Factor P(x) into linear factors given that k is a zero of P.

51) P(x) = 5x^{3} – 34x^{2} + 64x – 32; k = 2

A) P(x) = (x – 2)(x + 4)(5x – 4)

B) P(x) = (x – 2)(x + 4)(5x + 4)

C) P(x) = (x + 2)(x – 4)(5x – 4)

D) P(x) = (x – 2)(x – 4)(5x – 4)

52) P(x) = 3x^{3} + 34x^{2} + 115x + 100; k = -5

A) P(x) = (x – 5)(x + 5)(3x – 4)

B) P(x) = (x + 5)(x + 5)(3x + 4)

C) P(x) = (x – 5)(x – 5)(3x – 4)

D) P(x) = (x – 5)(x + 5)(3x + 4)

53) P(x) = 4x^{3} – 11x^{2} – 4x + 20; k = – (5/4)

A) P(x) = (x – 2)(x – 2)(4x – 5)

B) P(x) = (x + 2)(x + 2)(4x + 5)

C) P(x) = (x + 2)(x + 2)(4x – 5)

D) P(x) = (x – 2)(x – 2)(4x + 5)

54) P(x) = x^{3} – 4x^{2} – 49x + 196 ; k = 7

A) P(x) = x(x – 7)(x – 4)

B) P(x) = (x – 7)(x + 4)(x + 7)

C) P(x) = (x – 7)(x – 4)(x + 7)

D) P(x) = (x – 7)(x – 4)(x + 4)

55) P(x) = x^{3} – 48x – 128; k = -4

A) P(x) = (x – 4)^{2}(x – 8)

B) P(x) = (x + 8)(x + 4)(x – 4)

C) P(x) = (x + 4)^{2}(x + 8)

D) P(x) = (x + 4)^{2}(x – 8)

56) P(x) = 10x^{3} + 53x^{2} + 49x – 12; k = (1/5)

A) P(x) = (5x – 1)(2x – 3)(x – 4)

B) P(x) = (5x + 1)(2x + 1)(x – 12)

C) P(x) = (5x – 1)(2x + 3)(x + 4)

D) P(x) = (5x + 1)(2x + 3)(x – 4)

Divide.

57) (-20x^{3} – 17x^{2} + 17x + 5/4x + 1)

A) -5x^{2} – 3x + 5

B) x^{2} + 3x – 5

C) -5x^{2} + 5

D) x^{2} – 3x + 5

58) (x^{2} + 7x + 1/x + 2)

A) x + 6

B) (x + 5/x + 2)

C) x + 5 + (9/x + 2)

D) x + 5 + (-9/x + 2)

59) (4y^{4} + 6y^{3} + 3y – 1/2y^{2} + 1)

A) 2y^{2} + 3y – 1

B) 2y^{2} – 3y + 1

C) 2y^{2} + 3y

D) 2y^{2} – 1

60) (x^{4} + 4x^{2} + 5 /x^{2} + 1)

A) x^{2} + 3

B) x^{2} + 3 + (2/x^{2} + 1)

C) x^{2} + 3x + (1/2)

D) x^{2} + 3x + 2

61) (16y^{4} + 20y^{3} + 5y – 1/4y^{2} + 1)

A) 4y^{2} + 5y – 1

B) 4y^{2} – 5y + 1

C) 4y^{2} + 5y

D) 4y^{2} – 1

62) (-3x^{5} – x^{3} + 2x^{2} + 18x – 4/x^{2} – 2)

A) -3x^{3} – 7x + 2 + (4x/x^{2} – 2)

B) -3x^{3} – 7x + 2 + (4x – 8/x^{2} – 2)

C) -3x^{3} – 7x + 2 – (4x/x^{2} – 2)

D) -3x^{3} – 7x – 2 + (4x/x^{2} – 2)

63) (x^{4} + 4x^{3} + 5x^{2} + 4x + 3/x^{2} + 3x + 1)

A) x^{2} + x + 1

B) x^{2} + 6x + 24

C) x^{2} + 6x + 24 + (79x + 23/x^{2} + 3x + 1)

D) x^{2} + x + 1 + (2/x^{2} + 3x + 1)

64) (8x^{4} + 30x^{3} – 8x^{2} – 60x + 18/2x^{2} + 6x – 2)

A) 4x^{2} – 3x – 9

B) 4x^{2} + 4x – 9

C) 4x^{2} + 3x + 9

D) 4x^{2} + 3x – 9

65) (x^{2} – (1/5)x – 2) ÷ (5x – 1)

A) (1/5)x – 1

B) (1/5)x – 2

C) (1/5)x + (2/5x – 1)

D) (1/5)x – (2/5x – 1)

66) (-2x^{3} + 1x^{2} + (4/3)x) ÷ (6x^{2} + 3x + 1)

A) – (1/3)x + (1/3) + ((1/3)x – (2/3)/6x^{2} + 3x + 1)

B) – (1/3)x – (1/3) + ((2/3)x + (1/3)/6x^{2} + 3x + 1)

C) – (1/3)x + (1/3) + ((1/3)x + (2/3)/6x^{2} + 3x + 1)

D) – (1/3)x + (1/3) + ((2/3)x – (1/3)/6x^{2} + 3x + 1)

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Provide an appropriate response.

67) Suppose that a polynomial function P is defined in such a way that P(-5) = 7 and P(-5.3) = -1. What conclusion does the intermediate value theorem allow you to make?

68) Suppose that a polynomial function P is defined in such a way that P(-6) = 9 and P(2) = 5. Can we be certain that there is no zero between -6 and 2? Explain using a graph.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

69) Consider the graph of the polynomial function P(x) = x^{3} + 3x^{2} + -1x – 3.

What are the linear factors of P(x)?

A) x – 3, x – 1, x – 1

B) x + 1, x + 1, x + 4

C) x + 3, x – 1, x + 4

D) x + 3, x – 1, x + 1

70) Consider the graph of the polynomial function P(x) = x^{3} + 2x^{2} – 19x – 20.

What are the solutions of the equation P(x) = 0?

A) -4, 1, 5

B) -5, -1, 4

C) 1, 4

D) -5, -1

71) Consider the graph of the polynomial function P(x) = x^{3} + 5x^{2} – 12x – 36.

What are the zeros of the function P?

A) 2, 3, 6

B) -3, 2

C) -6, -3

D) -6, -2, 3

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

72) Consider the graph of the polynomial function P(x) = x^{3} + 4x^{2} – 11x – 30.

If P(x) is divided by x – 2, what is the remainder? What is P(2)?

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

73) Consider the graph of the polynomial function P(x) = x^{3} + 3x^{2} – 18x – 40.

Give the solution set of P(x) > 0 using interval notation.

A) (-5, -2)∪(4, ∞)

B) (-∞, -5)∪(-2, 4)

C) (-∞, -2)∪(4, 5)

D) (-4, -2)∪(5, ∞)

74) Consider the graph of the polynomial function P(x) = x^{3} + 4x^{2} – 11x – 30.

Give the solution set of P(x) < 0 using interval notation.

A) (-∞, 5)∪(2, 3)

B) (-5, -2)∪(3, ∞)

C) (-∞, -5)∪(-2, 3)

D) (-∞, -2)∪(3, 5)

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