Answers 135. 16 + 72i
136. − 60 − 24i
5. − 2
137. − 39 − 4i
138. 79 + 16i
8. g ( x) → +∞ as x → + ∞ and g ( x) → +∞ as
139. 47 − i
140. 27 − 36i
141. − 23 + 264i
142. 80 + 18i
6. 6848
x → −∞.
9. h( x) → −∞ as x → + ∞ and h( x) → +∞ as
x → −∞.
143. a. 23 lb 10.
b. $0.45
4
c. about $0.91
11.
y
y
2 2
Chapter 4
−4
4.1 Start Thinking
−2
2
−4
x4
−5
h(x) = x 3 − 2x + 3
4 x
2
5
g(x) =
7. 15,651
x
q(x) = x 4 − 2
5
12.
f(x) = x3
k(x) = 2x 2 + 3 − x 3
13.
f(x) = x 5 − 2x 3 + 1 y
y
−5
The graph of f ( x) = x3 is a curvy line that is moving upward from left to right as x increases. The graph of g ( x) = x 4 is similar to a parabola that opens up with its vertex at the origin. Both graphs have positive y-values when x is positive. When x is negative, the y-values of f ( x) are negative, and the y-values of
g ( x) are positive; The exponent is even; yes; (0, 0)
and (1, 1)
2
−2
2
2
−2
x
2
x
14. f is increasing when x > 3. f is decreasing when
x < 3. f is positive when x < 2 and x > 4. f is negative when 2 < x < 4.
15. f is increasing when x < −1.2 and x > 1.2. f is
4.1 Warm Up 1. − 20
2. 39
3. 10
4. 40
5. − 45.8
6. 33.96
2. up
decreasing when −1.2 < x < 1.2. f is positive when − 2 < x < 0 and x > 2. f is negative when x < − 2 and 0 < x < 2. 16. The degree is even and the leading coefficient is
4.1 Cumulative Review Warm Up 1. down
4
3. down
negative. 4. up
4.1 Practice A
1. not a polynomial function
1. polynomial function;
f ( x) = 5 x3 + 4 x 2 − 3x − 7, degree is 3, cubic,
leading coefficient is 5
2. polynomial function; f ( x) = 11x 2 + 12 x −
7,
degree is 2, quadratic, leading coefficient is 11 3. polynomial function;
2. not a polynomial function
1 2 5 x + 2 x − , degree 3 3 is 4, quartic, leading coefficient is 2 g ( x) = 2 x 4 −
3. polynomial function;
1 2 x + 2 x + 10, degree is 4, 3 quartic, leading coefficient is 1 g ( x) = x 4 − 4 x3 −
4. polynomial function; f ( x) = 8 x 2 −
4.1 Practice B
3x + 2,
14 x 3 −
4. not a polynomial function 5. 1841
6. − 47 9
7. − 85
degree is 2, quadratic, leading coefficient is 8 Copyright © Big Ideas Learning, LLC All rights reserved.
Algebra 2 Answers
A33
Answers 8. g ( x) → −∞ as x → + ∞ and g ( x) → −∞ as
15.
6
x → −∞.
2
x
−6
x → −∞.
−12 −18
q(x) = x 4 − x 3 − 5x 2 4
f
−4
9. h( x) → +∞ as x → + ∞ and h( x) → −∞ as
10.
y
y
−4
The degree is odd and the leading coefficient is positive.
4 x
2 −4
16. Sample answer: − 2 ≤ x ≤ 8; − 5 ≤ y ≤ 50
−8
y 40
−12
h
30
11.
y
h(x) = 4 − 2x 2 − x 4
20
2
−4
10
−2
2
4 x
2
4
x
−2
4.1 Enrichment and Extension
−4
1. y ≈ 1.17 x 4 − 14.33x3 + 60.83x 2 − 102.67 x + 56
12.
13. k(x) =
x5
−
2x 4
2. y ≈ 0.12 x 4 − 2.89 x3 + 23.51x 2 − 76.96 x + 84 f(x) =
+x−2
x6
y
−
3x 5 6
+
2x 3
+x+1
2
4 x
y
2
−4 −4
−2
14.
4 x
4
−2
4. y ≈ 1.39 × 10−4 x 4 − 0.03 x 3 + 2.78 x 2 − 88.57 x
+ 969.47
−6 −12
4.1 Puzzle Time
−18
COBWEBS 4.2 Start Thinking
y
−2
3. y ≈ − 0.29 x 4 + 1.07 x3 + 1.96 x 2 − 5.93x − 0.60
Sample answer:
f 2
4
x
−4 −8 −12
The degree is even and the leading coefficient is negative.
(x
+ 1)( x − 1) = x 2 − x + x − 1 = x 2 − 1;
(x
+ 3)( x − 3) = x 2 − 3x + 3x − 9 = x 2 − 9;
yes; In each example, the middle terms cancel out, leaving only two terms. The first term is the square of the first term in each binomial. The second term is the square of the second term in each binomial;
(x
+ 1)( x + 1) = x 2 + x + x + 1 = x 2 + 2 x + 1;
(x
− 3)( x − 3) = x 2 − 3x − 3x + 9 = x 2 − 6 x + 9;
no; The signs are the same inside the binomials, so the middle terms no longer cancel.
A34 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 4.2 Warm Up 1. 7
2. − x + 7
3. 28m + 21
4. 10r
5. − 3z 2 − 2 z
6. − 2m − 4 p
4.2 Cumulative Review Warm Up 1.
25 5 ; x + 4 2
3. 36; ( w − 6)
5. 16; ( x − 4)
2
2. 9; ( z + 3)
2
2
1. 8 x 7 + 15 x 6 − 2 x5 + x3 − 6 x + 2 2. 14 x 4 − 7 x3 − 4 x + 5 3. − 3x5 + 3x 4 − 8 x 2 + 10 x + 9
2
4. 9 x 4 + 5 x3 − 6 x 2 − 7 x + 11
625 25 ; x − 4. 4 2
2
729 27 ; s + 4 2
2
6.
