A1© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

Answers to Algebra 2 Unit 3 PracticeLeSSon 14-1 1. a. 0 , w , 40; (0, 40); {w | w ∈ , 0 , w , 40}

b.

w

V

40,000

30,000

20,000

10,000

20 40

Width

60

Vol

ume

c. (27, 37,926)

d. 27 units

2 a. h , 30 2 2r

b. V 5 pr2(30 2 2r)

c. 10 in.

d. 3,141.593 in.2

3. In real-life situations, the width must be greater than zero and the volume must be greater than zero.

4. C

5. Sample explanation: The formula for volume of a prism or a cylinder is the area of the base, which is a second-degree function, times the height.

LeSSon 14-2 6. Yes; f (x) 5 7x3 2 8x2 12x 2 5; degree 3; leading

coefficient is 7.

7. D

8. a. The leading coefficient is negative.

b. As x → 2∞, y → ∞ and as x → ∞, y → 2∞.

c. x-intercept: (2, 0), y-intercept: (0, 4)

d. relative min: (0, 4), relative max: (1, 5)

9. a.

x

y

25

215

210

25

5

5 10

b. x-intercepts: (20.55, 0), (0.61, 0), and (5.94, 0); y-intercept: (0, 1)

c. relative maximum: (0, 1), relative minimum: (4, 215)

10. Check student’s graph. The minimum number of times a cubic third-degree function can cross the x-axis is one. The maximum number of times a cubic third-degree function can cross the x-axis is three.

LeSSon 14-3 11. a. even

b. odd

12. The function is odd because it is symmetric about the origin.

13. Sample answer: f (x) 5 2x4 1 x2 13x; the function has an even degree (4) but not all of the exponents are even. The third term, 3x, has an odd exponent, 3x1.

14. Odd; an odd function must be an odd-degree polynomial. The end behavior of the graph of an odd function decreases on the left side of the graph and increases endlessly on the right side of the graph.

15. C

A2© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

LeSSon 15-1 16. a. 226 cakes

b. Hannah’s Cakes sold 20 more cakes in January.

17. a. R(t) 5 72t3 2 1240t2 1 5600t 1 500

b.

t

y

P(t)

C(t)

R(t)

5000

10,000

5 10

The relative maximum for C(t) is in February, while the relative maximums for P(t) and for R(t) are in March. The relative minimum for all three functions is in August.

c. The value of R(t) is equal to the sum of C(t) and P(t) for every value of t.

d. Subtract the value of P(t) from the value of R(t) to find the value of C(t), since R(t) represents the total revenue from both cakes and pastries.

18. B

19. a.

t

y

20,000

10,000

30,000

105

C(t)

R(t)

The domain is from January through December, 0 # t # 12.

b. Mari experiences a loss in January. The revenue for the bags steadily increases in December as does the cost to run the business. However, Mari does not experience a loss until the business cycle begins again in January of the next year.

c. The break-even point occurs in the beginning of February with revenue of about $8000.

d. P(t) 5 214t3 2 122t2 1 4410t 2 8800

t

y

20,000

10,000

30,000

105

P(t)

C(t)

R(t)

e. 0; (2, 0)

f. mid July, $11,533; the relative maximum

g. The profit is negative in January when the cost is greater than the revenue.

20. Answers will vary but should include reducing expenses to increase profit. Check students’ responses.

LeSSon 15-2 21. a. 11x4 2 2x3 1 10x2 2 8x 2 4

b. 2x5 2 3x4 2 5x3 1 7x2 1 7x 1 20

c. 21x4 1 8x2 2 9x 2 15

d. 22x3 1 16x2 1 10x 1 7

e. 213x3 2 5x2 1 2x 2 24

22. a. x3 1 6x2 2 29x 1 6

b. 3x4 22x3 2 15x2 1 49x 2 26

c. 25x4 2 20x3 2 10x2 1 28x 2 63

d. 14x4 2 53x3 2 65x2 2 91x 1 44

23. D

A3© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

24. a polynomial

25. a. V(x) 5 x(12 2 2x)(16 2 2x)

b. 4x3 2 56x2 1 192x

LeSSon 15-3 26. a. x 1 5

b. x2 1 1

c. x2 2 7 1 x30

31 d. 5x

27. a. x3 1 2x2 2 5x 1 1

b. x2 1 9

c. x2 2 3x 1 2

d. 2x4 1 5x2 2 7

28. B

29. a. Sample answer: I would use long division since one factor is in the form of x 1 k.

b. x2 1 9x 2 5

30. Step 1: Set up the division problem using only coefficients of the dividend and only the constant for the divisor. Include zero coefficients for any missing terms.

Step 2: Bring down the leading coefficient.

Step 3: Multiply the leading coefficient by the divisor, write the product under the second coefficient, and add.

Step 4: Repeat this process until there are no more coefficients.

