Unit 5 Review: Polynomials PDF

Unit 5 Review: Polynomials Write the equation of the polynomial function in intercept form and state the end behavior. 1. 2. 3. 4. Given the polynomial to the right, assume all real roots. Write a...

Unit 5 Review: Polynomials Write the equation of the polynomial function in intercept form and state the end behavior. 1. 2. 3. 4. Given the polynomial to the right, assume all real roots. Write a (3.8,4) possible function in intercept form with a=1 or a=-1. a) Intervals were function is Increasing b) Intervals were function is decreasing c) Intervals were function is positive d) Intervals were function is negative e) number of turning points f) number of local minimums g) domain and range h) number of local maximums i) is there a global max or min? Why? j) State the zeros and multiplicity (either 1, 2, or 3) k) what do you know about the degree? l) what do you know about the leading coefficient? Give the degree, leading coefficient, and end behavior 5. f(x) = 8x3 – 4x2 + 3x – 2 6. f(x) = -2(x-7)5(x+3)(x-4)2 7. f(x) = 4x2 – 5x3 - 3x7 – 2x + 1 Sketch the graph of the polynomial functions. 8. f(x) = (x + 1)3(x – 2)(x – 4) 9. f(x) = -x3(x + 5)2(x + 3) 10. f(x) = -x3 + 3x2 + 4x - 12 Graph the cubic function. List the transformations from the parent function and the point of inflection 11. f(x) = -2(x-3)3 – 1 12. f(x) = (½x-2)3 - 3 Use a calculator to graph the function. Identify the x-intercepts, where local/global minimums and maximums occur and determine the increasing and decreasing intervals. 13. f(x) = x ! + 2x " − 17x − 4 14. f(x) = x # − 5x $ + 2x " + x − 3 15. Find the zeros for f(x) = x7 – 29x5 + 100x3 and state the multiplicities. Will the graph have a global min or max? Factor completely 16. x4y – 125xy 17. 32 + 4x3y3 18. 54x3 – 16y6 19. x5 - 9x3 – x2 + 9 20. 4x3 + 12x2 – 9x - 27 Perform the indicated operation and simplify 21. (6x + 14) + (9x + 5) 22. (19x2 + 9x + 16) - (5x2 + 12x + 7) 23. (x + 5)(x2 – 6x + 3) 24. (2x+3)2 – (2x-1)(3x2-2x+1) 25. ( 3x 3 - 4 x2 + 2 x -10) ÷ ( x2 - 2 x + 2) 26. (3x 6 + 7 x4 + 4 x3 + 5) ÷ ( x4 + 3x3 + 4) 27. (2x2 + 5x – 3) ÷ (x + 3) 28. (3x3 – 13x2 – 20) ÷ (x – 5) !" ! #$" " %&" # %' 29. (8x − 6x − 2x + 2) ÷ (2x − 1) 4 2 2 30. " # %'"#( *Calculator Question* 31. A rectangular piece of cardboard with 12 cm length and 8 cm width will be cut by x cm in each corner to create a box. Write a polynomial in intercept form to represent the volume of the box *Calculator Question* 32. A company is designing a rectangular shipping box that must have a volume of 24 in3. The company has created a box with a width that is 2 inches less than its length. The height is 5 inches greater than the length. Write an equation that could be used to solve for the dimensions. 33. Find the possible rational zeros for f(x) = 10x5 – 7x4 + 3x3 - 12 34. Solve the equation by finding all roots x # − 2x $ − 14x " − 2x − 15 = 0. Are all of the roots x-intercepts? 35. State the possible roots. 10x # + 7x " − 3x − 18 = 0 36. Find all roots for f(x) = x4 -3x3 - 2x2 + 6x + 4. 37. Solve. Find all solutions: 0 = 4x3 + 16x2 - 22x - 10 38. Solve. Find all solutions: x3 - 2x2 + 5x – 10 = 0 39. Determine if x − 4 is a factor of f(x) = 2x $ − 7x " − 10x + 24 Write a function in standard form with least degree and leading coefficient of 1 with the given zeros: 40. −2, 1, 3 − √2 41. 3 (multiplicity of 2), 5 + i 42. 2 (multiplicity of 3), 4 + √5 43. -1, 4, 3 – 2i 44. Let f(x) = 2x7 – 8x5 + 7x2 + x – 10. Using what we’ve learned in this unit, describe as much as you can about the graph and its roots (you don’t need to solve for the roots) 45. What is the degree of a polynomial equation that has 3i as a root with a multiplicity of 3, and 2 − √3 as a root with multiplicity of 2 as the only roots? Briefly explain (don’t actually try and write the polynomial, unless you want to for fun…). 46. Is it possible to have a cubic function with two of the roots being 2i and √3 ? If not why?