4.2 Practice B
5. x 4 − 10 x3 + 13x 2 + 48 x + 12 6. −10 x 4 − 19 x3 + 7 x 2 + 14 x − 4
4.2 Practice A
7. 4 x 4 − 11x3 + 20 x 2 − 18 x + 12 8. 3 x 6 − 6 x 5 + x 4 + 3 x 3 − 40 x 2 − 25 x 9. The exponents were multiplied instead of added;
(
1. 4 x 2 + 7 x − 9
)
4 x 2 3 x 4 − 2 x 3 + 7 = 12 x 6 − 8 x 5 + 28 x 2
2. 7 x 5 + 5 x 4 + 3 x 2 − 3 x − 5 3. 5 x 4 + 2 x3 − 4 x 2 − 9
10. 6 x3 − 14 x 2 − 14 x + 6 11. 8 x 3 − 26 x 2 − 67 x − 15
4. − 4 x3 + 4 x 2 − 4 x + 2
12. −16 x3 + 12 x 2 + 28 x − 15
5. 8 x 4 + x3 − 3x 2 − 4 x + 6 6. 7 x 5 − 6 x 4 + 13 x 3 − 3 x 2 + 12 x + 8 7. 7 x 2 + 9 x − 8
13. 8 x3 − 30 x 2 + 13x + 30 14. 9 x 2 − 25 15. 36t 2 + 84t + 49
8. 15 x 4 + 35 x3 + 30 x 2
16. p 2 q 2 + 4 pq + 4
9. − 20 x 7 + 18 x6 + 14 x5 − 8 x 4
17. a. Sample answer: (3 x − 1)( x + 6)
10. − 24 x3 + 25 x 2 − 9 x + 2
b. 3 x 3 + 35 x 2 + 96 x − 36
11. − 3x3 − 20 x 2 − 21x − 54
4.2 Enrichment and Extension
12. The negative was distributed incorrectly;
(
)
− 3 x 2 4 x 2 − 5 x + 7 = −12 x 4 + 15 x 3 − 21x 2
2
1. a = 3, b = 4, c = 2 2. a = 1, b = 3, c = 2, d = 5
13. x3 − 13 x + 12
3. a = 0, b = 2, c = − 4, d = 10
14. x3 − 13x 2 + 24 x + 108
4. a = − 4, b = 3, c = 5, d = −13
15. 4 x3 + 8 x 2 − 15 x − 9
5. a = 7, b = 0, c = 1
16. 12 x3 − 25 x 2 − 87 x − 20
6. a = 9, b = − 30, c = − 5 or a = 9, b = 30,
17. x 2 − 64
18. y 2 + 8 y + 16
c = 5
19. 4 p 2 − 12 p + 9 Copyright © Big Ideas Learning, LLC All rights reserved.
Algebra 2 Answers
A35
Answers 4.3 Practice A
1
7.
1
1
1
2 3
1 1 1 8.
(x
3 6
4 5
1
1
10
6
15
42 x −5
2. 2 x + 3 +
5 x − 2
3. x + 1 +
3 x −9
4. 6 x − 1 +
2 x + 2
5. x + 9 +
28 x −3
6. 3 x − 8 −
12 x −1
1 4
10 20
1. x + 6 +
1 5
15
1 6
2
1
+ 1) = x 6 + 6 x 5 + 15 x 4 + 20 x 3 + 6
15 x 2 + 6 x + 1 9. ( 2 y − 2)
6
6
5
4
= 64 y − 384 y + 960 y −
1280 y + 960 y 2 − 384 y + 64
15 x + 2
7. 2 x − 5 +
2
8. x 2 − 3 x + 7 −
15 x + 3
3
10. (1 − y )
6
= 1 − 6 y + 15 y 2 − 20 y 3 +
−3 1
11. x12
( x2
4 x −1
2 0
7
− 3 3 −9
− 2) = x12 − 12 x10 + 60 x8 − 160 x 6 +
1
6
240 x 4 − 192 x 2 + 64 13. (bc + de)
6
= b 6 c 6 + 6b5 c 5 de + 15b 4 c 4 d 2 e 2 +
20b3c3 d 3e3 + 15b 2 c 2 d 4 e4 + 6bcd 5 e5 + d 6 e6
x3 + 2 x 2 + 7 2 = x2 − x + 3 − x +3 x +3 12. 28
4.3 Start Thinking
1. x + 3 +
+ 2)( x − 1); Inverse operations undo one another,
so if two binomials are multiplied to make a product, you can divide the product by one binomial to obtain the other binomial; no; Factoring will only work as division if there is no remainder. It is possible to divide polynomials that are not factorable. 4.3 Warm Up 1. 13(t + 3 y )
2. 3k ( k − 1)
3. ab (5ab − a + 11b)
4.
5. ( n − 11)( n − 2)
6. 3 ( x + 7)( x + 3)
(x
+ 5)( x − 5)
4.3 Cumulative Review Warm Up
3. g ( x) =
x 1 + 3 3
A36 Algebra 2 Answers
13. 41
14. 8
15. 18
4.3 Practice B
A WALKIE TALKIE
1. g ( x) = 3x − 2
−1 3 − 2
16. x 3 + x 2 − 3 x + 3; Multiply the result by x + 1.
4.2 Puzzle Time
(x
10. 5 x + 2 +
11. k = − 3;
15 y 4 − 6 y 5 + y 6
12.
50 x −5
9. x + 5 +
2. g ( x ) =
1 x +1 4
6 x − 4 2
2. 4 x 2 − 2 x + 9 −
17 x x2 + x − 4
3. 2 x 2 + 10 x + 14 +
7 x − 2
4. 4 x − 7 −
5. x 2 − 3 x +
6. x + 4 +
90 x x2 − 5x − 2
12 x +3
32 x − 4
7. 2 x 2 − 7 x + 7 −
4 x +1
8. x 3 + x 2 − 10 x + 29 −
102 x + 4
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 9. x 3 − 4 x 2 + 24 x − 140 +
820 x + 6
2.
y 4
x = 0.625 (0.625, 1.5625)
2
10. The powers in the quotient are too large by 1. The
remainder ( − 2) was not divided by ( x + 3); −3 1
2 0
−4
−2
2
4
x
y = −4x2 + 5x
7
− 3 3 −9 1 3
−1 3 − 2
3.
x = 2.3 y
2
x + 2x + 7 2 = x2 − x + 3 − x +3 x +3 12. −10
11. 7 14. − 95
y = 5x 2 − 23x + 8
8
−8
13. 305
8
16
x
−8
15. k = 4
−16
(2.3, −18.45)
4.3 Enrichment and Extension 1. y = −3 x − 3
2. y = x + 5
3. y = 2 x + 1
4. y = x + 6
5. y = − 2 x − 4
6. y = ax + b − ad
4.
4 2
−4
(−1, −2)
HE WAS ALWAYS WILLING TO LEND AN EAR
x = −1
yes; You can group the terms with coefficients of 3 and 21 together, or you can group the terms with coefficients of − 5 and 40 together; yes; You can group 3x and 21xy together, and you can group − 5 and 40y together.
5.
y 8
y = x2 − x + 3 −4
2. 2rs
y = x2 + 2x − 1
6
4.4 Warm Up 1. 3y
4 x
2
4.3 Puzzle Time
4.4 Start Thinking
y
(0.5, 2.75) x = 0.5
−2
6. 4. yz
5. ab
4.4 Cumulative Review Warm Up 1.
4 x
2
3. 3x y 4
6. xy y = 2x2 + 3x + 1
2
(−0.75, −0.125)
x = −0.5
2 x
y
x = −0.75 4
−8
−4
(−0.5, −6)
4
8 x
y = 4x2 + 4x − 5
Copyright © Big Ideas Learning, LLC All rights reserved.
4.4 Practice A 1. x ( x − 4)( x + 3)
2. 9 p5 ( p + 2)( p − 2)
3. 3n 4 ( n − 8)( n − 3)
4. 2k 2 ( k + 11)( k − 11)
5. w2 ( 2w + 3)( w − 5)
6. q 4 (3q + 4)( q − 7)
Algebra 2 Answers
A37
Answers 7.
(x
+ 3)( x 2 − 3 x + 9)
14. 9 g 2 + 25 (3 g + 5)(3 g − 5)
8.
(y
+ 10)( y 2 − 10 y + 100)
15. 2t 2 t 3 + 5 t 3 − 2
(
(
(
)
9. ( w − 5) w2 + 5w + 25 10.
− 3)( y 2 + 4)
(y
(
12. ( d + 5) 2d
(
(
11. ( q − 2) q 2 + 9
+ 3)
2
)(
13.
)
14. 6 p 2 − 5 6 p 2 + 5 16.
( y2
)
(x
)
− 6)( x − 3)( x + 3)
(
)(
15. n 2 + 4 n 2 + 7
)
)(
)
(
17. yes
18. no
19. no
20. no
(x
22. a.
+ 4); x ( x + 4)( 2 x − 1) = 2 x3 + 7 x 2 − 4 x
( x − ( − 4) )
2
+ y 2 = 32 ; ( h, k ) = ( − 4, 0),
r = 3 y
4
18. yes
19. yes
20. no
2
21. Sample answer: f ( x) = x + 5 x − 6 x; ( x − 1); 3
2
−6
x ( x − 1)( x + 6) = x3 + 5 x 2 − 6 x
−4
−2
3 x 3 − 17 x 2 − 9 x + 18 = ( x − 6)(3 x 2 + x − 3)
−4
b.
(x
)
23. a. a 2 + b 2 (5c − 3d ) b.
( xn
+ 3)
2
2
y
2
2
6
4
x
−2
1. 5t 3 (t + 8)(t − 8)
2. 2 p 4 ( p − 7)( p − 6)
3. 3x 2 ( x + 12)( x − 12)
4. a 4 (5a + 9)( a − 5)
5. j 7 ( 2 j − 3)(6 j − 5)
6. q8 (3q + 4)(5q + 6)
(
7. 2 p 6 ( p − 2) p 2 + 2 p + 4
(
(
2
c.
( x − 2)2 + ( y − (− 3)) (h, k ) = (2, − 3), r = 4 2
= 42 ;
y −2
)
9. 2 w (3w − 2) 9 w + 6 w + 4 4
−4
)
8. 25k 5 ( k + 4) k 2 − 4k + 16
(x
− 5) + y 2 = 2 2 ; ( h, k ) = (5, 0), r = 2 4
4.4 Practice B
10.
x −2
22. k = 9; f ( x) = 3x3 − 17 x 2 − 9 x + 18;
(
)
21. Sample answer: f ( x) = 2 x3 + 7 x 2 − 4 x;
+ 4)( y + 2)( y − 2)
17. no
)(
16. 5v 2 v 4 − 3 v 4 − 2
2
4
6 x
−2 −4
)
− 7)( x 2 + 5)
11. ( m − 2)( m + 4)( m − 4)
4.4 Enrichment and Extension 1. 27
2. −18
3. 3
12. ( w − 3)(3w + 2)(3w − 2)
4. − 4
5. −1
6. 6
13. ( s + 4)(5s + 1)(5s − 1)
7. x 5 − y 5
(
xy 3 +
A38 Algebra 2 Answers
) = ( x − y )( x 4 y4 )
+ x3 y + x 2 y 2 +
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers (
) = ( a + b)( a 6 − ab5 + b6 )
8. a 7 + b 7
a 2b4
− a 5b + a 4 b 2 − a 3b 3 +
4. n = − 3, n = −2, n = 3
(
) = (a 7 + b7 )(a7 − b7 ) = (a + b)(a − b)(a 6 − a 5b + a 4 b 2 − a 3b3 + a 2 b 4 − ab5 + b 6 )( a 6 + a 5b + a 4 b 2 + a 3b3 a 2 b 4 + ab5 + b 6 )
9. a14 − b14
10.
( x10 4
3
2
3
2
2
3
+ x y + x y + xy + y
4
2
+
2
y −2
2
−40
3
6. u = 0, u = ±
7
7. x = − 4, x = 0, x = 3
+ y )( x − y )( x − x y + x y − xy + y
(x
(x
5. p = 0, p = ±
− y10 ) = ( x 5 + y 5 )( x 5 − y 5 ) = 4
3. y = 0, y = 1
2. k = − 3, k = 0
4
)
)
4 x
f(x) = x 4 + x 3 − 12x 2
−60
8. x = − 2, x = 2
4.4 Puzzle Time
QUARTERBACK
g(x) = x 4 − 8x 2 + 16 18
4.5 Start Thinking
y
5 −10
10
6
f(x) = x3 − 6x2 −4
−2
−40
(0, 0) and (6, 0); The function simplifies to
0 = 0;
These points have x-values which yield a y-value of zero, meaning the graph crosses the x-axis. These are the only points that can be inserted into the function f ( x) = x3 − 6 x 2 to get this result.
h(x) = x 5 − 2x 4 − 15x3 y −2
4. z = 20
5. m = − 2
6. b = 20
4.5 Cumulative Review Warm Up
5 or x >
5
2. − 9 ≤ x ≤ − 3
5 < x <
4
x
−200
2. x = − 5
3. r = − 60
4. −
2
−100
1 4
1. x < −
4 x
9. x = − 3, x = 0, x = 5
4.5 Warm Up 1. t = −
2 −6
−300 −400
10. x = − 4, x = −1, x = 0 f(x) = −3x 3 − 15x 2 − 12x 40
3. x < − 5 or x > −1
5
5. x < − 3 or x > 1
y
20
−6
−2
2 x −20
6. 1 < x < 8
−40
4.5 Practice A 1. q = − 5, q = 0, q = 6
Copyright © Big Ideas Learning, LLC All rights reserved.
11. C
Algebra 2 Answers
A39
Answers 10. x = − 3, x = 3, x = 4
12. The factors of 18 include ±1 and ±18;
f ( x) = x + 3x − 8 x − 18; 3
2
f(x) = x 3 − 4x 2 − 9x + 36 y
Possible zeros: ±1, ± 2, ± 3, ± 6, ± 9, ±18 13. x = 3, x = − 3
20
14. x = −1, x = 2, x = − 3
−2
15. Sample answer: f ( x) = 4 x3 + 4 x 2 − 9 x − 9;
3 3 f = 0; f − = 0 2 2 16. a. k = −18
1 3. q = ± 2
4. w = ± 3
5. p = − 2, p = ± 5
6. y = ± 3, y = 8
7. x = − 2, x = 0, x = 6 f(x) = −5x 4 + 20x 3 + 60x 2 y
x
12. The factors of 2 include ± 2;
f ( x) = 2 x3 + 5 x 2 − 2 x − 6;
4.5 Practice B 2. h = 0, h = ±
4
11. B
b. k = − 31
3 1. x = − , x = 0 2
2 −20
2
1 3 Possible zeros: ±1, ± 2, ± 3, ± 6, ± , ± 2 2 13. x =
3 2
14. x = − 2, x = 3, x = 4 15. Sample answer:
f ( x) = 25 x3 − 50 x 2 − 49 x + 98; 7 7 f = 0; f − = 0 5 5
800
16. height is 11 cm, side length is 9 cm
400
−4
4.5 Enrichment and Extension
4
x
8. x = − 6, x = 0, x = 5 g(x) = −x 3 − x 2 + 30x
1. P ( x) = − 7 x 2 − 7 x + 14 2. P ( x) = x3 − 3x + 2
(
)
3. P ( x ) = a x 4 − x 3 − 6 x 2 , a can be any real
y 100
number 4
4. P ( x) = 3x 4 + 12 x3 − 3x 2 + 48 x − 60
x
−100
9. x = − 2, x = −1, x = 2 h(x) = x 3 + x 2 − 4x − 4 40
y
5. P ( x) = −3x3 − 3x 2 + 21x + 45
4.5 Puzzle Time
IT WAS A BREEZE WITH ONLY A FEW FOGGY PATCHES
20
−4
2
4 x
−20 −40
A40 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 4.6 Start Thinking
3.
f(x) = x
Sample answer:
4
g(x) = x + 6
Function
y
2
Number of x-intercepts
−6
−4
−2
x
f ( x) = x + 4
1
g ( x) = x 2 − 5
2
h ( x) = x 3 + 3 x 2 − x − 1
3
The graph of g ( x) = x + 6 is a horizontal
j ( x) = x 4 − x3 − 4 x 2 + 1
4
translation 6 units left of the parent absolute value function.
The degree of the function and the number of xintercepts are the same; no; Sometimes, there are solutions to polynomial functions that are imaginary numbers, which are not shown on the graph of the function.
−2 −4
4.
4
y
2
h(x) = −x −4
f(x) = x
−2
2
4 x
−2
4.6 Warm Up 1. 4
2. 3
3. 5
4. 6
5. 3
6. 5
4.6 Cumulative Review Warm Up 1.
y
f(x) = x −2
2
−4
g(x) = x − 5
The graph of g ( x) = x − 5 is a vertical translation 5 units down of the parent linear function.
2
−4
−2
reflection in the x-axis of the parent linear function. 4.6 Practice A 2. 3
3. 5
4. 6
5. − 3, − 2i, 2i, 3
6. − 3, −1, 1, 2
7. 0, 1, − 3i, 3i
8. − 3, −
x
−2
4
Sample answer: The graph of h( x) = − x is a
1. 4
2
2.
−4
2,
2, 3
9. 4; The graph shows 1 real zero, so the remaining
zeros must be imaginary. 10. 0; There are 4 zeros for this function. The graph
crosses the x-axis twice and touches the x-axis once at the repeated zero, leaving 0 imaginary zeros. 11. f ( x) = x3 + x 2 − 10 x + 8
y
f(x) = x2 2
4 x
−2 −4
The graph of f ( x) = x 2 is the parent quadratic function, so there was no transformation.
Copyright © Big Ideas Learning, LLC All rights reserved.
