefficient, and the term anxn is

the leading term.

Polynomial Functions

Graphs of Polynomial Functions and Nonpolynomial Functions

Graphs of Polynomials

Graphs are linesDegree 0 or 1 ex. f(x) = 3 or f(x) = x –

5

Graphs are parabolasDegree 2 ex. f(x) = x2 + 4x + 8

Graphs smooth curveDegree greater than

2 ex. f(x) = x3

These graphs will not have the following:Break or holeCorner or cusp

Even- and Odd-Degree Functions

The Leading-Term Test

Zero- another way of saying solution

Zeros of PolynomialsSolutionsPlace where graph crosses the x-axis

(x-intercepts)Zeros of the function

Place where f(x) = 0

Finding Zeros of a Polynomial

4 3 2( ) 5 21 18P x x x x x

Using the Graphing Calculator to Determine Zeros

Graph the following polynomial function and determine the zeros.

Before graphing, determine the end behavior and the numberof relative maxima/minima.

In factored form:P(x) = (x + 2)(x – 1)(x – 3)²

MultiplicityIf (x-c)k, k 1, is a factor of a polynomial function P(x) and:

K is oddThe graph crosses

the x-axis at (c, 0)

K is evenThe graph is tangent

to the x-axis at (c, 0)

Multiplicity

y = (x + 2)²(x − 1)³

 Answer.  

 −2 is a root of multiplicity 2,

and 1 is a root of multiplicity 3.  

These are the 5 roots:

−2,  −2,  1,  1,  1.

Multiplicity

y = x³(x + 2)4(x − 3)5

Answer.   

0 is a root of multiplicity 3,

-2 is a root of multiplicity 4,

and 3 is a root of multiplicity 5.  

1. Use the leading term to determine the end behavior.

2. Find all its real zeros (x-intercepts). Set y = 0.

3. Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.

4. Find the y-intercept. Set x = 0.

5. Use any additional information (i.e. turning points or multiplicity) to graph the function.

To Graph a Polynomial