CHAPTER 2 Polynomial and Rational Functionscrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch2.1-2.3_solutions.pdf92 Chapter 2 Polynomial and Rational Functions 32. is the vertex. Since

C H A P T E R 2Polynomial and Rational Functions

Section 2.1 Quadratic Functions . . . . . . . . . . . . . . . . . . . . . 88

Section 2.2 Polynomial Functions of Higher Degree . . . . . . . . . . 99

Section 2.3 Real Zeros of Polynomial Functions . . . . . . . . . . . . 112

Section 2.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . 126

Section 2.5 The Fundamental Theorem of Algebra . . . . . . . . . . . 132

Section 2.6 Rational Functions and Asymptotes . . . . . . . . . . . . 142

Section 2.7 Graphs of Rational Functions . . . . . . . . . . . . . . . 150

Section 2.8 Quadratic Models . . . . . . . . . . . . . . . . . . . . . . 165

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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C H A P T E R 2Polynomial and Rational Functions

Section 2.1 Quadratic Functions

88

1. opens upward and has vertexMatches graph (c).!2, 0".

f !x" ! !x " 2"2 2. opens downward and has vertexMatches graph (d).!0, 3".

f !x" ! 3 " x2

3. opens upward and has vertex Matches graph (b).

!0, 3". f !x" ! x2 # 3 4. opens downward and has vertexMatches graph (a).!4, 0".

f !x" ! "!x " 4"2

You should know the following facts about parabolas.

! is a quadratic function, and its graph is a parabola.

! If the parabola opens upward and the vertex is the minimum point. If the parabola opensdownward and the vertex is the maximum point.

! The vertex is

! To find the -intercepts (if any), solve

! The standard form of the equation of a parabola is

where

(a) The vertex is

(b) The axis is the vertical line x ! h.

!h, k".a $ 0.

f!x" ! a!x " h"2 # k

ax2 # bx # c ! 0.

x

!"b#2a, f!"b#2a"".

a < 0,a > 0,

f!x" ! ax2 # bx # c, a $ 0,

5.

(a) vertical shrink

(b) vertical shrink and vertical shift one unit downward

(c) vertical shrink and horizontalshift three units to the left

(d) horizontal shift threeunits to the left, vertical shrink, reflection in -axis, and vertical shift one unit downwardx

y ! "12 !x # 3"2 " 1,

y ! 12 !x # 3"2,

y ! 12 x2 " 1,

y ! 12 x2,

!9

!6

9

6

abc

d

6.

(a) vertical stretch

(b) vertical stretch, followed by a vertical shift upward one unit

(c) horizontal shift three units to theright, followed by a vertical stretch

(d) horizontal shift three unitsto the right, a vertical stretch, a reflection in thex-axis, and a vertical shift one unit upward

y ! "32!x " 3"2 # 1,

y ! 32!x " 3"2,

y ! 32x2 # 1,

y ! 32x2,

!9

!6

9

6

a

bc

d

Vocabulary Check1. nonnegative integer, real 2. quadratic, parabola 3. axis

4. positive, minimum 5. negative, maximum

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Section 2.1 Quadratic Functions 89

7.

Vertex:

-intercepts:

201510!10!15!20

30

25

5

x

y

!"5, 0", !5, 0"x

!0, 25"

f!x" ! 25 " x2 8.

Vertex:

Intercepts:

54321!3 !1!4!5

21

!2!3!4

!8

x

y

!±$7, 0"!0, "7"

f !x" ! x2 " 7 9.

Vertex:

-intercepts:

–4 –3 –1 1 2 3 4

–5

–3

–2

1

2

3

x

y

!±2$2, 0"x

!0, "4"

f !x" ! 12 x2 " 4

10.

Vertex:

Intercepts:

18

12108642

!4!6

1210642!2!4!6!10x

y

!±8, 0"!0, 16"

f !x" ! 16 " 14 x2 11.

Vertex:

-intercepts:

54321

!2!3!4

21!1!3!4!7!8x

y

!"4 ± $3, 0"x

!"4, "3"

f !x" ! !x # 4"2 " 3 12.

Vertex:

No -intercepts

6

12

18

24

30

42

36

6 12 18 24 30!6x

y

x

!6, 3"

f !x" ! !x " 6"2 # 3

13.

Vertex:

-intercepts:

–4 4 8 12 16

4

8

12

16

20

x

y

!4, 0"x

!4, 0"

h!x" ! x2 " 8x # 16 ! !x " 4"2 14.

Vertex:

Intercept:

!4 !3 !2 !1 1 2

1

2

3

4

5

6

x

y

!"1, 0"!"1, 0"

g!x" ! x2 # 2x # 1 ! !x # 1"2

15.

Vertex:

-intercepts: None

!2 !1 1 2 3

1

3

4

5

x

y

x

!12, 1"

f!x" ! x2 " x # 54 ! !x " 1

2"2# 1 16.

Vertex:

Intercepts:

–5 –4 –3 –2 –1 1 2

–3

–2

1

2

3

4

x

y

!"32 ± $2, 0"

!"32, "2"

f!x" ! x2 # 3x # 14 ! !x # 3

2"2 " 2

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90 Chapter 2 Polynomial and Rational Functions

17.

Vertex:

-intercepts:

–4 2 6

–4

–2

6

x

y

!1 " $6, 0", !1 # $6, 0"x

!1, 6"

f!x" ! "x2 # 2x # 5 ! "!x " 1"2 # 6 18.

Vertex:

Intercepts:

–6 –5 –3 –2 –1 1 2

–3

–2

1

2

4

5

x

y

!"2 ± $5, 0"!"2, 5"

! "!x # 2"2 # 5

! "1%!x # 2"2 " 5&

f!x" ! "x2 " 4x # 1 ! "1!x2 # 4x " 1"

19.

Vertex:

-intercept: None

–8 –4 4 8

10

20

x

y

x

!12, 20"

h!x" ! 4x2 " 4x # 21 ! 4!x " 12"2

# 20 20.

Vertex:

No -interceptsx

!1!2!3 1 2 3

1

3

4

5

6

x

y!14, 78"

! 2!x " 14"2 # 7

8

! 2!x " 14"2 " 1

8 # 1

! 2!x2 " 12 x " # 1

f!x" ! 2x2 " x # 1

21.

Vertex:

-intercepts:

!10

!6

8

6

!"3, 0", !1, 0"x

!"1, 4"

f!x" ! "!x2 # 2x " 3" ! "!x # 1"2 # 4 22.

Vertex:

Intercepts:

! "!x # 12"2

# 1214

! "!x2 # x # 14" # 1

4 # 30

f !x" ! "!x2 # x " 30"

!5, 0", !"6, 0"

!"12, 121

4 "

!27

!3

27

33

23.

Vertex:

-intercepts: !"4 ±$5, 0"x!13

!6

5

6!"4, "5"

g!x" ! x2 # 8x # 11 ! !x # 4"2 " 5

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24.

Vertex:

Intercepts:

! !x # 5"2 " 11

! !x2 # 10x # 25" " 11

f !x" ! x2 # 10x # 14

!"1.683, 0", !"8.317, 0"

!"5, "11"

!15

!12

6

2 25.

Vertex:

-intercept:

!5

!10

13

2

!4 ± 12$2, 0"x

!4, 1"

! "2!x " 4"2 # 1

! "2!x2 " 8x # 16 " 12"

! "2!x2 " 8x # 312 "

f !x" ! "2x2 # 16x " 31

26.

Vertex:

No -intercepts

! "4!x " 3"2 " 5

! "4!x2 " 6x # 9" # 36 " 41

f !x" ! "4x2 # 24x " 41

x

!3, "5"

!5

!50

10

5 27. is the vertex.

Since the graph passes through the point we have:

Thus, Note that ison the parabola.

!"3, 0"f !x" ! "!x # 1"2 # 4.

"1 ! a

0 ! 4a # 4

0 ! a!1 # 1"2 # 4

!1, 0",

f !x" ! a!x # 1"2 # 4

!"1, 4"

28. is the vertex.

Since the graph passes through we have:

Thus, y ! !x # 2"2 " 1.

1 ! a

4 ! 4a

3 ! 4a " 1

3 ! a!0 # 2"2 " 1

!0, 3",

f!x" ! a!x # 2"2 " 1

!"2, "1" 29. is the vertex.


f !x" ! 1!x # 2"2 # 5 ! !x # 2"2 # 5

1 ! a

4 ! 4a

9 ! a!0 # 2"2 # 5

!0, 9",

f!x" ! a!x # 2"2 # 5

!"2, 5"

30. is the vertex.


f!x" ! "2!x " 4"2 # 1

"2 ! a

"8 ! 4a

"7 ! 4a # 1

"7 ! a!6 " 4"2 # 1

!6, "7",

f!x" ! a!x " 4"2 # 1

!4, 1" 31. is the vertex.


f!x" ! 4!x " 1"2 " 2

4 ! a

16 ! 4a

14 ! 4a " 2

14 ! a!"1 " 1"2 " 2

!"1, 14",

f!x" ! a!x " 1"2 " 2

!1, "2"

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32. is the vertex.


f!x" ! 54 !x # 4"2 " 1

a ! 54

5 ! 4a

4 ! a!"2 # 4"2 " 1

!"2, 4",

f!x" ! a!x # 4"2 " 1

!"4, "1" 33. is the vertex.


f!x" ! "104125!x " 1

2"2# 1

"104125 ! a

"265 ! 25

4 a

"215 ! 25

4 a # 1

"215 ! a!"2 " 1

2"2# 1

!"2, "215 ",

f!x" ! a!x " 12"2

# 1

!12, 1"

34. is the vertex.

Since the graph passes through the point

we have:

f!x" ! "!x # 14"2

" 1

a ! "1

" 116 ! 1

16a

"1716 ! 1

16a " 1

"1716 ! a!0 # 1

4"2" 1

!"1, "1716 ",

f!x" ! a!x # 14"2

" 1

!"14, "1" 35.

-intercepts:

x ! 5 or x ! "1

0 ! !x " 5"!x # 1"

0 ! x2 " 4x " 5

!5, 0", !"1, 0"x

y ! x2 " 4x " 5

39.

-intercepts: !0, 0", !4, 0"x

x ! 0 or x ! 4

0 ! x!x " 4"

0 ! x2 " 4x

!4

!5

8

3

y ! x2 " 4x 40.

-intercepts:

x ! 0, x ! 5

0 ! x!"2x # 10"

0 ! "2x2 # 10x

!0, 0", !5, 0"x

!9

!1

15

15

y ! "2x2 # 10x 41.

-intercepts: !"52, 0", !6, 0"x

x ! "52 or x ! 6

0 ! !2x # 5"!x " 6"

0 ! 2x2 " 7x " 30

!20

!40

20

5

y ! 2x2 " 7x " 30

36.

-intercepts:

x ! 12, "3

0 ! !2x " 1"!x # 3"

0 ! 2x2 # 5x " 3

!12, 0", !"3, 0"x

y ! 2x2 # 5x " 3 37.

-intercept:

x ! "4

0 ! !x # 4"2

0 ! x2 # 8x # 16

!"4, 0"x

y ! x2 # 8x # 16 38.

-intercept:

x ! 3

0 ! !x " 3"2

0 ! x2 " 6x # 9

!3, 0"x

y ! x2 " 6x # 9

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42.

-intercepts:

x ! "7, 34

! !x # 7"!4x " 3"

0 ! 4x2 # 25x " 21

!"7, 0", !0.75, 0"x

!8

!70

2

10

y ! 4x2 # 25x " 21 43.

-intercepts: !"1, 0", !7, 0"x

x ! "1, 7

0 ! !x # 1"!x " 7"

0 ! x2 " 6x " 7

0 ! "12!x2 " 6x " 7"

!6

!3

12

9

y ! "12!x2 " 6x " 7" 44.

-intercepts:

x ! 3, "15

! !x " 3"!x # 15"

0 ! x2 # 12x " 45

0 ! 710 !x2 # 12x " 45"

!3, 0", !"15, 0"x

!18

!60

6

10

y ! 710 !x2 # 12x " 45"

45. opens upward

opens downward

Note: has -intercepts for all real numbers a $ 0.!"1, 0" and !3, 0"

xf!x" ! a!x # 1"!x " 3"

! "x2 # 2x # 3

! "!x2 " 2x " 3"

! "!x # 1"!x " 3"

g !x" ! "%x " !"1"&!x " 3",

! x2 " 2x " 3

! !x # 1"!x " 3"

f!x" ! %x " !"1"&!x " 3", 46.

Many correct answers.

opens upward.

opens downward.

"x2 # 10xf !x" ! "x!x " 10" !

x2 " 10xf !x" ! x!x " 10" !

f !x" ! a!x " 0"!x " 10" ! ax!x " 10"

47. opens upward

opens downward

Note: has -intercepts for all real numbers a $ 0.!"3, 0" and !"1

2, 0"xf!x" ! a!x # 3"!2x # 1"

! "2x2 " 7x " 3

g !x" ! "!2x2 # 7x # 3",

! 2x2 # 7x # 3

! !x # 3"!2x # 1"

! !x # 3"!x # 12"!2"

f!x" ! %x " !"3"&%x " !"12"&!2", 48.

opens downward

Many other answers possible.

g!x" ! "2x2 " x # 10

g!x" ! "f!x",

! 2x2 # x " 10, opens upward

! 2!x # 52"!x " 2"

f!x" ! 2%x " !"52"&!x " 2"

49. Let the first number and the second number. Then the sum is

The product is

The maximum value of the product occurs at the vertex of and is 3025. This happens when x ! y ! 55.P!x"

! "!x " 55"2 # 3025

! "%!x " 55"2 " 3025&

! "!x2 " 110x # 3025 " 3025"

P!x" ! "x2 # 110x

P!x" ! xy ! x!110 " x" ! 110x " x2.

x # y ! 110 ! y ! 110 " x.

y !x !

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51. Let be the first number and be the secondnumber. Then

The product is

Completing the square,

The maximum value of the product occurs at thevertex of the parabola and equals 72. This happenswhen and x ! 24 " 2!6" ! 12.y ! 6

P

! "2!y " 6"2 # 72.

! "2! y2 " 12y # 36" # 72

P ! "2y2 # 24y

P ! xy ! !24 " 2y"y ! 24y " 2y2.

x # 2y ! 24 ! x ! 24 " 2y.yx

52. Let first number and second number. Then The product is

The maximum value of the product is 147, and occurs when and y ! 13 !42 " 21" ! 7.x ! 21

! "13 !x " 21"2 # 147.

! "13 !x2 " 42x # 441" # 147

! "13 !x2 " 42x"

P!x" ! "13 x2 # 14x

14x " 13 x2.P!x" ! xy ! x1

3!42 " x" !

y ! 13 !42 " x".x # 3y ! 42,y !x !

(c) Distance traveled around track in one lap:

(e)

The area is maximum when and

y !200 " 2!50"

%!

100%

.

x ! 50

00

100

2000

y !200 " 2x

%

%y ! 200 " 2x

d ! %y # 2x ! 200

(d) Area of rectangular region:

The area is maximum when and

y !200 " 2!50"

%!

100%

.

x ! 50

! "2%

!x " 50"2 #5000

%

! "2%

!x2 " 100x # 2500 " 2500"

! "2%

!x2 " 100x"

!1%

!200x " 2x2"

A ! xy ! x'200 " 2x% (

53. (a)

y

x (b) Radius of semicircular ends of track:

Distance around two semicircular parts of track:

d ! 2%r ! 2%'12

y( ! %y

r !12

y

50. Let first number and second number.Then, The product is

The maximum value of the product occurs at thevertex of and is This happens whenx ! y ! S#2.

