4.2 Polynomial Functions of Higher Degree Objective: Identify zeros and multiplicities, determine end behavior, sketch polynomials

What are they? Polynomial functions model many real-world applications Stock market Number of new cases in the spread of an epidemic Objective: Identify real zeros and their multiplicity, end behavior of a polynomial, graph polynomials using zeros, multiplicity, and end behavior

A polynomial function is a function of the form where n is a nonnegative integer and each ai (i = 0, , n) is a real number. The polynomial function has a leading coefficient an and degree n. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial Function Leading Coefficient Degree -2 5 1 3 14 0 Polynomial Function

Graphs of Polynomial Functions Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. x y f (x) = x3 – 5x2 + 4x + 4 x y x y continuous not continuous continuous smooth not smooth polynomial not polynomial not polynomial Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions. Graphs of Polynomial Functions

If n is even, their graphs resemble the graph of f (x) = x2. Polynomial functions of the form f (x) = x n, n  1 are called power functions. f (x) = x5 x y f (x) = x4 x y f (x) = x2 f (x) = x3 If n is even, their graphs resemble the graph of f (x) = x2. If n is odd, their graphs resemble the graph of f (x) = x3. Power Functions

Graphing How do we graph polynomial functions when they become really complex like f(x) = x5 – 2x2 – x + 4 Use x-intercepts (setting f(x) = 0 and solving for x) – this is called finding zeros of a function Multiplicity of zeros End behavior

Real Zeros of Polynomial Functions X-intercepts: A real number a is a zero of a function y = f (x) if and only if f (a) = 0. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = 0. 3. x – a is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x). A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa. A polynomial function of degree n has at most n – 1 turning points and at most n zeros. Zeros of a Function

Illustration of 1-4 using f(x) = x2 - 1

Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2. Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2). The real zeros are x = –1, x = 0, and x = 2. y x f (x) = x4 – x3 – 2x2 These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0). Turning point The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible. Turning point Turning point Example: Real Zeros

You Try Find all the real zeros of f(x) = x3 + x2 – 2x

Repeated Zeros (Multiplicity of a Zero) If (x – a)n is a factor of a polynomial f, then a is called a zero of Multiplicity n of f. 1. If n is odd the graph of f (x) crosses the x-axis at (a, 0). 2. If n is even the graph of f (x) touches, but does not cross through, the x-axis at (a, 0). Ex: Find the zeros and determine their multiplicity of f (x) = (x – 2)3(x +1)4. x y Zero Multiplicity Behavior 2 3 odd crosses x-axis at (2, 0) –1 4 even touches x-axis at (–1, 0) Repeated Zeros

Examples: Find all the real zeros and state their multiplicities

Ex: Find a polynomial… Find a polynomial of minimum degree whose zeros are: -2 (multiplicity 1), 3 (multiplicity 1) and 1 (multiplicity 2)

Leading Coefficient Test As x grows positively or negatively without bound, the value f (x) of the polynomial function f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0) grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n is odd or even. x y x y an positive an negative n even n odd Leading Coefficient Test

In a nutshell…If you’re standing at the origin, degree Even Odd Leading Coefficient Positive Negative “x on the left” Starts at the top Starts at the bottom “x on the right” Ends at the top Ends at the bottom Graph

Example: Right-Hand and Left-Hand Behavior Example: Describe the right-hand and left-hand behavior for the graph of f(x) = –2x3 + 5x2 – x + 1. Negative -2 Leading Coefficient Odd 3 Degree x y f (x) = –2x3 + 5x2 – x + 1 Example: Right-Hand and Left-Hand Behavior

Bringing it all together…… For the polynomial f(x) = -(x + 1)2(x – 3) Determine the zeros, their multiplicity and whether they touch or cross the x-axis. What is the end behavior? What is the y-intercept? Sketch the graph.

For the given graph: Determine the zeros, their SMALLEST multiplicity and whether they touch or cross the x-axis Is the degree of the polynomial odd or even? What is the end behavior? What is the y-intercept? Write an equation for the polynomial function.