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40. This problem will prepare you for the Multi-Step Test Prep on page 436. The total number of lights in a triangular lighting rig is related to the triangular numbers, as shown at right. The sum of the first n triangular numbers is given by the polynomial function g(n) = __16 n 3 + __12 n 2 + __13 n. a. Find the sum of the first seven triangular numbers. b. Explain why n - 7 must be a factor of n 3 +3n 2 +2n - 504. c. Factor n 3 +3n 2 +2n - 504. Triangular numbers: 1, 3, 6, 10, 15, . . . Use the Factor Theorem to verify that each linear binomial is a factor of the given polynomial. Then use synthetic division to write the polynomial as a product. 41. (x - 2); P(x) = x 4 - 2x 3 + 5x 2 - 9x - 2. 42. (x - 1); P(x) = 4x 6 - 4x 5 - 2x 4 + 3x 3 - x 2 - 7x + 7. 43. (x + 2); P(x) = 2x 5 + 4x 4 - 6x 2 - 9x + 6. 44. (x - 4); P(x) = 2x 4 - 9x 3 + 7x 2 - 14x + 8. 45. Business The profit of a small business (in thousands of dollars) since it was founded can be modeled by the polynomial f (t) = -t 4 + 44t 3 - 612t 2 + 2592t, where t represents the number of years since 1980. a. Factor f (t) completely. b. What was the company’s profit in 1985? c. Find and interpret f (15). d. What can you say about the company’s long-term prospects? a207se_ Geometry Given the volume and height, find a polynomial expression for the area of the base in terms of x for each figure. (Hint: V = Bh) 46. V(x) = 2x 3 - 17x 2 + 27x + 18 47. V(x) = x 4 - 16 ��������� ��������� 48. V(x) = 3x 6 + 3x 3 ��������� 49. Write About It Describe how synthetic division can be used to factor a polynomial. 50. Which is a factor of x 3 + 2x 2 - 9x + 30? x+2 x-3 x+5 x-6 51. P (x) is a polynomial, and P (4) = P (-2) = P (-1) = 0. Which of the following could be P (x)? x 3 + 7x 2 + 14x + 8 x 2 + 3x + 2 -x 2 + 2x + 8 x 3 - x 2 - 10x - 8 52. Short Response Factor 4p 5 - 16p 3 - 20p completely. 434 Chapter 6 Polynomial Functions a207se_c06l04_0430_0435.indd 434 10/27/05 1:42:39 PM