Step 5: The numbers in the bottom row become the coefficients of the quotient. The number in the last column is the remainder. Write it over the divisor.

LeSSon 16-1 31. a. 495

b. 15

c. 15,504

d. 18,564

32. 3003

33. B

34. (a4 1 4a3b 1 6a2b2 1 4ab3 1 b4)

35. The sum of the exponents of the variables in each term plus 1 equals the number of terms. There are 5 1 1 or 6 terms.

LeSSon 16-2 36. a. 375

b. 15

c. 272,160

d. 257,344

37. a. 2945x4

b. 4,369,820x12

c. 275,000,000x3

d. 594,542,592x3

38. (x 2 3)5 5 x5 2 15x4 1 90x3 2 270x2 1 405x 2 243

39. B

40. 1280x3

LeSSon 17-1 41. a. (x 2 3)(x 2 4)

b. (3x 2 5)(x 1 2)

c. (3x2 1 5)(x 2 1)(x 1 1)

d. (x 1 9)(x 2 4)

42. a. (x 2 3)(2x2 1 5)

b. (x3 1 2)(3x 2 1)

c. (x 1 5)(x 2 3)(x 1 3)

d. (x 2 5)(x2 2 3)

43. a. (x 1 5)(x2 2 5x 1 25)

b. (x 2 2)(x2 1 2x 1 4)

c. (2x 1 6)(4x2 2 12x 1 36)

d. (4x 2 3)(16x2 1 12x 1 36)

44. a. (5x2 2 13)(5x2 1 13)

b. (x2 1 3)(x2 1 3)

c. (x3 2 5)(x3 2 5)

d. (2x5 2 9)(2x5 1 9)

45. D

A4© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

LeSSon 17-2 46. a. x 5 0, x 5 62i; 3 zeros

b. x 5 63, x 5 63i; 4 zeros

c. x 5 0 (double), x 5 4 (double); 4 zeros

d. x 5 1, x 5 612

; 3 zeros

47. . . . a polynomial f (x) of degree n $ 0 has exactly n linear factors, counting factors used more than once.

48. a. x3 2 x2 2 2x

b. x4 1 x3 2 7x2 2 x 1 6

c. x3 2 5x2 1 3x 1 9

d. x4 2 6x3 2 11x2 1 60x 1 100

49. C

50. a. x3 2 2x2 1 9x 2 18

b. x4 2 2x3 1 11x2 2 2x 1 10

c. x5 2 7x4 1 17x3 2 15x2

d. x4 2 116

LeSSon 18-1 51. a. I

b. V

c. III

d. II

e. IV

52. D

53. a.

x

y

25210

21000

2500

500

1000

5 10

x-intercepts: (27.6, 0), (0, 0), (0.02, 0), and (6.58, 0); y-intercept: (0, 0); relative minimums: (25.39, 2771) and (4.63, 2508); relative maximum: (0.01, 0.005)

b.

x

y

25210215220225

24000

22000

2000

4000

5 10 15 20 25

x-intercept: (219.13, 0); y-intercept: (0, 248); relative minimum: (212.67, 21064); relative maximum: (0, 248)

A5© 2015 College Board. All rights reserved. SpringBoard Algebra 2, Unit 3 Practice

54.

x

y

2326

260

230

30

60

3 6

55. a.

x

y

2224

240

220

20

40

2 4

b. x-intercepts: (22.026, 0), (0.106, 0), (1.28, 0) and (3.64, 0); y-intercept: (0, 1)

c. relative extrema: (21.27, 13.02), (2.822, 16.6), and (0.698, 22.76)

LeSSon 18-2

56. a. 61, 65, 613

, 653

b. 61, 62, 63, 612

, 632

, 626

57. a. 4

b. 4, 2, or 0

c. p(2x) 5 x5 1 6x4 1 3x3 2 5x2 2 3x 2 7

d. exactly 1.

58. a. Since there are two sign changes in h(x) and one sign change in h(2x), there are two or zero real positive roots and one negative real root.

b. Since there are three sign changes in j(x) and one sign change in j(2x), there are three or zero real positive roots and one negative real root.

59. a. zeros: (22, 0), (1, 0), and (3, 0)

b. y-intercept: 6

c. relative maximum: (21, 8)

relative minimum: (2, 24)

d.

x

y

25210

210

25

5

10

5 10

60. C

LeSSon 18-3 61. A; the factored form of h(x) is (x 2 2)(x 2 4),

so the zeros are x 5 2 and x 5 4.

62. x # 22 and 1 # x # 3

63. p(x); sample explanation: If you sketch each function, you will find that the range of m(x) is [0, ∞) while the range of p(x) is [225, ∞), so p(x) has the greater range.

64. a. 25 , x , 1 and x . 8

b. x # 22 and 3 # x # 4

65. B