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Presentation transcript:

Slide Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc. Chapter 4 Polynomial and Rational Functions

Slide Copyright © 2009 Pearson Education, Inc. CHAPTER 3: Polynomial and Rational Functions 4.1Polynomial Functions and Models 4.2Graphing Polynomial Functions 4.3Polynomial Division; The Remainder and Factor Theorems 4.4Theorems about Zeros of Polynomial Functions 4.5Rational Functions 4.6Polynomial and Rational Inequalities

Slide Copyright © 2009 Pearson Education, Inc. Classify the polynomial P(x) = 5 + 2x 2 + 6x 4 a. quadratic c. linear b. quartic d. cubic

Slide Copyright © 2009 Pearson Education, Inc. Classify the polynomial P(x) = 5 + 2x 2 + 6x 4 a. quadratic c. linear b. quartic d. cubic

Slide Copyright © 2009 Pearson Education, Inc. Determine the leading coefficient of the polynomial P(x) = 8x – 9x – x 3. a. 8 c.  1 b. 3 d. 5

Slide Copyright © 2009 Pearson Education, Inc. Determine the leading coefficient of the polynomial P(x) = 8x – 9x – x 3. a. 8 c.  1 b. 3 d. 5

Slide Copyright © 2009 Pearson Education, Inc. Determine the degree of the polynomial function P(x) = 5x 3 – 6x 2 + 2x + 6. a. 3 c. 5 b. 4 d. 6

Slide Copyright © 2009 Pearson Education, Inc. Determine the degree of the polynomial function P(x) = 5x 3 – 6x 2 + 2x + 6. a. 3 c. 5 b. 4 d. 6

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 3 – 3x 2 – x + 3? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 3 – 3x 2 – x + 3? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Find the zeros of the polynomial function and state the multiplicity of each. f(x) = (x + 3) 2 (x + 1) a. –3, multiplicity 2,  1 multiplicity 1 c. –3, multiplicity 2,  1 multiplicity 2 b. 3, multiplicity 2,  1 multiplicity 1 d. 3, multiplicity 3,  1 multiplicity 1

Slide Copyright © 2009 Pearson Education, Inc. Find the zeros of the polynomial function and state the multiplicity of each. f(x) = (x + 3) 2 (x + 1) a. –3, multiplicity 2,  1 multiplicity 1 c. –3, multiplicity 2,  1 multiplicity 2 b. 3, multiplicity 2,  1 multiplicity 1 d. 3, multiplicity 3,  1 multiplicity 1

Slide Copyright © 2009 Pearson Education, Inc. a. between  1 and 0 c. between 1 and 2 b. between 0 and 1 d. between 2 and 3 For f(x) =  2x 4 + 3, use the intermediate value theorem to determine which interval contains a zero of f.

Slide Copyright © 2009 Pearson Education, Inc. a. between  1 and 0 c. between 1 and 2 b. between 0 and 1 d. between 2 and 3 For f(x) =  2x 4 + 3, use the intermediate value theorem to determine which interval contains a zero of f.

Slide Copyright © 2009 Pearson Education, Inc. a. between  2 and  1 c. between 0 and 1 b. between  1 and 0 d. between 2 and 3 For f(x) =  2x 4 + 3x + 1, use the intermediate value theorem to determine which interval contains a zero of f.

Slide Copyright © 2009 Pearson Education, Inc. a. between  2 and  1 c. between 0 and 1 b. between  1 and 0 d. between 2 and 3 For f(x) =  2x 4 + 3x + 1, use the intermediate value theorem to determine which interval contains a zero of f.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 3 – x 2 – 4x + 4? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 3 – x 2 – 4x + 4? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 4 – x 2 – 4x + 4? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = x 4 – x 2 – 4x + 4? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = –2x 2 – 4x? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial function f(x) = –2x 2 – 4x? a.b. c.d.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Use long division to find the quotient and remainder when x 4 + 5x 2 – 3x + 2 is divided by x – 2.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Use long division to find the quotient and remainder when x 4 + 5x 2 – 3x + 2 is divided by x – 2.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Use synthetic division to find the quotient and remainder when 3x 3 – 6x is divided by x + 3.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Use synthetic division to find the quotient and remainder when 3x 3 – 6x is divided by x + 3.

