answers to algebra 2 unit 3 practice
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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