Download Use the Factor Theorem to verify that each linear binomial is a factor

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
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48. V(x) = 3x 6 + 3x 3
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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.
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Chapter 6 Polynomial Functions
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