12. f ( x) = x3 − 4 x 2 + x + 6 13. f ( x) = x3 − 2 x 2 − 3x + 6 14. Sample answer:
f ( x) = x5 − 2 x3 − 2 x 2 − 3 x − 2; Because i is a
zero, −i is also a zero. The graph touches the x-axis at −1 (has a multiplicity of 2) and the graph crosses the x-axis at 2.
Algebra 2 Answers
A41
Answers 15.
16.
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros
2
1
0
3
0
1
2
3
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros
1
1
2
4
4.6 Enrichment and Extension 1. a and b. f ( x) = ( x + 2)( x + 4)( x + 3)( x − 5) c. − 4, − 3, − 2, 5
(
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros
3
2
0
5
3
0
2
5
1
2
2
5
1
0
4
5
(
x − 4 +
)
3. 3
4. 6
5. −1, 1, 2, 2
6. 3, 3, 3, 3
7. − 2, − 2, −1, −1, 2
8. − 3, − 2, − 2, 2, 3
9. 2; Sample answer: There are four zeros for this
function. The graph crosses the x-axis twice, leaving two imaginary zeros. 10. 2; Sample answer: There are three zeros for this
function. The graph crosses the x-axis only once, leaving two imaginary zeros. 11. f ( x) = x − 8 x + 22 x − 20 2
12. f ( x) = x − 2 x + 6 x − 8 x + 8 4
)
3
3 3, 4 +
3
(
)
3. a. f ( x ) = ( x − 4)( 4 x − 3) x 2 − 4 x + 13 b. f ( x ) = ( x − 4)( 4 x − 3) x − ( 2 + 3i )
x − ( 2 − 3i ) c.
3 , 4, 2 − 3i, 2 + 3i 4
(
2. 5
3
4. a. f ( x ) = ( x + 2)( x + 1) x 2 − 6 x + 4
4.6 Practice B 1. 4
(
b. f ( x) = ( x + 5)( x − 2) x − 4 −
c. − 5, 2, 4 −
17.
)
2. a. f ( x ) = ( x + 5)( x − 2) x 2 − 8 x + 13
3
2
13. f ( x) = x3 − 3x 2 − 7 x + 21 14. Complex zeros come in pairs, so the remaining zero
cannot be complex.
(
b. f ( x) = ( x + 2)( x + 1) x − 3 +
(
x − 3 −
)
)
5
5
c. − 2, − 1, 3 −
5, 3 +
5
(
)
(
2 x −
5. a. f ( x ) = ( x + 3)( x − 5) x 2 − 2 b. f ( x ) = ( x + 3)( x − 5) x + c. − 3, −
)
2, 5,
)(
2
)
2
(
6. a. f ( x ) = ( x − 3)( x + 4)( 2 x − 1) x 2 + 9
)
b. f ( x) = ( x − 3)( x + 4)( 2 x − 1)( x + 3i )
(x
− 3i )
c. − 4, − 3i,
1 ,3 2
4.6 Puzzle Time
A PUP TENT
15. C
A42 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 4.7 Start Thinking
4.7 Cumulative Review Warm Up
Function Transformation Function g ( x) =
(x
− 2)
5
g ( x) =
(x
+ 2)
5
g ( x) = x5 − 2 g ( x) = 5
x + 2
Translation 2 units right Translation 2 units left
Translation 2 units down
Translation 2 units up
g ( x) = − x5
( 2 x )5 g ( x) = 5
g ( x) = 2x
Reflection in the x-axis
1. The graph of g ( x) = x 2 + 5 is a vertical
translation 5 units up of the graph of the parent quadratic function. g(x) = x2 + 5
g ( x) =
( 12 x)
Transformation
5
y
Horizontal shrink by a factor of 12 Horizontal stretch by a factor of 2 Vertical stretch by a factor of 2
The transformations of g ( x) = x5 behave in the same manner as other parent function transformations. Numbers added or subtracted inside parentheses translate the graph left or right, and numbers added or subtracted outside the parentheses translate the graph up or down. Numbers multiplied inside the parentheses horizontally stretch or shrink the graph, and numbers multiplied outside the parentheses vertically stretch or shrink the graph.
4
f(x) = x2 −8
−4
4
−8
2. The graph of g ( x ) = ( x − 1) is a horizontal 2
translation 1 unit right of the graph of the parent quadratic function. g(x) = (x − 1)2 4
f(x) = x2 −4
−2
2
4 x
−2 −4
2
1. Sample answer: The graph of g is a vertical stretch
by a factor of 5 of the graph of the parent function. 2. The graph of h is a reflection in the x-axis, followed
translation 2 units left of the graph of the parent quadratic function. g(x) = (x + 2)2
1 by a vertical shrink by a factor of of the graph of 3 the parent function. 3. Sample answer: The graph of g is a vertical stretch
4
−4
−2
2
4 x
−2 −4
4. The graph of f is a reflection in the x-axis, followed
1 of the graph of 2
y
f(x) = x2
by a factor of 2 of the graph of the parent function.
the parent function.
y
3. The graph of g ( x ) = ( x + 2) is a horizontal
4.7 Warm Up
by a vertical shrink by a factor of
8 x
−4
4. The graph of g ( x ) =
(x
− 5) + 3 is a horizontal 2
translation 5 units right, followed by a vertical translation 3 units up of the graph of the parent quadratic function. g(x) = (x − 5)2 + 3 8
f(x) = x2 −8
y
4
−4
4
8 x
−4 −8
Copyright © Big Ideas Learning, LLC All rights reserved.
Algebra 2 Answers
A43
Answers 4.7 Practice A 1. The graph of g is a vertical translation 2 units down
of the graph of the parent function f.