S2#4.P!x"

! "'x "S2(

2#

S2

4

! "'x2 " Sx #S2

4"

S2

4 ( ! "x2 # Sx

P!x" ! Sx " x2

P!x" ! xy ! x!S " x".

x # y ! S, y ! S " x.y !x !

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54. (a)

(b)

Maximum area if

(c)

Maximum if x ! 25, y ! 3313

00

50

2000

A !8x!50 " x"

3

x ! 25, y ! 3313

4x # 3y ! 200 ! y !13

!200 " 4x" ! A ! 2xy ! 2x 13

!200 " 4x" !8x3

!50 " x"

x y Area

2

4

6

8

10

12 2xy ! 121613%200 " 4!12"&

2xy ) 106713%200 " 4!10"&

2xy ! 89613%200 " 4!8"&

2xy ! 70413%200 " 4!6"&

2xy ) 49113%200 " 4!4"&

2xy ! 25613%200 " 4!2"&

x y Area

20

22

24

26

28

30 2xy ! 160013%200 " 4!30"&

2xy ) 164313%200 " 4!28"&

2xy ! 166413%200 " 4!26"&

2xy ! 166413%200 " 4!24"&

2xy ) 164313%200 " 4!22"&

2xy ! 160013%200 " 4!20"&

(d)

The maximum area occurs at the vertex and is5000 3 square feet. This happens when feetand feet. Thedimensions are feet by 33 feet.

(e) The results are the same.

132x ! 50

y ! !200 " 4!25""#3 ! 100#3x ! 25#

! "83

!x " 25"2 #5000

3

! "83

%!x " 25"2 " 625&

! "83

!x2 " 50x # 625 " 625"

! "83

!x2 " 50x"

A !83

x!50 " x"

55. (a)

(b) When feet.

(c) The vertex occurs at

The maximum height is

feet.

(d) Using a graphing utility, the zero of occurs ator 228.6 feet from the punter.x ) 228.6,

y

) 104.0

y !"162025 '3645

32 (2#

95 '3645

32 ( #32

x !"b2a

!"9#5

2!"16#2025" !364532

) 113.9.

y !32

x ! 0,

0 2500

120 56.

The maximum height of the dive occurs at

the vertex,

The height at is

The maximum height of the dive is 16 feet.

"49

!3"2 #249

!3" # 12 ! 16.

x ! 3

x !"b2a

! "24#9

2!" 4#9" ! 3.

y ! "49

x2 #249

x # 12

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57.

From the table, the minimum cost seems to be at

The minimum cost occurs at the vertex.

is the minimum cost.

Graphically, you could graph in the window and find the vertex!20, 700".

%0, 40& & %0, 1000&C ! 800 " 10x # 0.25x2

C!20" ! 700

x !"b2a

! "!"10"2!0.25" !

100.5

! 20

x ! 20.

C ! 800 " 10x # 0.25x2

x 10 15 20 25 30

C 725 706.25 700 706.25 725

58. (a)

(b) The parabola intersects at Thus, the maximum speed is 59.4 mph.Analytically,

Using the Quadratic Formula,

The maximum speed is the positive root,59.4 mph.

!"50 ± $82,732

4) "84.4, 59.4.

s !"50 ± $502 " 4!2"!"10,029"

2!2"

2s2 # 50s " 10,029 ! 0.

2s2 # 50s " 29 ! 10,000

0.002s2 # 0.05s " 0.029 ! 10

s ) 59.4.y ! 10

0.002s2 # 0.05s " 0.029

00

100

25

59. (a)

(b) The vertex occurs at

(c)

(d) Answers will vary.

! $14,400 thousand

R!24" ! "25!24"2 # 1200!24"

p !"b2a

!"12002!"25" ! $24.

! $13,500 thousand

R!30" ! "25!30"2 # 1200!30"

! $14,375 thousand

R!25" ! "25!25"2 # 1200!25"

! $14,000 thousand

R!20" ! "25!20"2 # 1200!20"

61.

(a)

(b) The maximum consumption per year of 4306 cigarettes per person per year occurred in 1960 Answers will vary.!t ! 0".

0 440

5000

!t ! 0 corresponds to 1960."

0 " t " 44C!t" ! 4306 " 3.4t " 1.32t2,

60. (a)

(b) The vertex occurs at

The price is $6.25 per pet.

(c) The maximum revenue is


R'254 ( ! "12'25

4 (2

# 150'254 ( ! $468.75.

p !"b2a

!"150

2!"12" !254

! 6.25.

R!8" ! "12!8"2 # 150!8" ! $432

R!6" ! "12!6"2 # 150!6" ! $468

R!4" ! "12!4"2 # 150!4" ! $408

(c) For 2000,

cigarettes per smoker per year,

cigarettes per smoker per day8909365

) 24

!2058"209,117,00048,306,000

) 8909

C(40) ! 2058.

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62.

(a) The vertex is or 1993.

(b) For 2004, and orClearly the model is not

accurate past 2003.

(c) Probably not. Answers will vary.

"$78,000,000.S ) "78,t ! 14

"b2a

!"218.1

2!"28.4" ) 3.8,

0 " t " 14S ! "28.40t2 # 218.1t # 2435,

65. The parabola opens downward and the vertex isMatches (c) and (d).!"2, "4".

66. The parabola opens upward and the vertex is Matches (a).!1, 3".

67. For is a

maximum when In this case, the

maximum value is Hence,

b ! ±20.

400 ! b2

"100 ! 300 " b2

25 ! "75 "b2

4!"1"

c "b2

4a.

x !"b2a

.

f!x" ! a'x #b2a(

2

# 'c "b2

4a(a < 0, 68. For is a

maximum when In this case, the maximum

value is Hence,

b ! ±16.

b2 ! 256

"192 ! 64 " b2

48 ! "16 "b2

4!"1"

c "b2

4a.

x !"b2a

.

f!x" ! a'x #b2a(

2

# 'c "b2

4a(a < 0,

69. For is a

minimum when In this case, the minimum

value is Hence,

b ! ±8.

b2 ! 64

40 ! 104 " b2

10 ! 26 "b2

4

c "b2

4a.

x !"b2a

.

f!x" ! a'x #b2a(

2

# 'c "b2

4a(a > 0, 70. For is a

minimum when In this case, the minimum

value is Hence,

b ! ±10.

b2 ! 100

"200 ! "100 " b2

"50 ! "25 "b2

4

c "b2

4a.

x !"b2a

.

f!x" ! a'x #b2a(

2

# 'c "b2

4a(a > 0,

63. True

impossible 12x2 ! "1,

"12x2 " 1 ! 0

64. True. For

For

In both cases, is the axis of symmetry.x ! "54

"b2a

!"302!12" !

"3024

!"54

.g!x",

"b2a

! ""10

2!"4" ! "108

! "54

.f !x",

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73.

Then and

!1.2, 6.8"

y ! 8 " 65 ! 34

5 .! x ! 65"2

3 x # y ! "23 x # !8 " x" ! 6 ! "5

3 x ! "2

x # y ! 8 ! y ! 8 " x

72.

on graph:

on graph:

From the first equation,

Thus, and hence and .y ! "x2 # 5x " 4b ! 5,0 ! 16a # 4!4 " a" " 4 ! 12a # 12 ! a ! "1

b ! 4 " a.

0 ! 16a # 4b " 4!4, 0"

0 ! a # b " 4!1, 0"

y ! ax2 # bx " 4

74.

The graphs intersect at !4, 2".

x ! 4

11x ! 44

12x " 40 ! x # 4

y ! 3x " 10 ! 14x # 1

75.

Thus, and are the points of intersection.!2, 5"!"3, 0"

x ! "3, x ! 2

!x # 3"!x " 2" ! 0

x2 # x " 6 ! 0

y ! x # 3 ! 9 " x2

76.

The graphs intersect at !2, 11".

x ! 2

!x " 2"!x2 # 2x # 8" ! 0

x3 # 4x " 16 ! 0

y ! x3 # 2x " 1 ! "2x # 15 77. Answers will vary. (Make a Decision)

71. Model (a) is preferable. means the parabola opens upward and profits are increasing for to the right of the vertex,

t # "b

!2a".

ta > 0

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Section 2.2 Polynomial Functions of Higher Degree

Section 2.2 Polynomial Functions of Higher Degree 99

! You should know the following basic principles about polynomials.

! is a polynomial function of degree n.

! If f is of odd degree and

(a) then

1.

2.

(b) then

1.

2.

! If f is of even degree and

(a) then

1.

2.

(b) then

1.

2.

! The following are equivalent for a polynomial function.

(a) is a zero of a function.

(b) is a solution of the polynomial equation

(c) is a factor of the polynomial.

(d) is an -intercept of the graph of

! A polynomial of degree has at most distinct zeros.

! If is a polynomial function such that and then takes on every value between andin the interval

! If you can find a value where a polynomial is positive and another value where it is negative, then there isat least one real zero between the values.

!a, b".f #b$f #a$ff#a$ ! f #b$,a < bf

nn

f.x#a, 0$#x " a$

f#x$ # 0.x # a

x # a

f#x$ ! "$ as x ! "$.

f#x$ ! "$ as x ! $.

an < 0,

f#x$ ! $ as x ! "$.

f#x$ ! $ as x ! $.

an > 0,

f#x$ ! $ as x ! "$.

f#x$ ! "$ as x ! $.

an < 0,

f#x$ ! "$ as x ! "$.

f#x$ ! $ as x ! $.

an > 0,

f#x$ # anxn % an"1xn"1 % . . . % a2x2 % a1x % a0, an ! 0,

1. is a line with -intercept Matches graph (f).

#0, 3$.y f#x$ # "2x % 3 2. is a parabola with intercepts and and opens upward. Matches graph (h).#4, 0$

#0, 0$ f#x$ # x2 " 4x

3. is a parabola with -interceptsand opens downward. Matches

graph (c).#0, 0$ and #"5

2, 0$x f#x$ # "2x2 " 5x 4. has intercepts

and Matches graph (a).

#"12 % 1

2%3, 0$.#"12 " 1

2%3, 0$#0, 1$, #1, 0$, f#x$ # 2x3 " 3x % 1

5. has intercepts

Matches graph (e).#0, 0$ and #±2%3 , 0$. f #x$ # "1

4 x4 % 3x2 6. has -intercept Matches graph (d).

#0, "43$.y f#x$ # "1

3x3 % x2 " 43

Vocabulary Check1. continuous 2. Leading Coefficient Test 3. relative extrema

4. solution, x-intercept 5. touches, crosses 6. Intermediate Value#x " a$,

n, n " 1,

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9.

(a)

Horizontal shift twounits to the right

(c)

Reflection in the -axis and a vertical

shrinkx

–4 –3 –2 2 3 4

–4

–3

–2

1

2

3

4

x

yf#x$ # "12x3

–3 –2 2 3 4 5

–4

–3

–2

1

2

3

4

x

yf#x$ # #x " 2$3

y # x3

(b)

Vertical shift two unitsdownward

(d)

Horizontal shift twounits to the right and a vertical shift two unitsdownward

–3 –2 1 2 4 5

–5

–4

–3

–2

1

2

3

x

yf#x$ # #x " 2$3 " 2

–4 –3 –2 2 3 4

–5

–4

1

2

3

x

yf#x$ # x3 " 2

10.

(a)

Horizontal shift five units to the left

(c)

Reflection in the -axis and then a vertical shift four units upward

x

–4 –3 –2 1 2 3 4

–2

1

2

3

5

6

x

–1

y

f#x$ # 4 " x4

543

6

21

!2!3!4

2 31!1!3!5!6!7 !2!4x

y

f#x$ # #x % 5$4

y # x4

(b)

Vertical shift five units downward

(d)

Horizontal shift one unit to the right and a vertical shrink

–4 –3 –2 –1 1 2 3 4

–2

x

y

f#x$ # 12#x " 1$4

4321

!6

54321!2!3!4!5x

y

f#x$ # x4 " 5

7. has intercepts Matches graph (g).

#0, 0$ and #"2, 0$.f#x$ # x4 % 2x3 8. has intercepts Matches (b).#"1, 0$, #3, 0$, #"3, 0$.

#0, 0$, #1, 0$,f#x$ # 15x5 " 2x3 % 9

5x

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11.

!12

!8

12

8

f

g

f#x$ # 3x3 " 9x % 1; g#x$ # 3x3 12.

!12

!8

12

8

gf

f#x$ # "13 #x3 " 3x % 2$, g#x$ # "1

3 x3

13.

!8

!20

8

12

f

g

f#x$ # "#x4 " 4x3 % 16x$; g#x$ # "x4 14.

!9

!4

9

8

g

f

f#x$ # 3x4 " 6x2, g#x$ # 3x4

15.

Degree: 4

Leading coefficient: 2

The degree is even and the leading coefficientis positive. The graph rises to the left and right.

f#x$ # 2x4 " 3x % 1 16.

Degree: 6

Leading coefficient:

The degree is even and the leading coefficient isnegative. The graph falls to the left and right.

"1

h#x$ # 1 " x6

17.

Degree: 2


The degree is even and the leading coefficientis negative. The graph falls to the left and right.

"3

g#x$ # 5 " 72x " 3x2 18.

Degree: 3


The degree is odd and the leading coefficient ispositive. The graph falls to the left and rises to the right.

13

f#x$ # 13 x3 % 5x

19. Degree: 5 (odd)


Falls to the left and rises to the right

63 # 2 > 0

20. Degree: 7 (odd)


Falls to the left and rises to the right

34 > 0

21.

Degree: 2


The degree is even and the leading coefficient isnegative. The graph falls to the left and right.

"23

h#t$ # "23 #t2 " 5t % 3$ 22.

Degree: 3


The degree is odd and the leading coefficient is negative. The graph rises to the left and falls to the right.

"78

f#s$ # "78 #s3 % 5s2 " 7s % 1$

23.

f#$x # ±5

f#x$ # #x % 5$#x " 5$

f#x$ # x2 " 25 24.

x # ±7

# #7 " x$#7 % x$

f#x$ # 49 " x2 25.

(multiplicity 2)h#$t # 3

h#t$ # #t " 3$2

h#t$ # t2 " 6t % 9

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35. (a)

(b)

(c)

t # ±1

# 12#t % 1$#t " 1$#t2 % 1$

g#t$ # 12t4 " 1

2

t # ±1

!6

!2

6

6 36.

(a)

(b) Zeros:

(c)

intercepts: #0, 0$, #±3, 0$x-

x # 0, ±3

0 # 14x3#x2 " 9$

0, ±3

!18

!12

18

12

y # 14x3#x2 " 9$

26.

x # "5 (multiplicity 2)

# #x % 5$2

f #x$ # x2 % 10x % 25 27.

x # "2, 1

# #x % 2$#x " 1$

f #x$ # x2 % x " 2 28.

x # 3, 4

# 2#x " 3$#x " 4$

# 2#x2 " 7x % 12$

f #x$ # 2x2 " 14x % 24

29.

t # 0, 2 (multiplicity 2)

# t#t " 2$2

f #t$ # t3 " 4t2 % 4t 30.

x # "4, 5, 0 (multiplicity 2)

# x2#x % 4$#x " 5$

# x2#x2 " x " 20$

f #x$ # x4 " x3 " 20x2

31.

