Name: _____________________________________________ Date: ____________________________ Period: ________ 4.1 Factoring Polynomials Completely Directions. (A) Factor each of the following polynomials completely., then (B) Find the roots 1. y = 2x3 + 3x2 – 32x + 15 2. y = x3-5x2 – 2x + 24 3. y = 9x3 – 15x2 + 7x – 1 4. y = x4 – 2x3 – 11x2 + 12x + 36 5. y = 2x4 – 9x3 + 11x2 – 4 6. y = x3 – 27 4.2 Zeroes of Polynomials Directions. Find the least degree polynomial that has the zeroes (roots) listed below. 1. x = 1, x = -2, x = 3 2. x = ½, x = 2/3, and x = -1 3. x = 2, x = 3, x = -4 (multiplicity of 2) 4. x = 3 (multiplicity of 4) 5. x = 5 (multiplicity of 2) and x = -5 (multiplicity of 2) Factor the following polynomial. T hen multiply the factors back together using the Box Method to check your answer. 6. y = 2x4 – 3x3 – 33x2 – 13x + 15 Factor the Polynomial Check the Answer 4.3 Descartes’ Rule of Signs Directions. For each polynomial unction, state (a) the possible number of positive and (b) the possible number of negative zeroes. 1. F(x) = 3x4 + 20x2 – 32 2. F(x) = 5x4 – 42x2 + 49 A) A) B) B) 3. F(x) = 4x3 – 12x2 – 5x + 1 4. F(x) = 2x4 – 3x3 + x A) A) B) B) 5. F(x) = 2x4 + 3x2 – 54 6. F(x) = x6 – 64 A) A) B) B) 7. F(x) = 9x6 – 3x5 + 33x4 – 11x3 + 18x2 – 6x 8. F(x) = 2x5 + 4x4 + 9x3 + 18x2 – 35x – 70 A) A) B) B) 9. F(x) = 16x6 – 32x4 – 25x2 + 50 10. F(x) = 27x9 – x6 – 27x3 + 1 A) A) B) B) 11. For the following polynomial, a) factor it and write it in terms of all linear factors. T hen, b) list the roots. 12. Invent a polynomial function that has 0 possible positive real zeroes and 5, 3, or 1 possible negative real zeroes. F(x) = x4 – 3x3 - x2 – 27x - 90 4.4 Sketching Graphs of Polynomial Functions Directions. For each polynomial function, find the x-intercepts, the y-intercept, and the end behavior of the graph. T hen make a rough sketch of the graph. 1. F(x) = x3 + 3x2 – 4 3. F(x) = 4x3 – 4x2 – 5x + 3 5. F(x) = x4 + 5x2 + 4 X-intercepts (touch/cross): 2. F(x) = x4 – 4x3 – x2 + 16x – 12 X-intercepts (touch/cross): Y-intercept: Y-intercept: End Behavior: End Behavior: X-intercepts (touch/cross): 4. F(x) = x3 – 27 X-intercepts (touch/cross): Y-intercept: Y-intercept: End Behavior: End Behavior: X-intercepts (touch/cross): 6. F(x) = 9x6 – 46x4 + 41x2 – 4 X-intercepts (touch/cross): Y-intercept: Y-intercept: End Behavior: End Behavior: 7. Review Question: Graph the following quadratic polynomial with 5 actual points on the graph. T hen find the x-intercepts and y-intercepts. F(x) = 2x2 – 12x + 12 X-intercepts (touch/cross): Y-intercept: End Behavior: