# Section 3.2 Polynomial Functions of Higher Degree

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Section 3.2 Polynomial Functions of Higher Degree

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What you should learnHow to use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functionsHow to use zeros of polynomial functions and the lead coefficient and end behavior to construct and graph polynomial functions.

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Polynomial FunctionA polynomial function is a function of the formwhere 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 FunctionLeading Coefficient Degree -2 5 1 3 14 0

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Graphs of Polynomial Functions Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions. continuousnot continuouscontinuoussmoothnot smoothpolynomialnot polynomialnot polynomial

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Power FunctionsPolynomial functions of the form f (x) = x n, n 1 are called power functions. 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. f (x) = x2 f (x) = x5f (x) = x4 f (x) = x3

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Example: Graph of f(x) = (x + 2)4 Example: Sketch the graph of f (x) = (x + 2)4 . This is a shift of the graph of y = x 4 two units to the left. This, in turn, is the reflection of the graph of y = x 4 in the x-axis. y = x4y = x4f (x) = (x + 2)4

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Leading Coefficient TestLeading Coefficient TestAs 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. n odd n even

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Example: Right-Hand and Left-Hand BehaviorExample: Describe the right-hand and left-hand behavior for the graph of f(x) = 2x3 + 5x2 x + 1.

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Zeros of a FunctionA real number a is a zero of a function y = f (x) if and only if f (a) = 0. 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.

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Example: Real ZerosExample: Find all the real zeros and turning points of the graph 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. These correspond to the x-intercepts (1, 0), (0, 0) and (2, 0). The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Repeated ZerosExample: Determine the multiplicity of the zeros of f (x) = (x 2)3(x +1)4. Zero Multiplicity Behavior 2 1 3 4 odd even crosses x-axis at (2, 0) touches x-axis at (1, 0)

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Example: Graph off(x) = 4x2 x4Example: Sketch the graph of f (x) = 4x2 x4. 1. Write the polynomial function in standard form: f (x) = x4 + 4x2 The leading coefficient is negative and the degree is even. 2. Find the zeros of the polynomial by factoring. f (x) = x4 + 4x2 = x2(x2 4) = x2(x + 2)(x 2) Zeros: x = 2, 2 multiplicity 1 x = 0 multiplicity 2 x-intercepts: (2, 0), (2, 0) crosses through (0, 0) touches only Example continued

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Copyright by Houghton Mifflin Company, Inc. All rights reserved.*Example ContinuedExample continued: Sketch the graph of f (x) = 4x2 x4. 3. Since f (x) = 4(x)2 (x)4 = 4x2 x4 = f (x), the graph is symmetrical about the y-axis.4. Plot additional points and their reflections in the y-axis: (1.5, 3.9) and (1.5, 3.9 ), ( 0.5, 0.94 ) and (0.5, 0.94) 5. Draw the graph.

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1-8 A Matchingf(x) = -2x + 3f(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1f(x) = - x4 + 3x2f(x) = -1/3 x3 + x2 4/3f(x) = x4 + 2x3f(x) = 1/5 x5 -2x3 + 9/5 xA

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1-8 B Matchingf(x) = -2x + 3f(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1A f(x) = -1/3 x3 + x2 4/3f(x) = x4 + 2x3f(x) = 1/5 x5 -2x3 + 9/5 xB

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1-8 C Matchingf(x) = -2x + 3f(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1 Af(x) = -1/3 x3 + x2 4/3f(x) = x4 + 2x3B C

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1-8 D MatchingC f(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1A f(x) = -1/3 x3 + x2 4/3f(x) = x4 + 2x3B D

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1-8 E Matching Cf(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1A f(x) = -1/3 x3 + x2 4/3DBE

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1-8 F Matching Cf(x) = x2 4xf(x) = -2x2 5xf(x) = 2x3 3x + 1A E DBF

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1-8 G Matching Cf(x) = x2 4xf(x) = -2x2 5xF A E DBG

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1-8 H Matching CG f(x) = -2x2 5xF A E DBH

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