& 0.5414, "5.5414

x #"5 ± %25 " 4#"3$

2# "

52

±%37

2

#12

#x2 % 5x " 3$

f #x$ #12

x2 %52

x "32

32.

x #25

, "2

#13

#5x " 2$#x % 2$

#13

#5x2 % 8x " 4$

f #x$ #53

x2 %83

x "43

33. (a)

(b)

(c)

x #4 ± %16 " 4

2# 2 ± %3

# 3#x2 " 4x % 1$

f #x$ # 3x2 " 12x % 3

x & 3.732, 0.268

!7

!10

11

2 34.

(a)

(b) Zeros:

(c)

#1 ± %2, 0$# & "0.414, 2.414$

x #2 ± %4 " 4#"1$

2# 1 ± %2

g#x$ # 5#x2 " 2x " 1$

"0.414, 2.414

!8

!11

10

1

g#x$ # 5x2 " 10x " 5

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37. (a)

(b)

(c)

x # 0, ±%2

# x#x2 % 3$#x2 " 2$

# x#x4 % x2 " 6$ f #x$ # x5 % x3 " 6x

x # 0, 1.414, "1.414

!6

!4

6

4 38.

(a)

(b) Zeros: 0,

(c)

#0, 0$, #±%3, 0$ t # 0, ±%3 #& 0, ±1.732$

# t#t2 " 3$2

# t#t4 " 6t2 % 9$

g#t$ # t5 " 6t3 % 9t

±1.732

!9

!6

9

6

g#t$ # t5 " 6t3 % 9t

39. (a)

(b)

(c)

x # ±%5

# 2#x2 % 4$#x % %5 $#x " %5 $ # 2#x4 " x2 " 20$

f #x$ # 2x4 " 2x2 " 40

2.236, "2.236

!10

!45

10

5 40.

(a)

(b) No real zeros

(c)

No real zeros

# 5#x2 % 1$#x2 % 2$ > 0

f #x$ # 5#x4 % 3x2 % 2$

!6

!5

6

50

f #x$ # 5x4 % 15x2 % 10

41. (a)

(b)

(c)

x # ±5, 4

# #x " 5$#x % 5$#x " 4$

# #x2 " 25$#x " 4$

# x2#x " 4$ " 25#x " 4$ f #x$ # x3 " 4x2 " 25x % 100

x # 4, 5, "5

!6

!10

6

130 42.

(a)

(b) Zeros:

(c)

x-intercepts: #"2, 0$, # 12, 0$

x # "2, 12

# #2x " 1$#2x " 1$#x % 2$

# #2x " 1$#2x2 % 3x " 2$

0 # 4x3 % 4x2 " 7x % 2

"2, 12

!9

!2

9

10

y # 4x3 % 4x2 " 7x % 2

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43. (a)

(b)

(c)

(multiplicity 2)x # 0 or x # 52

0 # x#2x " 5$2

0 # 4x3 " 20x2 % 25x

y # 4x3 " 20x2 % 25x

x # 0, 52

!2

!4

6

12 44.

(a)

(b) Zeros:

(c)

Zeros: 0,

#0, 0$, #±1, 0$, #±2, 0$

±1, ±2

# x#x " 2$#x % 2$#x " 1$#x % 1$

# x#x2 " 4$#x2 " 1$

# x#x4 " 5x2 % 4$

y # x5 " 5x3 % 4x

0, ±1, ±2

!9

!6

9

6

y # x5 " 5x3 % 4x

45.

Zeros:

Relative maximum:

Relative minimums:#1.225, "3.5$, #"1.225, "3.5$

#0, 1$

x & ± 0.421, ±1.680

!6

!4

6

4

f#x$ # 2x4 " 6x2 % 1 46.

Real zeros:

Relative maximums:

Relative minimum: #0, 5$

#"2.915, 19.688$#0.915, 5.646$,

"4.142, 1.934

!6

!10

4

25

f #x$ # "38x4 " x3 % 2x2 % 5

47.

Zeros:

Relative maximum:

Relative minimum: #0.324, 5.782$

#"0.324, 6.218$

x & "1.178

!9

!1

9

11

f#x$ # x5 % 3x3 " x % 6 48.

Real zero:

Relative maximum:

Relative minimum: #"1, "5$

#0.111, "2.942$

"1.819

!6

!7

6

1

f #x$ # "3x3 " 4x2 % x " 3

49.

Note: haszeros 0 and 4 for all nonzero real numbers a.

f #x$ # a#x " 0$#x " 4$ # ax#x " 4$

f #x$ # #x " 0$#x " 4$ # x2 " 4x 50. f #x$ # #x % 7$#x " 2$ # x2 % 5x " 14

51.

Note: has zeros 0,and for all nonzero real numbers a."3

"2,f #x$ # ax#x % 2$#x % 3$

f #x$ # #x " 0$#x % 2$#x % 3$ # x3 % 5x2 % 6x 52. f #x$ # #x " 0$#x " 2$#x " 5$ # x3 " 7x2 % 10x

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53.

Note: has zeros4, 3, and 0 for all nonzero real numbers a."3,

f#x$ # a#x4 " 4x3 " 9x2 % 36x$

# x4 " 4x3 " 9x2 % 36x

# #x " 4$#x2 " 9$x

f#x$ # #x " 4$#x % 3$#x " 3$#x " 0$ 54.

Note:has zeros 0, 1, 2 for all nonzero real numbers a.

"2, "1, f #x$ # a x#x % 2$#x % 1$#x " 1$#x " 2$

# x5 " 5x3 % 4x

# x#x4 " 5x2 % 4$

# x#x2 " 4$#x2 " 1$

# x#x % 2$#x % 1$#x " 1$#x " 2$

f #x$ # #x " #"2$$#x " #"1$$#x " 0$#x " 1$#x " 2$

55.

Note: has zeros and for all nonzero real numbers a.1 " %3

1 % %3f#x$ # a#x2 " 2x " 2$

# x2 " 2x " 2

# x2 " 2x % 1 " 3

# #x " 1$2 " #%3 $2

# !#x " 1$ " %3" !#x " 1$ % %3"f#x$ # !x " #1 % %3 $" !x " #1 " %3 $" 56.

Note:has zeros and for all nonzero realnumbers a.

6 " %36 % %3

f #x$ # a#x " #6 % %3$$#x " #6 " %3$$ # x2 " 12x % 33

# x2 " 12x % 36 " 3

# #x " 6$2 " 3

# ##x " 6$ " %3$##x " 6$ % %3$f #x$ # #x " #6 % %3$$#x " #6 " %3$$

57.

Note: has zeros

2, and for all nonzero realnumbers a.

4 " %54 % %5,

f #x$ # a#x " 2$!#x " 4$2 " 5"

# x3 " 10x2 % 27x " 22

# #x " 2$!#x " 4$2 " 5"

# #x " 2$!#x " 4$ " %5"!#x " 4$ % %5 " f #x$ # #x " 2$!x " #4 % %5 $"!x " #4 " %5 $" 58.

Note: has zeros

4, for all nonzero real numbers a.2 ± %7

f#x$ # a#x " 4$#x2 " 4x " 3$

# x3 " 8x2 % 13x % 12

# #x " 4$#x2 " 4x " 3$

# #x " 4$##x " 2$2 " 7$ # #x " 4$##x " 2$ " %7$##x " 2$ % %7$

f #x$ # #x " 4$#x " #2 % %7$$#x " #2 " %7$$

59.

Note: has zeros and for all nonzero real numbers a."1

"2, "2, f #x$ # a#x % 2$2#x % 1$

f #x$ # #x % 2$2#x % 1$ # x3 % 5x2 % 8x % 4 60.

Note: has zeros 3, 2, 2, 2for all nonzero real numbers a.

f #x$ # a#x " 3$#x " 2$3

# x4 " 9x3 % 30x2 " 44x % 24

f #x$ # #x " 3$#x " 2$3

61.

Note: has zeros 3, 3 for all nonzero real numbers a.

"4,"4,f #x$ # a#x % 4$2#x " 3$2

# x4 % 2x3 " 23x2 " 24x % 144

f #x$ # #x % 4$2#x " 3$2 62.

Note: has zeros 0, 0 for all nonzero real numbers a.

"5,"5,"5,f #x$ # a#x % 5$3x2

# x5 % 15x4 % 75x3 % 125x2

f #x$ # #x % 5$3#x " 0$2

63.

Note: has zerosrises to the left, and falls to the right."2,"1,"1,

a < 0,f #x$ # a#x % 1$2#x % 2$2,

# "x3 " 4x2 " 5x " 2

f #x$ # "#x % 1$2#x % 2$ 64.

Note: has zeros4, 4, falls to the left and falls to the right."1,"1,

a < 0,f #x$ # a#x % 1$2#x " 4$2,

# "x4 % 6x3 " x2 " 24x " 16

f #x$ # "#x % 1$2#x " 4$2

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65.

For example,

# "x3 % 3x " 2. f#x$ # "#x % 2$#x " 1$2

!3 !1

!2

!5!6!7

1 2 3 4 5 6 7

21

3

x

y

y = !x3 + 3x ! 2

66.

For example, f#x$ # #x % 2$#x % 1$#x " 1$2.

!4!6!8!10!12!14 2

2

4

6

8

10

x

y

y = x4 + x3 ! 3x2 ! x + 2

67.

!2!3 1 2 3 4 5 6 7

2345

x

y

y = x5 ! 5x2 ! x + 2

68.

!6!10!14!18 2

24

x

y

y = !x4 + x3 + 3x2 ! 6x

69. (a) The degree of is odd and the leading coefficient is 1. The graph falls to the left and rises to the right.

(b)

Zeros:

(c) and (d)

!2

24

!4!6!8

108642!4!6!8x

y

0, 3, "3

f #x$ # x3 " 9x # x#x2 " 9$ # x#x " 3$#x % 3$

f 70. (a) The degree of is even and the leading coefficient is 1. The graph rises to the left and rises to the right.

(b)

Zeros:

(c) and (d)4

3

2

1

!4

431!1!4x

y

0, 2, "2: #0, 0$, #±2, 0$

# x2#x " 2$#x % 2$

g#x$ # x4 " 4x2 # x2#x2 " 4$

g

71. (a) The degree of is odd and the leading coefficient is 1. The graph falls to the left and rises to the right.

(b)

Zeros: 0, 3

(c) and (d)

12345

542 61!2 !1!3!4x

y

f #x$ # x3 " 3x2 # x2#x " 3$

f 72. (a) The degree of is odd and the leading coefficient is 3. The graph falls to the left and rises to the right.

(b)

Zeros: 0, 8

(c) and (d)

2 3 4 5 6 7 9

!175!200!225

2550

y

x

f #x$ # 3x3 " 24x2 # 3x2#x " 8$

f©

Hou

ghto

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ifflin

Com

pany

. All

right

s re

serv

ed.


73. (a) The degree of is even and the leading coefficient is The graph falls to the left and falls to the right.

(b)

Zeros:

(c) and (d)

!4!8!12 4

2

6 8 1210x

y

#±%5, 0$#±2, 0$,±%5:±2,

f#x$ # "x4 % 9x2 " 20 # "#x2 " 4$#x2 " 5$

"1.f 74. (a) The degree is even and the leading coefficient

is The graph falls to the left and falls to the right.

(b)

Zeros: 2:

(c) and (d)

!2!3!4!5 1 3 4 5

121620

x

y

#2, 0$#"1, 0$,"1,

f #x$ # "x6 % 7x3 % 8 # "#x3 % 1$#x3 " 8$

"1.

75. (a) The degree is odd and the leading coefficient (c) and (d)is 1. The graph falls to the left and rises to the right.

(b)

Zeros: #"3, 0$#3, 0$,3, "3:

# #x " 3$#x % 3$2

# #x2 " 9$#x % 3$

x3 % 3x2 " 9x " 27 # x2#x % 3$ " 9#x % 3$!12!20 8 12 16 20

48

x

y

76. (a) The degree is odd and the leading coefficient (c) and (d)is 1. The graph falls to the left and rises to the right.

(b)

Zeros: #±2, 0$"2, 2:

# #x % 2$#x2 " 2x % 4$#x " 2$#x % 2$

# #x3 % 8$#x2 " 4$

x5 " 4x3 % 8x2 " 32 # x3#x2 " 4$ % 8#x2 " 4$ !1!3!4!5 1 3 4 5

!8

!32

48

x

y

77.

(a) Falls to left and falls to right;

(b)

zeros

(c) and (d)

!3!4!5 1 2 3 4 5

!4!5

12

!6!7!8

x

y

t # "2, "2, 2, 2 " #"2, 0$, #2, 0$;

g#t$ # "14#t4 " 8t2 % 16$ # "1

4#t2 " 4$2

#"14 < 0$

g#t$ # "14t4 % 2t2 " 4 78.

(a) Rises to right and rises to left;

(b)

Zeros:

(c) and (d)

!3 !2!4!5 1 2 3 4 5!1

123456789

x

y

#3, 0$#"1, 0$,

g#x$ # 110#x % 1$2#x " 3$2

# 110 > 0$

g#x$ # 110#x4 " 4x3 " 2x2 % 12x % 9$

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(c)

79.

(a)

The function has three zeros.They are in the intervals

and

(b) Zeros: "0.879, 1.347, 2.532

#2, 3$.#"1, 0$, #1, 2$

!5

!3

7

5

f #x$ # x3 " 3x2 % 3

(c)

81.

(a)

The function has two zeros.They are in the intervals

and

(b) Zeros: "1.585, 0.779

#0, 1$.#"2, "1$

!6

!5

6

3

g#x$ # 3x4 % 4x3 " 3

(c) x

0.2768

0.09515

"0.7356"1.54

"0.5795"1.55

"0.4184"1.56

"0.2524"1.57

"0.0812"1.58

"1.59

"1.6

y1 x

0.75

0.76

0.77

0.78 0.00866

0.79 0.14066

0.80 0.2768

0.81 0.41717

"0.1193

"0.2432

"0.3633

y1

x

0.0708

0.14514

0.21838

0.2905"0.84

"0.85

"0.86

"0.87

"0.0047"0.88

"0.0813"0.89

"0.159"0.9

y1 x

1.3 0.127

1.31 0.09979

1.32 0.07277

1.33 0.04594

1.34 0.0193

1.35

1.36 "0.0333

"0.0071

y1 x

2.5

2.51

2.52

2.53

2.54 0.03226

2.55 0.07388

2.56 0.11642

"0.0084

"0.0482

"0.087

"0.125

y1

80.

(a)

The function has three zeros.They are in the intervals

and

(b) Zeros: 0.642"0.832,"2.810,

#0, 1$.#"1, 0$,#"3, "2$,

!5 5

!6

4

f#x$ # "2x3 " 6x2 % 3

x

0.277

0.137

"0.269"2.79

"0.136"2.80

& 0"2.81

"2.82

"2.83

y1 x

0.010

0.068"0.82

"0.83

"0.048"0.84

"0.107"0.85

"0.166"0.86

y1 x

0.62 0.217

0.63 0.119

0.64 0.018

0.65

0.66 "0.189

"0.084

y1

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83.

No symmetry

Two -interceptsx

f #x$ # x2#x % 6$

!12

!5

8

35 84.

No symmetry

Two -intercepts #0, 0$, #4, 0$x

!3

!10

7

40

h#x$ # x3#x " 4$2 85.

