Sullivan PreCalculus Section 3.2 Polynomial Functions Objectives Identify Polynomials and Their Degree Graph Polynomial Functions Using Transformations Identify the Zeros of a Polynomial and Their Multiplicity Analyze the Graph of a Polynomial Function

A polynomial function is a function of the form where an , an-1 ,…, a1 , a0 are real numbers and n is a nonnegative integer. The domain consists of all real numbers. The degree of the polynomial is the largest power of x that appears.

Example: Determine which of the following are polynomials Example: Determine which of the following are polynomials. For those that are, state the degree.

A power function of degree n is a function of the form where a is a real number, a 0, and n > 0 is an integer.

Summary of Power Functions with Even Degree 1.) Symmetric with respect to the y-axis. 2.) Domain is the set of all real numbers. Range is the set of nonnegative real numbers. 3.) The graph always contains the points (0, 0); (1, 1); and (-1, 1). 4.) As the exponent increases in magnitude, the graph increases very rapidly as x increases, but for x near the origin the graph tends to flatten out and lie closer to the x-axis.

(1, 1) (-1, 1) (0, 0)

Summary of Power Functions with Odd Degree 1.) Symmetric with respect to the origin. 2.) Domain is the set of all real numbers. Range is the set of all real numbers. 3.) The graph always contains the points (0, 0); (1, 1); and (-1, -1). 4.) As the exponent increases in magnitude, the graph becomes more vertical when x > 1 or x < -1, but for -1 < x < 1, the graphs tends to flatten out and lie closer to the x-axis.

(0, 0) (1, 1) (-1, -1)

Graph the following function using transformations. (0,0) (1,1) (0,0) (1, -2)

(1,0) (2,-2) (1, 4) (2, 2)

Consider the polynomial: f x ( ) )( = + - 1 4 2 Solve the equation f (x) = 0 f x ( ) )( = + - 1 4 2 = 0 x + 1 = 0 OR x - 4 = 0 x = - 1 OR x = 4 If f is a polynomial function and r is a real number for which f (r) = 0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then a.) (r,0) is an x-intercept of the graph of f. b.) (x - r) is a factor of f.

If is a factor of a polynomial f and is not a factor of f, then r is called a zero of multiplicity m of f. Example: Find all real zeros of the following function and their multiplicity. x = 3 is a zero with multiplicity 2. x = - 7 is a zero with multiplicity 1. x = 1/2 is a zero with multiplicity 5.

If r is a Zero of Even Multiplicity Sign of f (x) does not change from one side to the other side of r. Graph touches x-axis at r. If r is a Zero of Odd Multiplicity Sign of f (x) changes from one side to the other side of r. Graph crosses x-axis at r.

Theorem: If f is a polynomial function of degree n, then f has at most n - 1 turning points. Theorem: For large values of x, either positive or negative, the graph of the polynomial resembles the graph of the power function

For the polynomial ( ) f x = + - 1 5 4 2 (a) Find the x- and y-intercepts of the graph of f. The x intercepts (zeros) are (-1, 0), (5,0), and (-4,0) To find the y - intercept, evaluate f(0) So, the y-intercept is (0,-20)

For the polynomial ( ) f x = + - 1 5 4 2 b.) Determine whether the graph crosses or touches the x-axis at each x-intercept. x = -4 is a zero of multiplicity 1 (crosses the x-axis) x = -1 is a zero of multiplicity 2 (touches the x-axis) x = 5 is a zero of multiplicity 1 (crosses the x-axis) c.) Find the power function that the graph of f resembles for large values of x.

d.) Determine the maximum number of turning points on the graph of f. For the polynomial ( ) f x = + - 1 5 4 2 d.) Determine the maximum number of turning points on the graph of f. At most 3 turning points. e.) Use the x-intercepts and test numbers to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph is below the x-axis. On the interval Test number: x = -5 f (-5) = 160 Graph of f: Above x-axis Point on graph: (-5, 160)

For the polynomial ( ) f x = + - 1 5 4 2 On the interval Test number: x = -2 f (-2) = -14 Graph of f: Below x-axis Point on graph: (-2, -14) On the interval Test number: x = 0 f (0) = -20 Graph of f: Below x-axis Point on graph: (0, -20)

For the polynomial ( ) f x = + - 1 5 4 2 On the interval Test number: x = 6 f (6) = 490 Graph of f: Above x-axis Point on graph: (6, 490) f.) Put all the information together, and connect the points with a smooth, continuous curve to obtain the graph of f.

(6, 490) (-1, 0) (-5, 160) (0, -20) (5, 0) (-4, 0) (-2, -14)