5. The graph of g is a vertical shrink by a factor of 32 ,
followed by a vertical translation 5 units down of the graph of the parent function f.
y
1
2
−4 −4
−2
g
f
−2
2
4 x
f
4 x
2
y
−4
g
−4
2. The graph of g is a horizontal translation 3 units left
of the graph of the parent function f. y
4 2
g
6. The graph of g is a vertical shrink by a factor of 12 ,
followed by a horizontal translation 2 units right of the graph of the parent function f. y
f
f −6
−2 −2
4
−4
2
−2
3. The graph of g is a reflection in x-axis, followed by
a vertical stretch by a factor of 5 of the graph of the parent function f. 4
4
2
7. g ( x ) =
(x
x
− 1) + 2 3
y y
2
−4
g
6
2 x
f
−2
f
4 2
g
4 x −4
g
4. The graph of g is a vertical stretch by a factor of 4,
followed by a vertical translation 3 units down of the graph of the parent function f.
−2
2
4 x
The graph of g is a horizontal translation 1 unit right of the graph of the parent function f. 8. g ( x) = 2 x 4 − 6 x + 2
y 2
y −4
−2
2
f
−2
f
4 x
g −4
−2
2 −2
4 x
g
−4
The graph of g is a vertical stretch by a factor of 2 of the graph of the parent function f.
A44 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 9. The parent function was translated 2 units down
instead of 2 units up. y
4. The graph of g is a vertical stretch by a factor of 3,
followed by a vertical translation 2 units up of the graph of the parent function f.
4
y
6
−4
4
4
x
−4
g
f −4
−2
10. g ( x ) = ( − x − 2) + 5 3
1 4 5 x − x − 3 3
11. g ( x ) =
12. a. W ( x) = 27 x3 + 12 x + 3
2 −2
5. The graph of g is a vertical shrink by a factor of 13 ,
followed by a vertical translation 3 units down of the graph of the parent function f.
b. Z ( x) = 46,656 x3 + 144 x + 3
y
4
g
2
f
4.7 Practice B 1. The graph of g is a horizontal translation 3 units
−4
−2
right, followed by a vertical translation 2 units down of the graph of the parent function f. 4
f
2
−4
6. The graph of g is a vertical shrink by a factor of 32 ,
g
2
6 x
followed by a horizontal translation 3 units left of the graph of the parent function f.
−2
6
−4
y
4
2. The graph of g is a horizontal translation 1 unit
right, followed by a vertical translation 4 units up of the graph of the parent function f.
−4
−2
1 4
7. g ( x ) = − x 3 + x 2 −
2
f
−2
4 x
3. The graph of g is a reflection in the x-axis, followed
by a vertical stretch by a factor of 3 of the graph of the parent function f.
−2
1 2
y 2
2
x
g
4
g
2 −2
6
4
f
2
g
y
−4
4 x
−2
y
−2
−4
4 x
g
4
−2
f 2
x
−8
y
f
2
4 x
The graph of g is a reflection in the x-axis, followed 1 by a vertical shrink by a factor of of the graph of 4 the parent function f.
−4
Copyright © Big Ideas Learning, LLC All rights reserved.
Algebra 2 Answers
A45
Answers 8. g ( x) = x 4 − x + 3
4.8 Start Thinking 15
y 6
g
f −4
−10
10
2
−15
−2
2
4 x
The graph of g is a reflection in the y-axis, followed by a vertical translation 2 units up of the graph of the parent function f. 9. The graph of g is a vertical stretch by a factor of 4,
1 , of the graph 4 of the parent function f ( x) = x 4 ; The graph of g is
not a vertical shrink by a factor of
a vertical stretch by a factor of 4, followed by a translation 3 units up of the graph of f. 1 3 1 10. g ( x ) = − x + x2 − 5 27 3
1. (5, − 33)
4 3
1 3
4. , 5
3. ( − 2, −19)
5. (8, 45)
6. ( 2, 10)
4.8 Cumulative Review Warm Up
2. g ( x ) =
(x
+ 2) − 2; ( − 2, − 2) 2
3. g ( x ) = ( x − 5) − 63; (5, − 63) 2
1 3 x + 9; W (6) = 9.25; When 864 x = 6 inches, the volume of the box is 9.25 cubic feet.
1 x3 + 9 23,328
4. h ( x ) = ( x − 10) − 191; (10, −191) 2
5. h ( x ) = ( x − 1) + 48; (1, 48) 2
6. f ( x ) = ( x − 3) − 6; (3, − 6) 2
4.7 Enrichment and Extension
4.8 Practice A
y = 2( x + 4) ; y = ( x + 3) − 2; 3
1. 3
3
4
3
2
y
x
−4
y = ( x − 5) − 2; y = ( x − 8) + 1 3
−2
2 x −4
−20
−8
4.7 Puzzle Time
FRYDAY
g(x) = (x − 1)2(x + 1)(x + 3)
y
y = ( x − 3) + 2; y = 2( x − 3) ; y = ( x − 5) ; 3
2.
f(x) = (x + 2)2(x − 3)
y = ( x + 3) − 6; y = x 3 ; y = ( 2 x ) ; 3
3
2. (3, 12)
2
12. a. W ( x ) =
3
4.8 Warm Up
1. f ( x ) = ( x + 3) − 29; ( − 3, − 29)
3 1 5 3 11. g ( x ) = − x 5 + x3 + x 2 + 2 2 2 2
b. Z ( x ) =
The shape of the graph of the function is a rounded “N”; There is one zero where − 2 < x < −1, another zero where −1 < x < 0, and a third zero where 1 < x < 2; no; The y-values are changing signs, but the table does not show an x-value when y = 0.
3.
h(x) = 2(x − 1)(x − 2)(x + 2) y 8
2
x
−8
A46 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 4.
h(x) = x 4 − 2x 2 + 3x
12.
f(x) = 3(x − 1)2(x + 1)2 y
y
8
4
4
−4 −2
2
4 x
2
x
−8
5. The function was graphed as if the zero x = − 3
x-intercepts: −1.89, 0; local maximum: none; local
had a multiplicity of 2 instead of the zero x = 1. f(x) = (x − 1)2(x + 3)
minimum: ( −1.26, − 4.43); increasing: x > −1.26;
y
decreasing: x < −1.26
8
13.
f(x) = x 4 − 4x 3 + 5x − 2 16
−4
−2
2
y
x
8
−4
−2
6. − 3, − 2, 2
7. − 7, −1, 1
8. 5
9. − 3, 32 , 3
10.