Symmetric about the -axis

Two -interceptsx

y

g#t$ # "12#t " 4$2#t % 4$2

!10

!150

10

10

86.

No symmetry.

Two -intercepts #"1, 0$, #3, 0$x

!6

!6

6

2

g#x$ # 18 #x % 1$2#x " 3$3 87.

Symmetric to origin

Three -interceptsx

# x#x % 2$#x " 2$

f #x$ # x3 " 4x

!9

!6

9

6 88.

Symmetric with respect to -axis

Three -intercepts #±%2, 0$

#0, 0$,x

y

!3

!2

3

2

f #x$ # x4 " 2x2

89.

Three -intercepts

No symmetry

x

g#x$ # 15 #x % 1$2#x " 3$#2x " 9$

!14

!6

16

14 90.

No symmetry; two -intercepts

!6

!3

6

24

x

h#x$ # 15#x % 2$2#3x " 5$2

(c) Because the function is even, we only need to verifythe positive zeros.

82.

(a)

The function has four zeros. They are in theintervals and

(b) Notice that is even. Hence, the zeros come insymmetric pairs. Zeros: ±3.130±0.452,

h

#"4, "3$.#"1, 0$#3, 4$,#0, 1$,

!10

!24

10

4

h#x$ # x4 " 10x2 % 2

x

0.42

0.43

0.44

0.45

0.46

0.47

0.48 "0.2509

"0.1602

"0.0712

0.01601

0.10148

0.18519

0.26712

y1 x

3.09

3.10

3.11

3.12

3.13

3.14

3.15 1.231

0.61571

0.01025

"0.5855

"1.171

"1.748

"2.315

y1

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92. (a)

(b) Domain: 0 < x < 6

# 8x#12 " x$#6 " x$

# #24 " 2x$#24 " 4x$x

V#x$ # length & width & height

93. The point of diminishing returns (where the graph changes from curving upward to curvingdownward) occurs when The point is which corresponds to spending$2,000,000 on advertising to obtain a revenue of $160 million.

#200, 160$x # 200.

94.

Point of Diminishing Returns:15.2 years

#15.2, 27.3$

00

35

60 95.

The model is a good fit.

4 150

250

96.

4 150

200 97. For 2010, and

thousand

thousand.

Answers will vary.

y2 & $285.0

y1 & $730.2

t # 20,

98. Answers will vary. 99. True. has only one zero, 0.f #x$ # x6

91. (a) Volume

Because the box is made from a square,

Thus:

# #36 " 2x$2x Volume # #length$2 & height

length # width.

# length & width & height (b) Domain:

18 > x > 0

"36 < "2x < 0

0 < 36 " 2x < 36

(c) Height, Length and Width Volume,

1

2

3

4

5

6

7

Maximum volume 3456 for x # 6

7!36 " 2#7$"2 # 338836 " 2#7$

6!36 " 2#6$"2 # 345636 " 2#6$

5!36 " 2#5$"2 # 338036 " 2#5$

4!36 " 2#4$"2 # 313636 " 2#4$

3!36 " 2#3$"2 # 270036 " 2#3$

2!36 " 2#2$"2 # 204836 " 2#2$

1!36 " 2#1$"2 # 115636 " 2#1$

Vx (d)

when is maximum.V#x$x # 6

5 73300

3500

(c)

Maximum occurs at x & 2.54.

1 2 3 4 5 6

120

240

360

480

600

720

x

V

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115.

or

or

or x # "12 x $ 1

!x # "12 and x # 1" !x $ "1

2 and x $ 1"

!2x % 1 # 0 and x " 1 # 0" !2x % 1 $ 0 and x " 1 $ 0"

#2x % 1$#x " 1$ $ 0

2x2 " x " 1 $ 0 210!2 !1

!2

x

1 2x2 " x $ 1

116.

and or and

and or and

impossible"26 # x < 7

x > 7"!x # "26x < 7"!x $ "26

x " 7 > 0"!x % 26 # 0x " 7 < 0"!x % 26 $ 0

x % 26x " 7

# 0

5x " 2 " 4#x " 7$

x " 7# 0

5x " 2x " 7

" 4 # 0

13 26 390!13!26!39x

7 5x " 2x " 7

# 4

111. ' fg (#"1.5$ #

f #"1.5$g#"1.5$ #

"2418

# "43

112. # f ' g$#"1$ # f #g#"1$$ # f#8$ # 109

113. #g ' f $#0$ # g# f #0$$ # g#"3$ # 8#"3$2 # 72 114.

"8 < x

3x " 15 < 4x " 720! 2! 4! 6! 8! 10

x3#x " 5$ < 4x " 7

100. True. The degree is odd and the leading coefficient is "1.

101. False. The graph touches at but does notcross the -axis there.x

x # 1,

102. False. The graph crosses the x-axis at and x # 0.

x # "3 103. True. The exponent of is odd #3$.#x % 2$

104. False. The graph rises to the left, and rises to theright.

105. The zeros are 0, 1, 1, and the graph rises to theright. Matches (b).

106. The zeros are 0, 0, 2, 2, and the graph falls to theright. Matches (e).

107. The zeros are 1, 1, and the graph rises tothe right. Matches (a).

"2,"2,

108.

# "59 % 128 # 69

# f % g$#"4$ # f#"4$ % g#"4$ 109.

# 72 " 39 # 33

#g " f $#3$ # g#3$ " f #3$ # 8#3$2 " !14#3$ " 3"

110. # "140849

& "28.7347# #"11$'8 ( 1649 (# f ' g$'"

47( # f'"

47(g'"

47(

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Section 2.3 Real Zeros of Polynomial Functions

You should know the following basic techniques and principles of polynomial division.

! The Division Algorithm (Long Division of Polynomials)

! Synthetic Division

! is equal to the remainder of divided by

! if and only if is a factor of

! The Rational Zero Test

! The Upper and Lower Bound Rule

f!x".!x ! k"f!k" " 0

!x ! k".f!x"f!k"

1.

2x2 # 10x # 12x # 3

" 2x # 4, x $ !3

0

4x # 12

4x # 12

2x2 # 6x

x # 3 ) 2x2 # 10x # 12

2x # 4 2.

5x2 ! 17x ! 12x ! 4

" 5x # 3, x $ 4

0

3x ! 12

3x ! 12

5x2 ! 20x

x ! 4 ) 5x2 ! 17 x ! 12

5x # 3

3.

x4 # 5x3 # 6x2 ! x ! 2x # 2

" x3 # 3x2 ! 1, x $ !2

0

!x ! 2

!x ! 2

3x3 # 6x2

3x3 # 6x2

x4 # 2x3

) x4 # 5x3 # 6x2 ! x ! 2x # 2

x3 # 3x2 ! 1 4.

x3 ! 4x2 ! 17x # 6x ! 3

" x2 ! x ! 20 !54

x ! 3

!54

!20x # 60

!20x # 6

!x2 # 3x

!x2 ! 17x

x3 ! 3x2

x ! 3 ) x3 ! 4x2 ! 17x # 6

x2 ! x ! 20

117.

or

or x ! !24 x " 8

x # 8 ! !16 x # 8 " 16

#x # 8# " 161680!8!16!24!32

x #x # 8# ! 1 " 15

Vocabulary Check1. is the dividend, is the divisor, is the quotient, and is the remainder.

2. improper, proper 3. synthetic division 4. Rational Zero

5. Descartes’s Rule, Signs 6. Remainder Theorem 7. upper bound, lower bound

r!x"q(x)d!x"f!x"

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Section 2.3 Real Zeros of Polynomial Functions 113

7.

7x3 # 3x # 2

" 7x2 ! 14x # 28 !53

x # 2

!53

28x # 56

28x # 3

!14x2 ! 28x

!14x2

7x3 # 14x2

x # 3 ) 7x3 # 0x2 # 0x # 3

7x2 ! 14x # 28 8.

8x4 ! 52x # 1

" 4x3 ! 2x2 # x !12

!9$2

2x # 1

!92

!x ! 12

!x ! 5

2x2 # x

2x2

!4x3 ! 2x2

!4x3

8x4 # 4x3

2x # 1 ) 8x4 # 0x3 # 0x2 # 0x ! 5

4x3 ! 2x2 # x ! 12

9.

6x3 # 10x2 # x # 82x2 # 1

" 3x # 5 !2x ! 32x2 # 1

!2x # 3

! !10x2 # 0x # 5" 10x2 ! 2x # 8

! !6x3 # 0x2 # 3x" 2x2 # 0x # 1 ) 6x3 # 10x2 # x # 8

3x # 5 10.

x4 # 3x2 # 1x2 ! 2x # 3

" x2 # 2x # 4 #2x ! 11

x2 ! 2x # 3

2x ! 11

4x2 ! 8x # 12

4x2 ! 6x # 1

2x3 ! 4x2 # 6x

2x3 # 0x

x4 ! 2x3 # 3x2

x2 ! 2x # 3 ) x4 # 0x3 # 3x2 # 0x # 1

x2 # 2x # 4

11.

x3 ! 9x2 # 1

" x !x # 9x2 # 1

!x ! 9

x3 # x

x2 # 1 ) x3 # 0x2 # 0x ! 9

x 12.

x5 # 7x3 ! 1

" x2 #x2 # 7x3 ! 1

x2 # 7

x5 ! x2

x3 ! 1" x5 # 0x4 # 0x3 # 0x2 # 0x # 7 x2

5.

4x3 ! 7x2 ! 11x # 54x # 5

" x2 ! 3x # 1, x $ !54

0

! !4x # 5" 4x # 5

! !!12x2 ! 15x" !12x2 ! 11x

! !4x3 # 5x2" 4x # 5 ) 4x3 ! 7x2 ! 11x # 5

x2 ! 3x # 1 6.

x $32

2x3 ! 3x2 ! 50x # 752x ! 3

" x2 ! 25,

0

!50x # 75

!50x # 75

2x3 ! 3x2

2x ! 3 ) 2x3 ! 3x2 ! 50x # 75

x2 ! 25

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15.

x $ 53x3 ! 17x2 # 15x ! 25

x ! 5" 3x2 ! 2x # 5,

5 3

3

!1715

!2

15!10

5

!2525

0

17.

6x3 # 7x2 ! x # 26x ! 3

" 6x2 # 25x # 74 #248

x ! 3

3 6

6

718

25

!175

74

26222

248

19.

9x3 ! 18x2 ! 16x # 32x ! 2

" 9x2 ! 16, x $ 2

2 9

9

!1818

0

!160

!16

32!32

0

21.

x3 # 512x # 8

" x2 ! 8x # 64, x $ !8

!8 1

1

0!8

!8

064

64

512!512

0

23.

x $ !12

4x3 # 16x2 ! 23x ! 15x # 1

2" 4x2 # 14x ! 30,

!12 4

4

16!2

14

!23!7

!30

!1515

0

16.

5x3 # 18x2 # 7x ! 6x # 3

" 5x2 # 3x ! 2, x $ !3

!3 5

5

18!15

3

7!9

!2

!66

0

18.

2x3 # 14x2 ! 20x # 7x # 6

" 2x2 # 2x ! 32 #199

x # 6

!6 2

2

14!12

2

!20!12

!32

7192

199

24.

3x3 ! 4x2 # 5

x ! 32

" 3x2 #12

x #34

#49

8x ! 12

32 3

3

!492

12

034

34

598

498

20.

5x3 # 6x # 8x # 2

" 5x2 ! 10x # 26 !44

x # 2

!2 5

5

0!10

!10

620

26

8!52

!44

22.

x3 ! 729x ! 9

" x2 # 9x # 81, x $ 9

9 1

1

09

9

081

81

!729729

0

13.

2x3 ! 4x2 ! 15x # 5!x ! 1"2 " 2x !

17x ! 5!x ! 1"2

!17x # 5

2x3 ! 4x2 # 2x

x2 ! 2x # 1 ) 2x3 ! 4x2 ! 15x # 5

2x 14.

x4

!x ! 1"3 " x # 3 #6x2 ! 8x # 3

!x ! 1"3

6x2 ! 8x # 3

3x3 ! 9x2 # 9x ! 3

3x3 ! 3x2 # x

x4 ! 3x3 # 3x2 ! x

x3 ! 3x2 # 3x ! 1 ) x4

x # 3

!x ! 1"3 " x3 ! 3x2 # 3x ! 1

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27.

" y1

!9

!6

9

6

"x4 ! 3x2 ! 1

x2 # 5

"x4 ! 8x2 # 5x2 ! 40 # 39

x2 # 5

"!x2 ! 8"!x2 # 5" # 39

x2 # 5

y2 " x2 ! 8 #39

x2 # 528.

" y1 !6 6

!2

6

"x4 # x2 ! 1

x2 # 1

"x2!x2 # 1" ! 1

x2 # 1

y2 " x2 !1

x2 # 1

30.

f !!23" " 34

3

f !x" " !x # 23"!15x3 ! 6x # 4" # 34

3

!23 15

15

10

!10

0

!6

0

!6

0

4

4

14

!83

343

f !x" " 15x4 # 10x3 ! 6x2 # 14, k " !23

32.

f !!%5" " 6

f !x" " !x # %5"!x2 # !2 ! %5"x ! 2%5" # 6

!%5 1

1

2! %5

2 ! %5

!55 ! 2%5

! 2%5

!410

6

29.

f!4" " !0"!26" # 3 " 3

f!x" " !x ! 4"!x2 # 3x ! 2" # 3

4 1

1

!14

3

!1412

!2

11!8

3

f !x" " x3 ! x2 ! 14x # 11, k " 4

31.

f !%2 " " 0!4 # 6%2 " ! 8 " !8

f !x" " !x ! %2 "!x2 # !3 # %2 "x # 3%2 " ! 8

%2 1

1

3%2

3 # %2

!2

2 # 3%2

3%2

!14

6

!8

25.

!15

!10

15

10

" y1

"x2

x # 2

"x2 ! 4 # 4

x # 2

"!x ! 2"!x # 2" # 4

x # 2

y2 " x ! 2 #4

x # 226.

!18 12

!12

8

" y1

"x2 # 2x ! 1

x # 3

"x2 # 2x ! 3 # 2

x # 3

"!x ! 1"!x # 3" # 2

x # 3

y2 " x ! 1 #2

x # 3

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35.(a) 1 2

2

02

2

!72

!5

3!5

!2 " f !1"

f !x" " 2x3 ! 7x # 3

(c) 12 2

2

0

1

1

!712

!132

3

!134

!14 " f !1

2"

(b) !2 2

2

0!4

!4

!78

1

3!2

1 " f !!2"

(d) 2 2

2

04

4

!78

1

32

5 " f !2"

36.

(a)

(b)

(c)

(d) !1 2

2

0!2

!2

32

5

0!5

!5

!15

4

0!4

!4

34

7 " g!!1"

3 2

2

06

6

318

21

063

63

!1189

188

0564

564

31692

1695 " g!3"

1 2

2

02

2

32

5

05

5

!15

4

04

4

34

7 " g!1"

2 2

2

04

4

38

11

022

22

!144

43

086

86

3172

175 " g!2"

g!x" " 2x6 # 3x4 ! x2 # 3

37.