Slide Copyright © 2009 Pearson Education, Inc. a.  2 c.  4 b. 8 d.  5 Use synthetic division to determine which number is a zero of P(x) = x 3 – x 2 – 22x + 40.

Slide Copyright © 2009 Pearson Education, Inc. a.  2 c.  4 b. 8 d.  5 Use synthetic division to determine which number is a zero of P(x) = x 3 – x 2 – 22x + 40.

Slide Copyright © 2009 Pearson Education, Inc. a.  1276 c. 174 b d.  1326 Use synthetic division to find P(  5) for P(x) =  2x 4 – 2x 2 + 5x – 1.

Slide Copyright © 2009 Pearson Education, Inc. a.  1276 c. 174 b d.  1326 Use synthetic division to find P(  5) for P(x) =  2x 4 – 2x 2 + 5x – 1.

Slide Copyright © 2009 Pearson Education, Inc. a.  5 c. 2 b. 1 d.  10 Use synthetic division to find determine which number is a zero of P(x) = x 3 – 6x 2 + 3x + 10.

Slide Copyright © 2009 Pearson Education, Inc. a.  5 c. 2 b. 1 d.  10 Use synthetic division to find determine which number is a zero of P(x) = x 3 – 6x 2 + 3x + 10.

Slide Copyright © 2009 Pearson Education, Inc. a. c.  3 + 2i b.  4 d.  3  2i Suppose that a polynomial function of degree 5 with rational coefficients 4, and 3 – 2i as zeros. Find one other zero.

Slide Copyright © 2009 Pearson Education, Inc. a. c.  3 + 2i b.  4 d.  3  2i Suppose that a polynomial function of degree 5 with rational coefficients 4, and 3 – 2i as zeros. Find one other zero.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Find a polynomial function of lowest degree with rational coefficients and  3 and 4i as some of its zeros.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Find a polynomial function of lowest degree with rational coefficients and  3 and 4i as some of its zeros.

Slide Copyright © 2009 Pearson Education, Inc. a. c.  5 b.  2 d. Use the rational zeros theorem to determine which number cannot be a zero of P(x) = 10x 4 + 6x 2 – 5x + 2.

Slide Copyright © 2009 Pearson Education, Inc. a. c.  5 b.  2 d. Use the rational zeros theorem to determine which number cannot be a zero of P(x) = 10x 4 + 6x 2 – 5x + 2.

Slide Copyright © 2009 Pearson Education, Inc. a. 1 c. 5, 3 or 1 b. 3 or 1 d. 2 or 0 How many negative real zeros does Descartes’ rule of signs indicate g(x) =  x 5 + 4x 4 – 2x 3 + 3x 2 – 6 has?

Slide Copyright © 2009 Pearson Education, Inc. a. 1 c. 5, 3 or 1 b. 3 or 1 d. 2 or 0 How many negative real zeros does Descartes’ rule of signs indicate g(x) =  x 5 + 4x 4 – 2x 3 + 3x 2 – 6 has?

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d.  3 Use the rational zeros theorem to determine which number cannot be a zero of P(x) = 4x 4 + 3x 2 + x – 3.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d.  3 Use the rational zeros theorem to determine which number cannot be a zero of P(x) = 4x 4 + 3x 2 + x – 3.

Slide Copyright © 2009 Pearson Education, Inc. a. y = 0 c. x =  4 b. x = 4 d. Find the vertical asymptote for

Slide Copyright © 2009 Pearson Education, Inc. a. y = 0 c. x =  4 b. x = 4 d. Find the vertical asymptote for

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial a.b. c.d. function

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial a.b. c.d. function

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial a.b. c.d. function

Slide Copyright © 2009 Pearson Education, Inc. Which graph represents the polynomial a.b. c.d. function

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve (x + 4)(x – 2)(x – 6) ≤ 0.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve (x + 4)(x – 2)(x – 6) ≤ 0.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve 3x 2 < 17x – 10.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve 3x 2 < 17x – 10.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve 3x 2 >  x + 10.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve 3x 2 >  x + 10.

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve

Slide Copyright © 2009 Pearson Education, Inc. a. c. b. d. Solve