8
x-intercepts: −1.15, 0.48, 1, 3.67; local maximum:
(0.74, 0.38); local minimum: (− 0.59, 4.01), (2.85, −14.37); increasing: − 0.59 < x < 0.74, x > 2.85; decreasing: x < − 0.59, 0.74 < x < 2.85
4 x
f(x) = 2x 3 − 5x 2 + 3
14. ( − 0.82, 1.09), (0.82, −1.09); The point
(− 0.82, 1.09) is a local maximum. The point
x-intercepts: − 0.69,1, 2.19; local maximum: (0, 3);
(0.82, −1.09) is a local minimum; The real zeros
local minimum: (1.67, −1.63); increasing:
are −1.41, 0, and 1.41; The minimum degree is 3.
x < 0, x > 1.67; decreasing: 0 < x < 1.67 8
15. (1.50, 3.69) ; The point (1.50, 3.69) is a local
y
maximum; The real zeros are − 0.89 and 2.19; The minimum degree is 4.
g(x) = −x 4 + 2x 4
−4
−2
x
−16
y
−2
11.
4
−8
4
−4
2
2
4 x
4.8 Practice B 1.
f(x) = 4(x + 3)2(x − 2)2
2.
1 2
g(x) = (x − 4)(x + 3)(x − 6)
y
y
x-intercepts: 0, 1.26; local maximum:
(0.790, 1.19); local minimum: none; increasing:
80
x < 0.79; decreasing: x > 0.79
40
−4
Copyright © Big Ideas Learning, LLC All rights reserved.
−2
20 10
2
4 x
−4
8
Algebra 2 Answers
A47
x
Answers 3.
1 5
h(x) = (x − 3)(x − 4)(x + 8)
4.
f(x) = (x − 2)(x 2 + x + 2)
y
16
40
12.
h(x) = x 5 − 3x 2 − 9x − 2
y
y −2
8
x −6
−4
−2
20
4 x
−12
−8
10
−16
−4
x-intercepts: −1.36, − 0.24, 2; local maximum:
(− 0.92, 3.08); local minimum: (1.36, −15.14);
8 x
4
increasing: x < − 0.92, 1.36 < x; decreasing: − 0.92 < x < 1.36
5. The function was graphed as if the zero x = 0 had
a multiplicity of 1 instead of a multiplicity of 2. 13.
y 4
f(x) = x 4 − 3x 3 + 3x 2 + x − 2 20
2
f(x) = x2(x + 2)3 −4
2
y
10
4 x −2
−2
2
4 x
−10
−4
1 6. 2
5 7. − , − 2, 1 2
3 8. − 3, , 3 4
3 9. − 2
10.
x-intercepts: − 0.7, 1; local maximum: none; local
minimum: ( − 0.14, − 2.07); increasing: − 0.14 < x; decreasing: x < − 0.14 14. a. about 1.6 in. b. about 67.6 in.3
f(x) = 0.5x 3 − 3x 2 + 1.5 3
c. length ≈ 8.8 in., width ≈ 4.8 in., height ≈ 1.6 in.
y 8
4
4.8 Enrichment and Extension 1. x = 3.16 2. x = 1.52
x
5. x = 1.92
4. x = − 2.99
x-intercepts: − 0.67, 0.76, 5.91; local maximum:
(0, 1.5); local minimum: (4, −14.5);
increasing:
x < 0, x > 4; decreasing: 0 < x < 4 11.
3. x = −1.31
4.8 Puzzle Time
SHIFTY 4.9 Start Thinking
Sample answer: g(x) = 0.4x 3 − 3x 3
8
y
−2
2
x
−8
−4
y
4
8 x
−2
x-intercepts: − 2.74, 0, 2.74; local maximum:
(−1.58, 3.16); local minimum: (1.58, − 3.16);
quadratic; parabola; cubic
increasing: x < −1.58, x > 1.58; decreasing: −1.58 < x < 1.58
A48 Algebra 2 Answers
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 3. arithmetic; d = − 9
4.9 Warm Up 1. 4
2. −
1 3
3. 4
4. 1
4.9 Cumulative Review Warm Up 1. (1, 6)
2. ( − 3, −11)
1. f ( x) = x3 − 3x 2 − x + 3
1 2
3. 3; f ( x ) =
3 2 x + 3x − 4 2
1 3 5 x + x 6 6
7. tn = 4n
11. tn =
2 1 n + 3 3
6. not arithmetic 8. tn = − 2 n − 1 10. tn = − 2 n + 14 12. tn = −1.3n + 9.9
4.9 Puzzle Time
ANCHOR Cumulative Review
1 13 4. 2; f ( x ) = x 2 + x +8 8 4 5. 3; f ( x ) =
3 5
9. tn = −11n + 29
4.9 Practice A
2. f ( x) = − x3 +
5. arithmetic; d =
4. not arithmetic
1 3 8 x − 5 x 2 + x + 32 3 3
1 2 7 x + x−2 2 2 b. yes; The cumulative number of customers will continue to increase.
6. a. f ( x) =
4.9 Practice B 1. f ( x) =
1 3 3 x − 3x2 + x + 5 2 2
2. f ( x ) =
2 3 14 x − x − 4 3 3
1. 17j
2. − 4 p
3. 5q + 3
4. 3m + 5
5. − 3b + 19
6. − 2 x + 2
7. 4c + 28
8. − 5r + 15
9. 8 z − 44
10. 9 a − 38
11. 38 x − 9
12. 19 d − 10
13. x = 4
14. y = 3
15. b = 2
16. m = 5
17. a = 4
18. p = 6
19. w = 14
20. x = 2
21. y = − 5
22. s = 2
3. 2; f ( x) =
5 2 23 x − x −1 2 2
23. x = − 2
24. x = 2
4. 3; f ( x ) =
1 3 31 x − x + 39 2 2
25. x = 2
26. x = −11
5. 4; f ( x ) =
1 4 7 95 2 149 x − x3 + x − x + 56 6 3 6 3
27. x = − 9
28. x = − 3
6. a. f ( x ) =
1 3 1 17 x − x2 + x − 2 6 2 6
b. no; The wave height will decrease eventually.