(a) 3 1

1

!53

!2

!7!6

!13

4!39

!35 " h!3"

h!x" " x3 ! 5x2 ! 7x # 4

(c) !2 1

1

!5!2

!7

!714

7

4!14

!10 " h!!2"

(b) 2 1

1

!52

!3

!7!6

!13

4!26

!22 " h!2"

(d) !5 1

1

!5!5

!10

!750

43

4!215

!211 " h!!5"

33.

f !1 ! %3 " " 0

f !x" " !x ! 1 # %3 "&4x2 ! !2 # 4%3 "x ! !2 # 2%3 "'

1 ! %3 4

4

!64 ! 4%3

!2 ! 4%3

!1210 ! 2%3

!2 ! 2%3

!44

0

34.

f!2 # %2" " 0

f !x" " !x ! !2 # %2""!!3x2 # !2 ! 3%2"x # 8 ! 4%2"

2 # %2 !3

!3

8!6 ! 3%2

2 ! 3%2

10!2 ! 4%2

8 ! 4%2

!88

0

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39.

Zeros: 2, !3, 1

" !x ! 2"!x # 3"!x ! 1"

x3 ! 7x # 6 " !x ! 2"!x2 # 2x ! 3"

2 1

1

02

2

!74

!3

6!6

0

41.

Zeros: 12, 2, 5

2x3 " !2x ! 1"!x ! 2"!x ! 5"

2x3 " !x ! 12 "!2x2 ! 14x # 20"

2x3 ! 15x2 # 27x ! 10

12 2

2

!151

!14

27!7

20

!1010

0

40.

Zeros: !4, !2, 6

" !x # 4"!x ! 6"!x # 2"

x3 ! 28x ! 48 " !x # 4"!x2 ! 4x ! 12"

!4 1

1

0!4

!4

!2816

!12

!4848

0

42.

Zeros: 23, 34, 14

" !3x ! 2"!4x ! 3"!4x ! 1"

48x3 ! 80x2 # 41x ! 6 " !x ! 23"!48x2 ! 48x # 9"

23 48

48

!8032

!48

41!32

9

!66

0

43. (a)

(b)

Remaining factors:

(c)

(d) Real zeros: 1

(e)

!8

!3

7

7

12,!2,

f (x) " (x # 2"!2x ! 1"!x ! 1"

!x ! 1"!2x ! 1",

2x2 ! 3x # 1 " !2x ! 1"!x ! 1"

!2 2

2

1!4

!3

!56

1

2!2

0

44. (a)

(b)

Remaining factors:

(c)

(d) Real zeros: 2

(e)

!6

!20

6

30

13,!3,

f (x) " !x # 3"!3x ! 1"!x ! 2"

!x ! 2"!3x ! 1",

3x2 ! 7x # 2 " !3x ! 1"!x ! 2"

!3 3

3

2!9

!7

!1921

2

6!6

0

38.

(a) (b)

(c) (d) !10 4

4

!16!40

!56

7560

567

0!5670

!5670

2056,700

56,720 " f !!10"

5 4

4

!1620

4

720

27

0135

135

20675

695 " f !5"

!2 4

4

!16!8

!24

748

55

0!110

!110

20220

240 " f !!2"

1 4

4

!164

!12

7!12

!5

0!5

!5

20!5

15 " f !1"

f!x" " 4x4 ! 16x3 # 7x2 # 20

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46. (a)

4 8

8

!3032

2

!118

!3

12!12

0

!2 8

8

!14!16

!30

!7160

!11

!1022

12

24!24

0

47. (a)

(b)

Remaining factors:

(c)

(d) Real zeros:

(e)

!9

!20

3

320

!723,!1

2,

f !x" " !2x # 1"!3x ! 2"!x # 7"

!x # 7"!3x ! 2",

6x2 # 38x ! 28 " !3x ! 2"!2x # 14"

!12 6

6

41

!3

38

!9

!19

!28

!14

14

0

48. (a)

(b)

Remaining factors:

(c)

(d) Real zeros:

(e)

!6

!10

6

15

± %512,

f !x" " !2x ! 1"!x # %5"!x ! %5"!x # %5"!x ! %5",

2x2 ! 10 " 2!x ! %5"!x # %5"

12 2

2

!1

1

0

!10

0

!10

5

!5

0

(b)

Remaining factors:

(c)

(d) Real zeros:

(e)!6

!400

6

50

!2, 4, !34, 12

f !x" " !x # 2"!x ! 4"!4x # 3"!2x ! 1"

!4x # 3", !2x ! 1"

8x2 # 2x ! 3 " !4x # 3"!2x ! 1"

49.

factor of

factor of 1

Possible rational zeros:

Rational zeros: !3±1,

f !x" " x2!x # 3" ! !x # 3" " !x # 3"!x2 ! 1"

±3±1,

q "

!3p "

f !x" " x3 # 3x2 ! x ! 3 50.

factor of 16

factor of 1


Rational zeros: 4, ±2

f !x" " x2!x ! 4" ! 4!x ! 4" " !x ! 4"!x2 ! 4"

±16±8,±4,±2,±1,

q "

p "

f !x" " x3 ! 4x2 ! 4x # 16

45. (a)

!4 1

1

1!4

!3

!1012

2

8!8

0

5 1

1

!45

1

!155

!10

58!50

8

!4040

0

(b)

Remaining factors:

(c)

(d) Real zeros:

(e)!7

!200

7

20

5, !4, 2, 1

f !x" " !x ! 5"!x # 4"!x ! 2"!x ! 1"

!x ! 2", !x ! 1"

x2 ! 3x # 2 " !x ! 2"!x ! 1"

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55.

Using a graphing utility and synthetic division,and are rational zeros. Hence,

!y # 6"!y ! 1"2!2y ! 1" " 0 # y " !6, 1, 12.

!61$2, 1,

2y4 # 7y3 ! 26y2 # 23y ! 6 " 054.

Using a graphing utility and synthetic division,and are rational zeros. Hence,

!x ! 6"!x # 5"!x2 # 1" " 0 # x " !5, 6.x " !5x " 6

x4 ! x3 ! 29x2 ! x ! 30 " 0

56.

The real zeros are !2, 0, 1.

x!x ! 1"!x # 2"!x ! 1"!x ! 1" " 0

x!x ! 1"!x # 2"!x2 ! 2x # 1" " 0

!2 1

1

0!2

!2

!34

1

2!2

0

1 1

1

!11

0

!30

!3

5!3

2

!22

0

x!x4 ! x3 ! 3x2 # 5x ! 2" " 0

x5 ! x4 ! 3x3 # 5x2 ! 2x " 0

57.

Using a graphing utility and synthetic division,4, are rational zeros. Hence,

!32.1

2,!3,x " 4,

!x ! 4"!x # 3"!2x ! 1"!2x # 3" " 0 #!3

212,!3,

4x4 ! 55x2 ! 45x # 36 " 0 58.

Using a graphing utility and synthetic division,and 3 are rational zeros. Hence,

3.32,!2,x " !5

2,

!2x # 5"!x # 2"!2x ! 3"!x ! 3" " 0 #

32,!2,

!52,

4x4 ! 43x2 ! 9x # 90 " 0

53.


The only real zeros are and 2. You can verify this by graphing the function f!z" " z4 ! z3 ! 2z ! 4.!1

z4 ! z3 ! 2z ! 4 " !z # 1"!z ! 2"!z2 # 2" " 0

2 1

1

!22

0

20

2

!44

0

!1 1

1

!1!1

!2

02

2

!2!2

!4

!44

0

±1, ±2, ±4

z4 ! z3 ! 2z ! 4 " 0

51.

factor of

factor of 2


Using synthetic division, 3, and 5 are zeros.

Rational zeros: 3, 5, 32!1,

f !x" " !x # 1"!x ! 3"!x ! 5"!2x ! 3"

!1,

±452±15

2 ,±92,±5

2,±32,±1

2,

±45,±15,±9,±5,±3,±1,

q "

!45p "

f !x" " 2x4 ! 17x3 # 35x2 # 9x ! 45 52.

factor of

factor of 4


Using synthetic division, 1 and 2 are zeros.

Rational zeros: 2±12,±1,

f !x" " !x # 1"!x ! 1"!x ! 2"!2x ! 1"!2x # 1"

!1,

±14±1

2,±1,±2,

q "

!2p "

f !x" " 4x5 ! 8x4 ! 5x3 # 10x2 # x ! 2

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63.

(a)

From the calculator we have and

(b)

(c)

The exact roots are x " 0, 3, 4, ±%2.

" x!x ! 3"!x ! 4"!x ! %2 "!x # %2 "h!x" " x!x ! 3"!x ! 4"!x2 ! 2"

4 1

1

!44

0

!20

!2

8!8

0

3 1

1

!73

!4

10!12

!2

14!6

8

!2424

0

x ( ±1.414.x " 0, 3, 4

h!x" " x!x4 ! 7x3 # 10x2 # 14x ! 24"

h!x" " x5 ! 7x4 # 10x3 # 14x2 ! 24x

65.

4 variations in sign 4, 2 or 0 positive real zeros

0 variations in sign 0 negative real zeros # f !!x" " 2x4 # x3 # 6x2 # x # 5

# f !x" " 2x4 ! x3 # 6x2 ! x # 5

67.

2 variations in sign 2 or 0 positive real zeros

1 variation in sign 1 negative real zero # g!!x" " !4x3 # 5x # 8

# g!x" " 4x3 ! 5x # 8

64.

(a)

(b)

" !x ! 3"!x # 3"!3x ! 1"!2x ! 3"

g!x" " !x ! 3"!x # 3"!6x2 ! 11x # 3"

!3 6

6

7!18

!11

!3033

3

9!9

0

3 6

6

!1118

7

!5121

!30

99!90

9

!2727

0

x " ±3.0, 1.5, 0.333

g!x" " 6x4 ! 11x3 ! 51x2 # 99x ! 27

66.

3 sign changes or 1 positive zeros

1 sign change negative zero # 1

f !!x" " 3x4 ! 5x3 ! 6x2 ! 8x ! 3

# 3

f !x" " 3x4 # 5x3 ! 6x2 # 8x ! 3

68.

1 sign change positive zero

No sign change no negative zeros #

g!!x" " !2x3 ! 4x2 ! 5

# 1

g!x" " 2x3 ! 4x2 ! 5

59.

Using a graphing utility and synthetic division, 1,and are rational zeros. Hence,

!52.3

2,!2,!1,x " 1,!x ! 1"!x # 1"!x # 2"!2x ! 3"!2x # 5" " 0 #

!52

32,!2,!1,

4x5 # 12x4 ! 11x3 ! 42x2 # 7x # 30 " 0 60.

Using a graphing utility and synthetic division, 1,and are rational zeros. Hence,

x " 1, !1, !1, 32, !52.

!x ! 1"!x # 1"2!2x ! 3"!2x # 5" " 0 #!5

232,!1,!1,

4x5 # 8x4 ! 15x3 ! 23x2 # 11x # 15 " 0

61.

(a) Zeros:

(b)is a zero.

(c)

" !t # 2"&t ! !%3 # 2"'&t # !%3 ! 2"' h!t" " !t # 2"!t2 ! 4t # 1"

t " !2!2 1

1

!2!2

!4

!78

1

2!2

0

!2, 3.732, 0.268

h!t" " t3 ! 2t2 ! 7t # 2 62.

(a) Zeros: 6, 5.236, 0.764

(b)

" !s ! 6"!s ! 3 ! %5"!s ! 3 # %5"f!s" " !s ! 6"!s2 ! 6s # 4"

6 1

1

!126

!6

40!36

4

!2424

0

f!s" " s3 ! 12s2 # 40s ! 24

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72. (a)


2 sign changes or 2 negative zeros

(b)

(c)

(d) Zeros: ±2, ±12

!6

!15

6

9

±14, ±1

2, ±1, ±2, ±4

# 0

f !!x" " 4x4 ! 17x2 # 4

# 0

f !x" " 4x4 ! 17x2 # 471.

(a) has 3 variations in sign 3 or 1 positivereal zeros.

has 1variation in sign 1 negative real zero.

(b) Possible rational zeros:

(c)

(d) Real zeros: !12, 1, 2, 4

!4

!8

8

16

±8±4,±2,±1,±12,

# f !!x" " !2x4 ! 13x3 ! 21x2 ! 2x # 8

# f !x"

f !x" " !2x4 # 13x3 ! 21x2 # 2x # 8

73.

(a) has 2 variations in sign 2 or 0 positivereal zeros.

has 1variation in sign 1 negative real zero.


(c)

(d) Real zeros: 1, 34

, !18

!4

!2

4

6

±3±32,±3

4,±38,± 3

16,± 332,±1,±1

2,

±14,±1

8,± 116,± 1

32,

# f !!x" " !32x3 ! 52x2 ! 17x # 3

# f !x"

f !x" " 32x3 ! 52x2 # 17x # 3 74. (a)

1 sign change 1 positive zero

2 sign changes or 2 negative zeros

(b)

(c)

(d) Zeros: !2, 18

±%145

8

!8

!24

8

8

±34, ±9

4

±1, ±2, ±3, ±6, ±9, ±18, ±12, ±3

2, ±92, ±1

4

# 0

f !!x" " !4x3 # 7x2 # 11x ! 18

#

f !x" " 4x3 # 7x2 ! 11x ! 18

70. (a)


0 sign changes negative zeros

(b)

(c)

(d) Zeros: 23, 2, 4

!6

!4

12

8

±13, ±2

3, ±43, ±8

3, ±163 ; ±1, ±2, ±4, ±8, ±16

# 0

f !!x" " 3x3 # 20x2 # 36x # 16

# 3

f !x" " !3x3 # 20x2 ! 36x # 1669.

(a) has 1 variation in sign 1 positive realzero.

has 2 variationsin sign 2 or 0 negative real zeros.


(c)

(d) Real zeros: !2, !1, 2

!6

!7

6

1

±1, ±2, ±4

# f !!x" " !x3 # x2 # 4x ! 4

# f !x"

f !x" " x3 # x2 ! 4x ! 4

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77.

5 is an upper bound.

is a lower bound.

Real zeros: 2!2,

!3

!3 1

1

!4!3

!7

021

21

16!63

!47

!16141

125

5 1

5

!425

21

0105

105

16525

541

!162705

2689

f !x" " x4 ! 4x3 # 16x ! 16 78.


is a lower bound.

Real zeros: 0.380, 1.435

!4

!4 2

2

0!8

!8

032

32

!8!128

!136

3544

547

3 2

2

06

6

018

18

!854

46

3138

141

f !x" " 2x4 ! 8x # 3

79.

The rational zeros are and ±2.±32

" 14!2x # 3"!2x ! 3"!x # 2"!x ! 2"

" 14!4x2 ! 9"!x2 ! 4"

" 14!4x4 ! 25x2 # 36"

P!x" " x4 ! 254 x2 # 9

81.

The rational zeros are 14 and ±1.

" 14!4x ! 1"!x # 1"!x ! 1"

" 14!4x ! 1"!x2 ! 1"

" 14&x2!4x ! 1" ! 1!4x ! 1"'

" 14!4x3 ! x2 ! 4x # 1"

f !x" " x3 ! 14x2 ! x # 1

4

80.


Rational zeros: !3, 12, 4

" 12!x ! 4"!2x ! 1"!x # 3"

f !x" " 12!x ! 4"!2x2 # 5x ! 3"

4 2

2

!38

5

!2320

!3

12!12

0

±12, ±3

2±1, ±2, ±3, ±4, ±6, ±12,

f !x" " 12!2x3 ! 3x2 ! 23x # 12"

82.


