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Warm-Up 5/3/13 Homework: Review 3.1-3.2 (due Mon) HW 3.3A #1-15 odds (due Tues) Find the zeros and tell if the graph touches or crosses the x-axis. Tell the end behaviors. 1)f(x) = 2(x-5)(x+4) 2 2)f(x) = 5x 3 +7x 2 -x +9 Use your calculator! Answers: 1. x=5 mult=1, crosses x-axis, x=-4 mult= 2 touches x-axis and turns around. falls to the left and rises to the right. (odd exp) 2. Falls to the left and rises to the right.

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Homework Answers: Pg. 323 (2-32 even) 2. Polynomial function, degree 4 4. Polynomial function, degree 7 6. Not a Polynomial function 8. Not a Polynomial function 10. Polynomial function, degree 2 12. Not a Polynomial function because graph is not smooth. 14. Polynomial function 16. C 18. d 20. f(x) = 11x 3 -6x 2 + x + 3; graph falls left and rises to the right. (odd) 22. f(x) = 11x 4 -6x 2 + x + 3; graph rises to the left and to the right. (even) 24. f(x) = -11x 4 -6x 2 + x + 3; graph falls left and to the right. (Even and neg) 26. f(x) = 3(x+5)(x+2) 2 x = -5 has multiplicity 1; The graph crosses the x-axis. x = -2 has multiplicity of 2; The graph touches the x-axis and turns around.

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Homework Answers cont: Pg. 323 (2-32 even) 28. f(x) = -3(x + ½)(x-4) 3 ; x = -1/2 has multiplicity 1; Graph crosses the x-axis. x = 4 has multiplicity 3; graph crosses the x-axis 30. f(x) = x 3 +4x 2 +4x; x(x+2) 2 ; x = 0 has multiplicity 1; Graph crosses the x-axis. x = - 2 has multiplicity 2; graph touches the x-axis and turns around. 32. f(x) = x 3 +5x 2 -9x-45; (x-3)(x+3)(x+5); x = 3,-3,-5 have multiplicity 1; Graph crosses the x-axis.

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Announcements: Quiz on Monday, May 6th Lesson 3.3 Objective: Be able to use long and synthetic division to divide polynomials, evaluate a polynomial by using the Remainder Theorem, and solve a polynomial Equation by using the Factor Theorem.

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Lesson 3.3 Dividing Polynomials: Remainder and Factor Theorems Long Division of Polynomials and the Division Algorithm EXAMPLE 1: Divide: x X 2 + 9x --- 5x + 45 + 5 5x + 45 --- 0 = x + 5

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Example 2: Divide Answer: Remainder: 7x-5/3x 2 – 2x

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You try: Use long division to divide. 2x 2 + 7x + 14 + 21x-10 x 2 -2x Answer:

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Synthetic Division: Shortcut to dividing polynomials of c =1. Example 3:

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Synthetic Division cont:

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You try: Use synthetic division to divide. Answer: 5x 2 - 10x + 26 - 44 x + 2

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The Factor Theorem: Let f(x) be a polynomial. a. If f(c) =0, then x-c is a factor of f(x). b. If x-c is a factor of f(x), then f(c) = 0. Example 4: Solve the equation 2x 3 – 3x 2 -11x + 6 = 0 given that 3 is a zero of f(x) = 2x 3 -3x 2 -11x + 6. Step 1: Use synthetic division and solve. 3| 2 -3 -11 6 6 9 -6 2 3 -2 0 The remainder is 0, which means that x-3 is a factor of 2x 3 -3x 2 -11x + 6

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What are the factors of 2x 3 -3x 2 -11x + 6? Using synthetic division, we found the factors to be (x-3)(2x 2 + 3x-2) =0. Finish factoring: (x-3) (2x-1) (x+2) =0 X=3, x=1/2, x=-2 Find the x- intercepts. The solution set is {-2, 1/2, 3}

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You try: f(x)= 15x 3 +14x 2 -3x -2=0, given that -1 is a zero of f(x), find all factors. Answers: -1| 15 14 -3 -2 -15 1 2 15 -1 -2 0 Factors: (x+1) (15x 2 –x -2)=0 (x+1) (5x-2)(3x+1)=0 x = -1, x= 2/5, x = -1/3 Solution Set {-1, -1/3, 2/5}

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Based on the Factor Theorem, the following statements are useful in solving polynomial equations.

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Summary: Explain how the Factor Theorem can be used to determine if x-1 is a factor of x 3 -2x 2 – 11x + 12.

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