After 14 seconds, the height of the wave will not be 397 inches.
29. a. y = 35 + 1.20 x b. $35 c. $44.60 d. when you are downloading 46 songs or fewer or
when you are downloading 72 songs or more 30. x + 7
31. n − 5
32. 3m − 9
33.
4.9 Enrichment and Extension 1. arithmetic; d = 3
2. not arithmetic
Copyright © Big Ideas Learning, LLC All rights reserved.
2 9
1 c +1 5
Algebra 2 Answers
A49
Answers 34. 17 − 4 y
35. − 5 + 4 p
36. 5 j − 14
37.
64. x = 7 ±
1 t − 4m 2
38.
(x
+ 6)( x + 4)
39.
(x
− 11)( x + 3)
40.
(x
+ 12)( x − 7)
41.
(x
+ 11)( x + 5)
42.
(x
− 7)( x − 2)
43.
(x
− 10)( x − 5)
(x
− 6)(9 x + 1)
44. (5 x + 7)( x + 8)
45.
46. ( 2 x − 5)( x + 4)
47. (3x + 4)( x + 3)
48. ( 2 x − 3)( x + 1)
49. ( 2 x + 7)( x − 4)
50. a. 252 ft2
66. x =
68. x =
70.
51. a. p = 2( x + 4) + 2( 2 x − 23) or p = 6 x − 38 b. a = ( x + 4)( 2 x − 23) or a = 2 x
2
72.
74.
− 15 x − 92
c. length = 23 ft, width = 15 ft d. 345 ft2
75.
52. x = − 2 and x = 3 76.
53. x = −1 and x = 9 54. x = − 7 and x = − 5
77.
55. x = 3
59. x = −
7 and x = 3 2
62. x =
17 2
A50 Algebra 2 Answers
9± 4 5 2
69. x =
105 4
21 ±
461 5
9+i 3 9−i 3 and 2 2
3+
13 2
5+
and
101 2
3−
and
13 2
5−
101 2
5 + 3i 15 5 − 3i 15 and 8 8 7 +
301 6
and
7 −
301 6
11 + i 39 11 − i 39 and 10 10 9+
137 4
and
9−
137 4
b. 2 students 80. The graph of g is a translation 4 units right of the
graph of f. 81. The graph of g is a translation 8 units up of the
graph of f. 82. The graph of g is a translation 3 units right and 5
units up of the graph of f.
1 9 and x = 2 4
7±
3
−9 ±
79. a. 4 students
1 60. x = − and x = 4 5 61. x = −
67. x =
b. 4 students
57. x = 1 and x = 7
9 4 and x = − 2 3
123
78. a. 1 student
56. x = −11 and x = 4
58. x = −
12 ±
− 9 ± 97 2
71. − 4 and 5
73.
b. $829.08
65. x =
47
83. The graph of g is a translation 1 unit left and 4 units
down of the graph of f. 63. x =
−13 ± 173 2
84. The graph of g is a reflection in the x-axis, followed
by a translation 3 units down of the graph of f.
Copyright © Big Ideas Learning, LLC All rights reserved.
Answers 85. The graph of g is a reflection in the x-axis, followed
by a translation 1 unit right and 5 units up of the graph of f. 86. The graph of g is a reflection in the x-axis, followed
by a vertical stretch by a factor of 5 and a translation 2 units down of the graph of f.
5.1 Warm Up 1. k 5
2. 24u10 v3
3. 25a 6 b 20 c 2
4. 729x8 y10 z 5
5. 8g 9 h5 j13
6. 2xy 9
5.1 Cumulative Review Warm Up 1. y = 0.06 ( x − 4) + 1 2
1 87. The graph of g is a vertical shrink by a factor of , 3 followed by a translation 5 units left and 3 units down of the graph of f. 88. g ( x) = − x + 3 89. g ( x) =
1 x +3 +1 4
90. g ( x) = 4
x − 2 + 16
91. g ( x) = x + 2 + 1
13 92. g ( x ) = − x + 2
2. y = − 6 ( x + 4) + 8 2
5.1 Practice A 1. 5
2. ± 7
3. ± 3
4. 3
5. 2
6. 8
7. 125
8. 100
9. 2
10. 7
11. 5
12. 0.44
13. 2.47
14. 59,049
15. 0.03
16. 6.05 in.
17. 5.6 m
18. x = ± 4
19. x = 5
20. x = 12.32, − 0.32
93. g ( x ) = − ( x + 2) + 8
21. x = 3
22. x = − 2.57
94. a. 41 ft
23. x = ±1.57
24. 2.57%
2
b. 44 ft c. 2 ft; The ball was initially hit from 2 feet above
the ground. d. maximum
8.
Chapter 5 5.1 Start Thinking Example
Expanded Form
Simplest Form
x2 + x2
x• x+ x• x
2x2
x • x x8 x5
2
−2
4
1. 7 4. 216
e. ( 4.75, 47.125); 4.75 sec
4
5.1 Practice B
(x • (x •
x • x • x) • x • x • x)
x• x• x• x• x• x• x• x x• x• x• x• x
1 = ; Because 22 = 4, they are reciprocals. 4
Copyright © Big Ideas Learning, LLC All rights reserved.
x
8
1 1024
3. − 3
2. no real roots 5. 8 9.
1 81
7. − 3125
6. 4 10. 0.51
11. 7.42
12. 8,869.01
13. 0.07
14. 25.64
15. −12.41
16. 7.6 in.
17. 7 m
18. x = ±1.78
19. x = − 2.97
20. x = ± 3
21. x = 3.42
22. x = ± 5
23. x = − 7
24. a. about 0.72 au b. 12 years
x3
5.1 Enrichment and Extension 1. n = 6
2. n = 2
3. n = 5
4. n = −1
5. n = − 4 or n = 4
6. n = 8
Algebra 2 Answers
A51