Rational zeros: !2, !13, 12

" 16!z # 2"!3z # 1"!2z ! 1"

f !x" " 16!z # 2"!6z2 ! z ! 1"

!2 6

6

11!12

!1

!32

!1

!22

0

±1, ±2, ±12, ±1

3, ±23, ±1

6

f !z" " 16!6z3 # 11z2 ! 3z ! 2"

75.


is a lower bound.

Real zeros: 1.937, 3.705

!1

!1 1

1

!4!1

!5

05

5

0!5

!5

155

20

4 1

1

!44

0

00

0

00

0

150

15

f !x" " x4 ! 4x3 # 15 76.


is a lower bound.

Real zeros: 0.611, 3.041!2.152,

!3

!3 2

2

!3!6

!9

!1227

15

8!45

!37

4 2

2

!38

5

!1220

8

832

40

f !x" " 2x3 ! 3x2 ! 12x # 8

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87.

Using the graph and synthetic division, is a zero:

is a zero of the cubic, so

For the quadratic term, use the Quadratic Formula.

The real zeros are 1, 2 ± %3.!12,

x "4 ± %16 ! 4

2" 2 ± %3

y " !2x # 1"!x ! 1"!x2 ! 4x # 1".

x " 1

y " !x # 12"!2x3 ! 10x2 # 10x ! 2"

!12 2

2

!9

!1

!10

5

5

10

3

!5

!2

!1

1

0

!1$2

y " 2x4 ! 9x3 # 5x2 # 3x ! 1 88.

Using the graph and synthetic division, 1 and are zeros:

Using the Quadratic Formula:

The real zeros are 1, 3 ± 2%2.!2,

x "6 ± %36 ! 4

2" 3 ± 2%2

y " !x ! 1"!x # 2"!x2 ! 6x # 1"

!2

y " x4 ! 5x3 ! 7x2 # 13x ! 2

89.

Using the graph and synthetic division, and are zeros:


The real zeros are 4 ± %17.3$2,!1,

x "8 ± %64 # 4

2" 4 ± %17

y " !!x # 1"!2x ! 3"!x2 ! 8x ! 1"

3$2!1

y " !2x4 # 17x3 ! 3x2 ! 25x ! 3 90.

Using the graph and synthetic division, 2 and are zeros:


The real zeros are 2, 2 ± %6.!1,

x "4 ± %16 # 8

2" 2 ± %6

y " !!x ! 2"!x # 1"!x2 ! 4x ! 2"

!1

y " !x4 # 5x3 ! 10x ! 4

83.

Rational zeros: 1

Irrational zeros: 0

Matches (d).

!x " 1"

" !x ! 1"!x2 # x # 1"

f!x" " x3 ! 1

85.

Rational zeros: 3

Irrational zeros: 0

Matches (b).

!x " 0, ±1"

f !x" " x3 ! x " x!x # 1"!x ! 1"

84.

Rational zeros: 0

Irrational zeros:

Matches (a).

1, !x " 3%2 "

" !x ! 3%2 "!x2 # 3%2x # 3%4 " f !x" " x3 ! 2

86.

Rational zeros:

Irrational zeros:

Matches (c).

2, !x " ±%2 "1, !x " 0"

" x!x # %2 "!x ! %2 " " x!x2 ! 2"

f !x" " x3 ! 2x

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94.

(a) (b) The second air-fuel ratio of 16.89 can be obtainedby finding the second point where the curves y and

intersect.

(c) Solve or

By synthetic division:

(d) The positive zero of the quadratic can be found using the Quadratic Formula.

x "75.75 ! %!!75.75"2 ! 4!!5.05"!2720.75"

2!!5.05" ( 16.89

!5.05x2 ! 75.75x # 2720.75

15 !5.05

!5.05

0!75.75

!75.75

3857!1136.25

2720.75

!40,811.2540,811.25

0

!5.05x3 # 3857x ! 40,811.25 " 0.!5.05x3 # 3857x ! 38,411.25 " 2400

y1 " 2400

130

18

2700

y " !5.05x3 # 3857x ! 38,411.25, 13 ! x ! 18

93. (a) Combined length and width:

Volume

(b)

Dimensions with maximum volume:20 % 20 % 40

00

30

18,000

" 4x2!30 ! x"

" x2!120 ! 4x"

" l & w & h " x2y

4x # y " 120 # y " 120 ! 4x

(c)

Using the Quadratic Formula, or

The value of is not possible because it is negative.15 ! 15%5

2

15 ± 15%52

.x " 15

!x ! 15"!x2 ! 15x ! 225" " 0

15 1

1

!3015

!15

0!225

!225

3375!3375

0

x3 ! 30x2 # 3375 " 0

4x3 ! 120x2 # 13,500 " 0

13,500 " 4x2!30 ! x"

91. (a)

(b)

(c) The model fits the data well.

(d) For 2010, and:

Hence, the population will be about 307.8 million, which seems reasonable.

21 0.0058

0.0058

0.50.1218

0.6218

1.3813.0578

14.4378

4.6303.1938

307.7938

t " 21

!2 220

300

P!t" " 0.0058t3 # 0.500t2 # 1.38t # 4.6 92.

(a)

(b) For 1980, and mines.

For 1990, and:

3102 mines in 1990

(c) Answers will vary.

10 0.232

0.232

!2.112.32

0.21

!261.82.1

!259.7

5699!2597

3102

t " 10

C " 5699t " 0

0 240

6000

C " 0.232t3 ! 2.11t2 ! 261.8t # 5699

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101.

Hence, 56 ! 8k " 0 # k " 7.

4 1

1

!k4

4 ! k

2k16 ! 4k

16 ! 2k

!864 ! 8k

56 ! 8k

102. Use synthetic division:

Since the remainder should be 0, k " 5.15 ! 3k

3 1

1

!k3

3 ! k

2k9 ! 3k

9 ! k

!1227 ! 3k

15 ! 3k

105.

x " !53

, 53

!3x # 5"!3x ! 5" " 0

9x2 ! 25 " 0 106.

x " ±%21

4

x2 "2116

16x2 ! 21 " 0

107.

x " !32

#%32

, !32

!%32

"!3 ±%3

2

"!6 ±%12

4

x "!6 ±%62 ! 4!2"!3"

2!2"

2x2 # 6x # 3 " 0 108.

x "54

, 32

!4x ! 5"!2x ! 3" " 0

8x2 ! 22x # 15 " 0

103. (a)

(b)

(c)

In general,

xn ! 1x ! 1

" xn!1 # xn!2 # . . . # x # 1, x $ 1.

x4 ! 1x ! 1

" x3 # x2 # x # 1, x $ 1

x3 ! 1x ! 1

" x2 # x # 1, x $ 1

x2 ! 1x ! 1

" x # 1, x $ 1 104. You can check polynomial division by multiplyingthe quotient by the divisor and adding the remainder. This should yield the original dividendif the multiplication was performed correctly.

97. The zeros are 1, 1, and The graph falls to the right.

Since

y " !2!x ! 1"2!x # 2" " !2x3 # 6x ! 4

a " !2.f !0" " !4,

a < 0y " a!x ! 1"2!x # 2"

!2. 98. The zeros are 1, and The graph rises to the right.

Since

y " 2!x ! 1"!x # 1"!x # 2" " 2x3 # 4x2 ! 2x ! 4

a " 2.f !0" " !4,

y " a!x ! 1"!x # 1"!x # 2", a > 0

!2.!1,

95. False, is a zero of f.!47 96.

True

12 6

6

13

4

!922

!90

45!45

0

1840

184

492

96

!4848

0

99. f !x" " !!x # 1"!x ! 1"!x # 2"!x ! 2" 100. f !x" " 2!x # 2"!x ! 1"!x ! 2"

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! You should know how to work with complex numbers.

! Operations on complex numbers

(a) Addition:

(b) Subtraction:

(c) Multiplication:

(d) Division:

! The complex conjugate of

! The additive inverse of

! The multiplicative inverse of

! !!a " !a i for a > 0.

a ! bia2 # b2.

a # bi is

a # bi is !a ! bi.

"a # bi#"a ! bi# " a2 # b2

a # bi is a ! bi:

a # bic # di

"a # bic # di $ c ! di

c ! di"

ac # bdc2 # d2 #

bc ! adc2 # d2 i

"a # bi#"c # di# " "ac ! bd# # "ad # bc#i"a # bi# ! "c # di# " "a ! c# # "b ! d#i

"a # bi# # "c # di# " "a # c# # "b # d#i

Section 2.4 Complex Numbers

1.

b " 4

a " !9

a # bi " !9 # 4i 3.

b # 3 " 8 ! b " 5

a ! 1 " 5 ! a " 6

"a ! 1# # "b # 3#i " 5 # 8i2.

b " 5

a " 12

a # bi " 12 # 5i

Vocabulary Check1. (a) ii (b) iii (c) i 2. 3. complex,

4. real, imaginary 5. Mandelbrot Set

a # bi!!1, !1

109.

[Answer not unique]

f"x# " "x ! 0#"x # 12# " x2 # 12x 110.

[Answer not unique]

" x3 ! 6x2 ! 19x # 24

f "x# " "x ! 1#"x # 3#"x ! 8#

112.

[Answer not unique]

" x2 ! 4x # 1

" x2 ! 4x # 4 ! 3

" "x ! 2#2 ! 3

" $"x ! 2# ! !3%$"x ! 2# # !3 % f "x# " $x ! "2 # !3 #%$x ! "2 ! !3 #%111.

[Answer not unique]

" x4 ! 6x3 # 3x2 # 10x

" "x2 # x#"x2 ! 7x # 10#

f"x# " "x ! 0#"x # 1#"x ! 2#"x ! 5#

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Section 2.4 Complex Numbers 127

18. " 10 # 7!2i" "7 # 3# # "3!2 # 4!2 #i"7 #!!18 # # "3 # !!32# " "7 # 3!2 i# # "3 # 4!2 i#

19. 13i ! "14 ! 7i# " 13i ! 14 # 7i " !14 # 20i

20. 22 # "!5 # 8i# ! 10i " "22 ! 5# # "8 ! 10#i " 17 ! 2i

21.

"196

#376

i

"9 # 10

6#

15 # 226

i

&32

#52

i' # &53

#113

i' " &32

#53' # &5

2#

113 'i 22.

" !112

#4730

i

&34

#75

i' ! &56

!16

i' " &34

!56' # &7

5#

16' i

23. "1.6 # 3.2i# # "!5.8 # 4.3i# " !4.2 # 7.5i

24.

( !2.4 # 17.75i

" !2.4 # &12.8 #7!2

2 'i

!"!3.7 ! 12.8i# ! "6.1 ! !!24.5# " 3.7 # 12.8i ! 6.1 #!492

i

25.

" !12 i 2 " "2!3#"!1# " !2!3

!!6 $ !!2 " "!6 i#"!2 i# 26.

" !50 i2 " 5!2"!1# " !5!2

!!5 $ !!10 " "!5 i#"!10 i#

27. "!!10 #2" "!10 i #2

" 10i2 " !10 28. "!!75 #2" "!75 i#2

" 75i2 " !75

17. "!1 # !!8# # "8 ! !!50# " 7 # 2!2i ! 5!2i " 7 ! 3!2i

4.

a " 0

a # 6 " 6

b " !52

2b " !5

"a # 6# # 2bi " 6 ! 5i 5.

" 5 # 4i

5 # !!16 " 5 # !16"!1# 6.

" 2 ! 3i

2 ! !!9 " 2 ! !9"!1#

7. !6 " !6 # 0i 8. 8 " 8 # 0i 9. !5i # i2 " !5i ! 1 " !1 ! 5i

10.

" 3 # i

!3i2 # i " !3"!1# # i 11. "!!75#2 " !75 12.

" !11

"!!4#2! 7 " !4 ! 7

15.

" !3 # 3i

"4 # i# ! "7 ! 2i# " "4 ! 7# # "1 # 2#i 16.

" 14 ! 8i

"11 ! 2i# ! "!3 # 6i# " "11 # 3# # "!2 ! 6#i

13. !!0.09 " !0.09 i " 0.3i 14. !!0.0004 " 0.02i

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37. is the complex conjugate of

"4 # 3i#"4 ! 3i# " 16 # 9 " 25

4 # 3i.4 ! 3i 38. The conjugate of is

"7 ! 5i#"7 # 5i# " 49 # 25 " 74

7 # 5i.7 ! 5i


"!6 ! !5i#"!6 # !5i# " 36 # 5 " 41

!6 ! !5i.!6 # !5i 40. The conjugate of is

"!3 # !2i#"!3 ! !2i# " 9 # 2 " 11

!3 ! !2i.!3 # !2i


"!20i#"!!20i# " 20

!!20 "!20i.!!20i 42. The conjugate of is

!!13 "!!13i# " "!13i#"!!13i# " 13

!!13i.!!13 " !13i


"3 ! !2i#"3 # !2i# " 9 # 2 " 11

3 ! !!2 " 3 ! !2i.3 # !2i

44. The conjugate of is

"1 # 2!2i#"1 ! 2!2i# " 1 # 8 " 9

1 ! 2!2i.1 # !!8 " 1 # 2!2i

45.6i

"6i

$!i!i

"!6i!i2

"!6i

1" !6i 46.

!52i $

ii

"!5i!2

"52

i

47.2

4 ! 5i"

24 ! 5i

$4 # 5i4 # 5i

"8 # 10i16 # 25

"841

#1041

i 48.3

1 ! i$

1 # i1 # i

"3 # 3i1 ! i2 "

3 # 3i2

"32

#32

i

35. "4 # 5i#2 ! "4 ! 5i#2 " $"4 # 5i# # "4 ! 5i#%$"4 # 5i# ! "4 ! 5i#% " 8"10i# " 80i

36.

" !8i

" 1 ! 4i # 4i2 ! 1 ! 4i ! 4i2

"1 ! 2i#2 ! "1 # 2i#2 " 1 ! 4i # 4i2 ! "1 # 4i # 4i2#

30.

" 6 ! 22i

" 12 ! 22i ! 6

"6 ! 2i#"2 ! 3i# " 12 ! 18i ! 4i # 6i2

31.

" !20 # 32i

" 32i # 20"!1#

4i"8 # 5i# " 32i # 20i2 32. !3i"6 ! i# " !18i ! 3 " !3 ! 18i

33. "!14 # !10 i#"!14 ! !10 i# " 14 ! 10i2 " 14 # 10 " 24

34.

" "21 # 5!2 # # "7!5 ! 3!10 #i " 21 # !50 # 7!5 i ! 3!10 i

" 21 ! 3!10 i # 7!5i ! !50 i2

"3 # !!5 #"7 ! !!10 # " "3 # !5 i#"7 ! !10 i#

29.

" 5 # i

" 3 # i # 2

"1 # i#"3 ! 2i# " 3 ! 2i # 3i ! 2i2

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53.

" !12

!52

i

"!1 ! 5i

2

"2 ! 2i ! 3 ! 3i

1 # 1

21 # i

!3

1 ! i"

2"1 ! i# ! 3"1 # i#"1 # i#"1 ! i#

54.

"125

#95

i

"12 # 9i

5

"4i ! 2i2 # 10 # 5i

4 ! i2

2i

2 # i#

52 ! i

"2i"2 ! i#

"2 # i#"2 ! i# #5"2 # i#

"2 # i#"2 ! i#

55.

