Find a polynomial function ƒ1x2 of least degree with real coefficients having zeros as given. 23. -1, 4, 7 24. 8, 2, 3 25. 23, -23, 2, 3 26. -2 + 25, -2 - 25, -2, 1 27. 2, 4, -i 28. 0, 5, 1 + 2i Find all rational zeros of each function. 29. ƒ1x2 = 2x3 - 9x2 - 6x + 5 30. ƒ1x2 = 8x4 - 14x3 - 29x2 - 4x + 3 Show that each polynomial function has a real zero as described in parts (a) and (b). In Exercises 31 and 32, also work part (c). 31. ƒ1x2 = 3x3 - 8x2 + x + 2 (a) between -1 and 0 (b) between 2 and 3 (c) Find the zero in part (b) to three decimal places. 32. ƒ1x2 = 4x3 - 37x2 + 50x + 60 (a) between 2 and 3 (b) between 7 and 8 (c) Find the zero in part (b) to three decimal places. 33. ƒ1x2 = 6x4 + 13x3 - 11x2 - 3x + 5 (a) no zero greater than 1 (b) no zero less than -3 Solve each problem. 34. Use Descartes’ rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of ƒ1x2 = x3 + 3x2 - 4x - 2. 35. Is x + 1 a factor of ƒ1x2 = x3 + 2x2 + 3x + 2? 36. Find a polynomial function ƒ with real coefficients of degree 4 with 3, 1, and -1 + 3i as zeros, and ƒ122 = -36. 37. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ122 = 16. 38. Find all zeros of ƒ1x2 = x4 - 3x3 - 8x2 + 22x - 24, given that 1 + i is a zero. 39. Find all zeros of ƒ1x2 = 2x4 - x3 + 7x2 - 4x - 4, given that 1 and -2i are zeros. 40. Find a value of k such that x - 4 is a factor of ƒ1x2 = x3 - 2x2 + kx + 4. 41. Find a value of k such that when the polynomial x3 - 3x2 + kx - 4 is divided by x - 2, the remainder is 5. 42. Give the maximum number of turning points of the graph of each function. (a) ƒ1x2 = x5 - 9x2 (b) ƒ1x2 = 4x3 - 6x2 + 2 43. Concept Check Give an example of a cubic polynomial function having exactly one real zero, and then sketch its graph. 44. Concept Check Give an example of a fourth-degree polynomial function having exactly two distinct real zeros, and then sketch its graph. 22. Concept Check Suppose the polynomial function ƒ has a zero at x = -3. Which of the following statements must be true? A. 13, 02 is an x-intercept of the graph of ƒ. B. 10, 32 is a y-intercept of the graph of ƒ. C. x - 3 is a factor of ƒ1x2. D. ƒ1-32 = 0 427 CHAPTER 3 Review Exercises
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