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Warm-Up 5/3/13

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

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Warm-Up 5/3/13

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  1. Warm-Up 5/3/13 • Find the zeros and tell if the graph touches or crosses the x-axis. Tell the end behaviors. • f(x) = 2(x-5)(x+4)2 • f(x) = 5x3 +7x2 -x +9 Use your calculator! Homework: Review 3.1-3.2 (due Mon) HW 3.3A #1-15 odds (due Tues) 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.

  2. Homework Answers: Pg. 323 (2-32 even) 2. Polynomial function, degree 4 4. Polynomial function, degree 7 • Not a Polynomial function • Not a Polynomial function 10. Polynomial function, degree 2 12. Not a Polynomial function because graph is not smooth. • Polynomial function • C 18. d • f(x) = 11x3 -6x2 + x + 3; graph falls left and rises to the right. (odd) • f(x) = 11x4 -6x2 + x + 3; graph rises to the left and to the right. (even) • f(x) = -11x4 -6x2 + x + 3; graph falls left and to the right. (Even and neg) • 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.

  3. 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) = x3+4x2+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) = x3+5x2-9x-45; (x-3)(x+3)(x+5); x = 3,-3,-5 have multiplicity 1; Graph crosses the x-axis.

  4. 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.

  5. Lesson 3.3 Dividing Polynomials: Remainder and Factor Theorems • Long Division of Polynomials and the Division Algorithm EXAMPLE 1: Divide: = x + 5 x + 5 --- --- X2 + 9x 5x + 45 --- --- 5x + 45 0

  6. Example 2: Divide Answer: Remainder: 7x-5/3x2 – 2x

  7. You try: Use long division to divide. Answer: 2x2 + 7x + 14 + 21x-10 x2-2x

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

  9. Synthetic Division cont:

  10. You try: Use synthetic division to divide. Answer: 5x2 - 10x + 26 - 44 x + 2

  11. 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 2x3 – 3x2-11x + 6 = 0 given that 3 is a zero of f(x) = 2x3 -3x2-11x + 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 2x3 -3x2-11x + 6

  12. What are the factors of 2x3 -3x2-11x + 6? • Using synthetic division, we found the factors to be (x-3)(2x2+ 3x-2) =0. • Finish factoring: (x-3) (2x-1) (x+2) =0 • Find the x- intercepts. X=3, x=1/2, x=-2 • The solution set is {-2, 1/2, 3}

  13. You try: f(x)= 15x3+14x2 -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) (15x2 –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}

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

  15. Summary: • Explain how the Factor Theorem can be used to determine if x-1 is a factor of x3 -2x2 – 11x + 12.

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