"62949

#297949

i

"62 # 297i

949

"!100 # 72i # 225i # 162

252 # 182

"!4 # 9i25 # 18i $

25 ! 18i25 ! 18i

"!4 # 9i

9 # 18i # 16

i

3 ! 2i#

2i3 # 8i

"3i # 8i2 # 6i ! 4i2

"3 ! 2i#"3 # 8i# 56.

"517

!2017

i

"!i # 1

1!

12 # 3i16 # 1

1 # i

i!

34 ! i

"1 # i

i $!i!i

!3

4 ! i $4 # i4 # i

57. !6i3 # i2 " !6i2i # i2 " !6"!1#i # "!1# " 6i ! 1 " !1 # 6i

58. 4i2 ! 2i3 " !4 # 2i

49.

"3 # 4i

5"

35

#45

i

"4 # 4i # i2

4 # 1

2 # i2 ! i

"2 # i2 ! i

$2 # i2 # i

50.

"22 # 9i

5"

225

#95

i

8 ! 7i1 ! 2i

$1 # 2i1 # 2i

"8 # 16i ! 7i ! 14i2

1 ! 4i2

51.

"!401681

!9

1681i

"!40 ! 9i81 # 402

"i

!9 ! 40i$

!9 # 40i!9 # 40i

i

"4 ! 5i#2 "i

16 ! 25 ! 40i52.

"60169

!25169

i

"!25i # 6025 # 144

5i

"2 # 3i#2 "5i

!5 # 12i $!5 ! 12i!5 ! 12i

59.

" !375!3 i

"!!75 #3" "5!3i#3

" 53"!3 #3 i3 " 125"3!3#"!i# 60. "!!2 #6" "!2 i#6

" 8i 6 " 8i4i2 " !8

61.1i3 "

1i3 $

ii

"ii4 "

i1

" i 62.1

"2i#3 "1

8i3 "1

!8i$

8i8i

"8i

!64i2 "18

i

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67. 5i

68. !3 69. 2 70. !4i

71.

321

!2!3!4!5!6!7

7654321!1!2!3

Realaxis

Imaginaryaxis

4 ! 5i 72.

Realaxis

Imaginaryaxis

54321

!2!3!4!5

21!1!2!3!4!5!6!7!8

!7 # 2i 73.

Realaxis4321!1!2!3!4

4

3

2

1

!2

!3

!4

Imaginaryaxis

3i

74.

54321!1!2!3!4

5

34

21

!2!3!4!5

Realaxis

Imaginaryaxis

!5i 75. 1

Realaxis4321!1!2!3

4

3

2

1

!2

!3

!4

Imaginaryaxis

76.

3

2

1

!2

!3

!4

!5

Realaxis

5

4

321!1!2!3!4!5!6!7

Imaginaryaxis

!6

77. The complex number is in the Mandelbrot Set since for the corresponding Mandelbrot sequence is

which is bounded. Or in decimal form

!0.164291 # 0.477783i, !0.201285 # 0.343009i.0.5i, !0.25 # 0.5i, !0.1875 # 0.25i, !0.02734 # 0.40625i,

!864,513,055

4,294,967,296#

46,037,845134,217,728

i!10,76765,536

#19574096

i,!7

256#

1332

i,12

i, !14

#12

i, !316

#14

i,

c " 12i,1

2i,

64. (a)

(c) "2i#4 " 24i4 " 16"1# " 16

24 " 16 (b)

(d) "!2i#4 " "!2#4i4 " 16"1# " 16

"!2#4 " 16

66. !1 ! 2i65. 4 # 3i

63.

The three numbers are cube roots of 8.

" 8

" !1 ! 3!3 i # 9 # 3!3i

" !1 ! 3!3 i ! 9i2 ! 3!3i 3

"! 1 ! !3i#3" "!1#3 # 3"!1#2"!!3i# # 3"!1#"!!3i#2

# "!!3i#3

" 8

" !1 # 3!3i # 9 ! 3!3i

" !1 # 3!3 i ! 9i2 # 3!3i 3

"! 1 # !3i#3" "!1#3 # 3"!1#2"!3i# # 3"!1#"!3i#2

# "!3i#3

"2#3 " 8

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78. 2

Not bounded. is not in the Mandelbrot Set.c " 2

4.4 % 1012

14462 # 2 " 2,090,918

382 # 2 " 1446

62 # 2 " 38

22 # 2 " 6

79.

"212 ! 66i

68( 3.118 ! 0.971i

z "23 ! 14i8 ! 2i &8 # 2i

8 # 2i'

"8 ! 2i

23 ! 14i

""3 ! 4i# # "5 # 2i#

"5 # 2i#"3 ! 4i#

1z

"1z1

#1z2

"1

5 # 2i#

13 ! 4i

z2 " 3 ! 4i

z1 " 5 # 2i

80.

"11240 # 4630i

877( 12.816 # 5.279i z "

340 # 230i29 # 6i &29 ! 6i

29 ! 6i'

"29 # 6i

340 # 230i

""20 ! 10i# # "9 # 16i#

"9 # 16i#"20 ! 10i#

1z

"1z1

#1z2

"1

9 # 16i#

120 ! 10i

z2 " 20 ! 10i

z1 " 16i # 9

81. False. A real number is equal to its conjugate.a # 0i " a

82. False. i44 # i150 ! i74 ! i109 # i61 " 1 ! 1 # 1 ! i # i " 1

83. False. For example, which is not an imaginary number."1 # 2i# # "1 ! 2i# " 2,

85. True. Let and Then

" z1 z2 .

" a1 # b1i a2 # b2i

" "a1 ! b1i#"a2 ! b2i#

" "a1a2 ! b1b2 # ! "a1b2 # b1a2 #i

" "a1a2 ! b1b2# # "a1b2 # b1a2#i

z1z2 " "a1 # b1i#"a2 # b2i#

z2 " a2 # b2i.z1 " a1 # b1i 86. True. Let and Then

" z1 # z2 .

" a1 # b1i # a2 # b2i

" "a1 ! b1i# # "a2 ! b2i#

" "a1 # a2 # ! "b1 # b2 #i

" "a1 # a2# # "b1 # b2#i

z1 # z2 " "a1 # b1i# # "a2 # b2i#

z2 " a2 # b2i.z1 " a1 # b1i

84. False. For example, which is not an imaginary number."i#"i# " !1,

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7.

Zeros:

has no real zeros and the graph of has no -intercepts.xf!x" f !x"

±#2i, ±#2i

f !x" ! x4 " 4x2 " 4 ! !x2 " 2"2


Section 2.5 The Fundamental Theorem of Algebra

! You should know that if is a polynomial of degree then has at least one zero in the complexnumber system. (Fundamental Theorem of Algebra)

! You should know that if is a complex zero of a polynomial with real coefficients, then isalso a complex zero of

! You should know the difference between a factor that is irreducible over the rationals such as anda factor that is irreducible over the reals such as x2 " 9".!

x2 # 7"!f.

a # bif,a " bi

fn > 0,f

1.

The three zeros are and x ! #3.x ! 0x ! 0,

f !x" ! x2!x " 3" 2.

Zeros: 2, #4, #4, #4

g!x" ! !x # 2"!x " 4"3

3.

Zeros: ±4i#9,

f !x" ! !x " 9"!x " 4i"!x # 4i" 4.

The four zeros are 2, 3i, #3i.t ! 3,

h!t" ! !t # 3"!t # 2"!t # 3i"!t " 3i"

5.

Zeros:

The only real zero of is This corresponds to the -intercept of on the graph.!4, 0"xx ! 4.f!x"

4, ± i

f !x" ! x3 # 4x2 " x # 4 ! x2!x # 4" " 1!x # 4" ! !x # 4"!x2 " 1"

6.

The zeros are and 4. This corresponds to the -intercepts of and on the graph.!4, 0"!#2, 0", !2, 0",xx ! 2, #2,

! !x " 2"!x # 2"!x # 4"

! !x2 # 4"!x # 4"

! x2!x # 4" # 4!x # 4"

f !x" ! x3 # 4x2 # 4x " 16

Vocabulary Check

1. Fundamental Theorem, Algebra 2. Linear Factorization Theorem

3. irreducible, reals 4. complex conjugate

87.

! 16x2 # 25

!4x # 5"!4x " 5" ! 16x2 # 20x " 20x # 25 88.

! x3 " 6x2 " 12x " 8

!x " 2"3 ! x3 " 3x22 " 3x!2"2 " 23

89.

! 3x2 " 232 x # 2

!3x # 12"!x " 4" ! 3x2 # 1

2 x " 12x # 2 90. !2x # 5"2 ! 4x2 # 20x " 25

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Section 2.5 The Fundamental Theorem of Algebra 133

11.

has no rational zeros. By the Quadratic Formula,the zeros are

! !x # 6 # #10 "!x # 6 " #10 " f !x" ! $x # !6 " #10 "%$x # !6 # #10 "%

x !12 ± #!#12"2 # 4!26"

2! 6 ± #10.

f

f !x" ! x2 # 12x " 26 12.


! !x " 3 # #11"!x " 3 " #11"f !x" ! !x # !#3 " #11""!x # !#3 # #11""

x !#6 ± #62 # 4!#2"

2! #3 ± #11.

f

f !x" ! x2 " 6x # 2

13.

The zeros of are x ! ±5i.f!x"

! !x " 5i"!x # 5i"

f!x" ! x2 " 25 14.

Zeros:

f !x" ! !x " 6i"!x # 6i"

±6i

f !x" ! x2 " 36

15.

Zeros: ±32, ±3

2i

! !2x # 3"!2x " 3"!2x " 3i"!2x # 3i"

! !4x2 # 9"!4x2 " 9"

f !x" ! 16x4 # 81 16.

Zeros: ±53, ±5

3i

! !3y " 5i"!3y # 5i"!3y " 5"!3y # 5"

! !9y2 " 25"!9y2 # 25"

f !y" ! 81y4 # 625

17.

f !z" ! &z #12

"#223 i

2 '&z #12

##223 i

2 '

!12

±#223

2i

!1 ±##223

2

z !1 ±#1 # 4!56"

2

f !z" ! z2 # z " 56 18.

Zeros:

h!x" ! !x # 2 " #7 "!x # 2 # #7 "2 ± #7

x !4 ± #16 " 12

2! 2 ± #7

h!x" ! x2 # 4x # 3

8.

Zeros:

The only real zeros are This corresponds to the -intercepts of and on the graph.!2, 0"!#2, 0"xx ! #2, 2.

±2, ± i

! !x " 2"!x # 2"!x2 " 1"

! !x2 # 4"!x2 " 1"

f !x" ! x4 # 3x2 # 4

9.


! !x # 2 # #3 "!x # 2 " #3 "h!x" ! $x # !2 " #3"%$x # !2 # #3"%

x !4 ± #16 # 4

2! 2 ± #3.

h

h!x" ! x2 # 4x " 1 10.

Zeros:

g!x" ! !x " 5 " #2 "!x " 5 # #2 "

x !#10 ± #8

2! #5 ± #2

g!x" ! x2 " 10x " 23

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24.

Zeros:

f !x" ! !x " 1"!x " 5 " 2i"!x " 5 # 2i"

x ! #1, #10 ± #16 i

2! #5 ± 2i

#1 1

1

11#1

10

39#10

29

29#29

0

f !x" ! x3 " 11x2 " 39x " 2923.


By the Quadratic Formula, the zeros ofare

The zeros of are

! !t " 5"!t # 4 # 3i"!t # 4 " 3i"

f!t" ! $t # !#5"%$t # !4 " 3i"%$t # !4 # 3i"%

t ! #5 and t ! 4 ± 3i.f!t"

t !8 ± #64 # 100

2! 4 ± 3i.

t2 # 8t " 25

#5 1

1

#3#5

#8

#1540

25

125#125

0

±1, ±5, ±25, ±125

f!t" ! t3 # 3t2 # 15t " 125

25.


By the Quadratic Formula, the zeros of are those of

Zeros:

! !5x " 1"!x # 1 # #5 i"!x # 1 " #5 i" f !x" ! 5&x "15'!x # !1 " #5 i""!x # !1 # #5 i""

#15

, 1 ±#5 i

x !2 ±#4 # 4!6"

2! 1 ±#5 i

x2 # 2x " 6:5x2 # 10x " 30

#15 5

5

#9#1

#10

282

30

6#6

0

±15

±1,±25

,±2,±35

,±3,±65

,±6,

f !x" ! 5x3 # 9x2 " 28x " 6

19.

The zeros of are x ! ± i and x ! ±3i.f!x"

! !x " i"!x # i"!x " 3i"!x # 3i"

! !x2 " 1"!x2 " 9"

f!x" ! x4 " 10x2 " 9 20.

Zeros:

f !x" ! !x " 2i"!x # 2i"!x " 5i"!x # 5i"

x ! ±2i, ±5i

! !x2 " 25"!x2 " 4"

f !x" ! x4 " 29x2 " 100

21.

Using synthetic division, is a zero:

The zeros are #4i.4i,53,

! !3x # 5"!x " 4i"!x # 4i"

! !3x # 5"!x2 " 16"

f !x" ! !x # 53"!3x2 " 48"

53 3

3

#55

0

480

48

#8080

0

53

f !x" ! 3x3 # 5x2 " 48x # 80 22.

Using synthetic division, is a zero:

The zeros are #5i.5i,23,

! !3x # 2"!x " 5i"!x # 5i"

! !3x # 2"!x2 " 25"

f !x" ! !x # 23"!3x3 " 75"

23 3

3

#22

0

750

75

#5050

0

23

f !x" ! 3x3 # 2x2 " 75x # 50

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29. (a)By the Quadratic Formula,

The zeros are and

(b)

(c) -intercepts: and

(d)

0

!4

15

6

!7 # #3, 0"!7 " #3, 0"x

! !x # 7 # #3"!x # 7 " #3" f !x" ! $x # !7 " #3"%$x # !7 # #3"%

7 # #3.7 " #3

x !14 ± #!#14"2 # 4!46"

2! 7 ± #3.

f !x" ! x2 # 14x " 46.

30. (a)

By the Quadratic Formula,

The zeros are and

(b)

(c) -intercepts:

(d)

!2

!10

13

40

!6 " #2, 0"!6 # #2, 0"x

! !x # 6 # #2"!x # 6 " #2"f !x" ! !x # !6 " #2""!x # !6 # #2""

6 # #2.6 " #2

x !12 ± #!#12"2 # 4!34"

2! 6 ± #2.

f !x" ! x2 # 12x " 34 31. (a)

The zeros are

(b)

(c) -intercept:

(d)

!4 4

!50

40

&32

, 0'x

f !x" ! !2x # 3"!x " 2i"!x # 2i"

±2i.32

,

! !2x # 3"!x2 " 4"

f !x" ! 2x3 # 3x2 " 8x # 12

26.

Factoring the quadratic,

Zeros:

f !s" ! !3s " 2"!s # 1 " #3i"!s # 1 # #3i"

#23

, 1 ± #3i

s !2 ± #4 # 16

2! 1 ± #3i.

! !3s " 2"!s2 # 2s " 4"

f !s" ! 3s3 # 4s2 " 8s " 8 27.


The zeros of are 2, 2, and ±2i.g

! !x # 2"2!x " 2i"!x # 2i"

g!x" ! !x # 2"!x # 2"!x2 " 4"

2 1

1

#22

0

40

4

#88

0

0

2 1 #42

8#4

#168

16#16

±1, ±2, ±4, ±8, ±16

g!x" ! x4 # 4x3 " 8x2 # 16x " 16

28.

Zeros:

h!x" ! !x " 3"2!x " i"!x # i"

x ! #3, ± i

#3 1

1

3#3

0

10

1

3#3

0

0

#3 1 6#3

10#9

6#3

9#9

h!x" ! x4 " 6x3 " 10x2 " 6x " 9

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35. (a)

The zeros are

(b)

(c) No -intercepts

(d)

!5

!10

5

200

x

! !x " 3i"!x # 3i"!x " 4i"!x # 4i"

f !x" ! !x2 " 9"!x2 " 16"

±3i, ±4i.

! !x2 " 9"!x2 " 16"

f !x" ! x4 " 25x2 " 144 36. (a)

The zeros are and 4.

(b)

(c) -intercept:

(d)

!4

!5

8

40

!4, 0"x

f !x" ! !x2 " 1"!x # 4"2

#i, 4i,

! !x2 " 1"!x # 4"2

! !x2 " 1"!x2 # 8x " 16"

f!x" ! x4 # 8x3 " 17x2 # 8x " 16

37.

Note that where isany nonzero real number, has zeros 2, ± i.

af (x) ! a!x3 # 2x2 " x # 2",

! !x3 # 2x2 " x # 2"

! !x # 2"!x2 " 1"

f !x) ! !x # 2"!x # i"!x " i" 38.

Note that where is any nonzero real number, has zeros 3, ±4i.

af (x) ! a!x3 # 3x2 " 16x # 48",

! x3 # 3x2 " 16x # 48

! !x # 3"!x2 " 16"

f !x) ! !x # 3"!x # 4i"!x " 4i"

32. (a)

The zeros are

(b) f !x" ! !2x # 5"!x " 3i"!x # 3i"

±3i.52,

! !2x # 5"!x2 " 9"

f !x" ! 2x3 # 5x2 " 18x # 45

33. (a)

Use the Quadratic Formula to find the zeros of

The zeros are and

(b)

(c) -intercept:

(d)

!10

!300

5

300

!#6, 0"x

f !x" ! !x " 6"!x # 3 " 4i"!x # 3 # 4i"

3 # 4i.#6, 3 " 4i,

x !6 ± #!#6"2 # 4!25"

2! 3 ± 4i.

x2 # 6x " 25.

! !x " 6"!x2 # 6x " 25"

f !x" ! x3 # 11x " 150 34. (a)

Use the Quadratic Formula to find the zeros of

The zeros are and

(b)

(c) -intercept:

(d)

!20

!10

10

10

!#2, 0"x

f !x" ! !x " 2"!x " 4 " i"!x " 4 # i"

#4 # i.#4 " i,#2,

!#8 ± ##4

2! #4 " i

x !#8 ± #82 # 4!17"

2

x2 " 8x " 17.

! !x " 2"!x2 " 8x " 17"

f !x" ! x3 " 10x2 " 33x " 34

(c) -intercept:

(d)

!4 5

!80

40

!52, 0"x

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44. (a)

(b)

! #2x4 " 2x3 " 2x2 " 2x " 4

f !x" ! #2!x2 # x # 2"!x2 " 1"

f !x" ! #2!x " 1"!x # 2"!x # i"!x " i"

f !1" ! 8 ! a!2"!#1"!2" ! a ! #2

! a!x " 1"!x # 2"!x2 " 1"

f !x" ! a!x " 1"! # 2"!x # i"!x " i" 45. (a)

(b)

! #2x3 " 6x2 # 10x # 18

f !x" ! #2!x " 1"!x2 # 4x " 9"

f !x" ! #2!x " 1"!x # 2 # #5i"!x # 2 " #5i"f !#2" ! 42 ! a!#1"!4 " 8 " 9" ! a ! #2

! a!x " 1"!x2 # 4x " 9"

! a!x " 1"!x2 # 4x " 4 " 5"

f !x" ! a!x " 1"!x # 2 # #5i"!x # 2 " #5i"

46. (a)

f !x" ! #2!x " 2"!x # 2 # 2#2i"!x # 2 " 2#2i"f !#1" ! #34 ! a!1"!17" ! a ! #2

! a!x " 2"!x2 # 4x " 12"

! a!x " 2"!x2 # 4x " 4 " 8"

f !x" ! a!x " 2"!x # 2 # 2#2i"!x # 2 " 2#2i" (b)

! #2x3 " 4x2 # 8x # 48

f !x" ! #2!x " 2"!x2 # 4x " 12"

39.

Note that where is any nonzero real number, has zeros 2, 2,4 ± i.

af (x) ! a!x4 # 12x3 " 53x2 # 100x " 68",

! x4 # 12x3 " 53x2 # 100x " 68

! !x2 # 4x " 4"!x2 # 8x " 17"

! !x # 2"2!x # 8x " 16 " 1"

f !x) ! !x # 2"2!x # 4 # i"!x # 4 " i) 40. Because is a zero, so is

Note that where is any nonzero real number, has zeros

2 ± 5i.#1,#1,a

f (x) ! a!x4 # 2x3 " 22x2 " 54x " 29",

! x4 # 2x3 " 22x2 " 54x " 29

! !x2 " 2x " 1"!x2 # 4x " 29"

! !x " 1"2!x2 # 4x " 4 " 25"

f !x) ! !x " 1"2!x # 2 # 5i"!x # 2 " 5i"

2 # 5i.2 " 5i

41. Because is a zero, so is

Note that where isany nonzero real number, has zeros 0, 1 ± #2i.#5,

af !x" ! a!x4 " 3x3 # 7x2 " 15x",

! x4 " 3x3 # 7x2 " 15x

! !x2 " 5x"!x2 # 2x " 3"

! !x2 " 5x"!x2 # 2x " 1 " 2"

f !x" ! !x # 0"!x " 5"!x # 1 # #2i"!x # 1 " #2i"1 # #2i.1 " #2i 42. Because is a zero, so is

Note that where is any nonzero real number, has zeros 0, 4,1 ± #2i.

af !x" ! a!x4 # 6x3 " 11x2 # 12x",

! x4 # 6x3 " 11x2 # 12x

! !x2 # 4x)(x2 # 2x " 1 " 2"

f !x" ! !x # 0"!x # 4"!x # 1 # #2i"!x # 1 " #2i"1 ± #2i.1 " #2i

43. (a)

f !x" ! #!x # 1"!x " 2"!x # 2i"!x " 2i"

f !#1" ! 10 ! a!#2"!1"!5" ! a ! #1

! a!x # 1"!x " 2"!x2 " 4"

f !x" ! a!x # 1"!x " 2"!x # 2i"!x " 2i" (b)

! #x4 # x3 # 2x2 # 4x " 8

! #!x2 " x # 2"!x2 " 4"

f !x" ! #!x # 1"!x " 2"!x2 " 4"

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52.

Since 3i is a zero, so is .

The zeros of f are and #1.3i, #3i

#3i 1

1

1 " 3i#3i

1

3i#3i

0

3i 1

1

13i

1 " 3i

9#9 " 3i

3i

9#9

0

#3i

f !x" ! x3 " x2 " 9x " 9 53. Since is a zero,so is

The zero of is

The zeros of are #3, 5 ± 2i.f

x ! #3.x " 3

5 # 2i 1

1

#2 " 2i5 # 2i

3

#15 " 6i15 # 6i

0

5 " 2i 1

1

#75 " 2i

#2 " 2i

#1#14 " 6i

#15 " 6i

87#87

0

5 # 2i.5 " 2ig!x" ! x3 # 7x2 # x " 87.

47.

(a)

(b)

(c) f !x" ! !x # #7"!x " #7"!x " i"!x # i"

f !x" ! !x # #7"!x " #7"!x2 " 1"

f !x" ! !x2 # 7"!x2 " 1"

f !x" ! x4 # 6x2 # 7 48.

(a)

(b)

(c) f !x" ! !x " 3i"!x # 3i"!x " #3"!x # #3"f !x" ! !x2 " 9"!x " #3"!x # #3"f !x" ! !x2 " 9"!x2 # 3"

f !x" ! x4 " 6x2 # 27

51.

Since is a zero, so is

The zero of The zeros of

are x ! #32 and x ! ±5i.

f2x " 3 is x ! #32.

#5i 2

2

3 " 10i#10i

3

15i#15i

0

5i 2

2

310i

3 " 10i

50#50 " 15i

15i

75#75

0

#5i.5i

f!x" ! 2x3 " 3x2 " 50x " 75 Alternate Solution

Since are zeros of is a factor of By long division we have:

Thus, and the zeros of areand x ! #3

2.x ! ±5iff!x" ! !x2 " 25"!2x " 3"

0

3x2 " 0x " 75

3x2 " 0x " 75

2x3 " 0x2 " 50x

x2 " 0x " 25 ) 2x3 " 3x2 " 50x " 75

2x " 3

f!x".x2 " 25!x " 5i"!x # 5i" !f!x",x ! ±5i

49.

(a)

(b)

(c) !x " #6"!x # #6"!x # 1 # #2i"!x # 1 " #2i"f !x" !

f !x" ! !x " #6"!x # #6"!x2 # 2x " 3"

f !x" ! !x2 # 6"!x2 # 2x " 3"

f !x" ! x4 # 2x3 # 3x2 " 12x # 18

50.

(a)

(b)

(c) f !x" ! !x " 2i"!x # 2i"&x #3 " #29

2 '&x #3 # #29

2 ' f !x" ! !x2 " 4"&x #

3 " #292 '&x #

3 # #292 '

f !x" ! !x2 " 4"!x2 # 3x # 5"

f !x" ! x4 # 3x3 # x2 # 12x # 20

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56.


The zero of is The zeros of f are x ! #2, #1 ± 3i.x ! #2.x " 2

#1 " 3i 1

1

3 # 3i#1 " 3i

2

2 # 6i#2 " 6i

0

#1 # 3i 1

1

4#1 # 3i

3 # 3i

14#12 # 6i

2 # 6i

20#20

0

#1 " 3i.#1 # 3i

f !x" ! x3 " 4x2 " 14x " 20

54.


The zero of The zeros of are

Alternate Solution

Since are zeros of

is a factor of By long division we have:

Thus, and the zeros of g are x ! #3 ± i and x ! 14.g!x" ! !x2 " 6x " 10"!4x # 1"

0

#x2 # 6x # 10

#x2 # 6x # 10

4x3 " 24x2 " 40x

x2 " 6x " 10 ) 4x3 " 23x2 " 34x # 10

4x # 1

g!x".

! !x " 3"2 # i2 ! x2 " 6x " 10

$x # !#3 " i"% $x # !#3 # i"% ! $!x # 3" # i%$!x " 3" " i%

g!x",#3 ± i

x ! #3 ± i and x ! 14.g!x"4x # 1 is x ! 1

4.

#3 # i 4

4

11 " 4i#12 # 4i

#1

#3 # i3 " i

0

#3 " i 4

4

23#12 " 4i

11 " 4i

34#37 # i

#3 # i

#1010

0

#3 # i.#3 " i

g!x" ! 4x3 " 23x2 " 34x # 10

55. Since is a zero, so is

The zero of is The zeros of are x ! #23, 1 ± #3i.hx ! #2

3.3x " 2

1 " #3i 3

3

#1 # 3#3i3 " 3#3i

2

#2 # 2#3i2 " 2#3i

0

1 # #3i 3

3

#43 # 3#3i

#1 # 3#3i

8#10 # 2#3i

#2 # 2#3i

8#8

0

1 " #3i.1 # #3ih!x" ! 3x3 # 4x2 " 8x " 8.

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61.

(a) The root feature yields the real root 0.75,and the complex roots

(b) By synthetic division:

The complex roots of are

x !8 ± #64 # 4!8"!12"

2!8" !12

±#52

i.

8x2 # 8x " 12

34 8

8

#146

#8

18#6

12

#99

0

0.5 ± 1.118i.

h!x" ! 8x3 # 14x2 " 18x # 960.

(a) Zeros:

(b)

has zeros #1 ± 3i.x2 " 2x " 10

#2 1

1

4#2

2

14#4

10

20#20

0

x ! #2

#2, #1 ± 3i

f !x" ! x3 " 4x2 " 14x " 20


The zero of is The zeros of are x ! 34, 12!1 ± #5i".hx ! 3

4.8x # 6

12!1 " #5i" 8

8

#10 # 4#5i4 " 4#5i

#6

3 " 3#5i#3 # 3#5i

0

12!1 # #5i" 8

8

#144 # 4#5i

#10 # 4#5i

18#15 " 3#5i

3 " 3#5i

#99

0

12!1 " #5i".1

2!1 # #5i"h!x" ! 8x3 # 14x2 " 18x # 9.

59.

(a) The root feature yields the real roots 1 and 2, and the complex roots

(b) By synthetic division:

The complex roots of are x !#6 ± #62 # 4!11"

2! #3 ± #2i.x2 " 6x " 11

2 1

1

42

6

#112

11

#2222

0

1 1

1

31

4

#54

#1

#21#1

#22

22#22

0

#3 ± 1.414i.

f !x" ! x4 " 3x3 # 5x2 # 21x " 22

58.


The zero of is The zeros of f are x ! 3, #2 ± #2i

5.x ! 3.25x # 75

25

25

#65 " 5#2i#10 # 5#2i

#75

#30 # 15#2i30 " 15#2i

0

0

25 #55#10 " 5#2i

#5424 # 15#2i

#1818

#2 # #2 i5

.15

!#2 " #2 i" !#2 " #2 i

5

f !x" ! 25x3 # 55x2 # 54x # 18

#2 " #2 i5

#2 # #2 i5

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63.

Since the roots are imaginary, the ball never willreach a height of 64 feet. You can verify this graphically by observing that and do not intersect.y2 ! 64

y1 ! #16t2 " 48t

t !#48 ± #1792i

#32

#16t2 " 48t # 64 ! 0

0 " t " 3 #16t2 " 48t ! 64,

64. No. Setting yields a quadratic with no real roots.

#0.0001x2 " 60x # 9,150,000 ! 0

P ! R # C ! xp # C ! x!140 # 0.0001x" # !80x " 150,000" ! 9,000,000

69.

Vertex:

Intercepts:

2015105!5

!5

!10

!15

!20

x

y

!8, 0", !#1, 0", !0, #8"

f !x" ! !x # 8"!x " 1"

!72, #81

4 " ! !x # 7

2"2# 81

4

f !x" ! x2 # 7x # 8 ! !x2 # 7x " 494 " # 8 # 49

4

65. False, a third degree polynomial must have at leastone real zero.

66. True. The complex conjugate of the zero is also a zero.

4 " 3i

67.

(a) has two real zeros each of multiplicity 2 for

(b) has two real zeros and two complex zeros ifk < 0.f

f !x" ! x4 # 4x2 " 4 ! !x2 # 2"2.k ! 4:f

f !x" ! x4 # 4x2 " k 68. Answers will vary.

62.

(a) Zeros:

(b)

has zeros #2 ± #2i

5.25x2 " 20x " 6

3 25

25

#5575

20

#5460

6

#1818

0

3, #0.4 ± 0.2828i

f !x" ! 25x3 # 55x2 # 54x # 18

70.

Vertex:

Intercepts:

76

321

!2!3

65421!1!3!4x

y

!3, 0", !#2, 0", !0, 6"

f !x" ! #!x2 # x # 6" ! #!x # 3"!x " 2"

!12, 25

4 " ! #!x # 1

2"2" 25

4

! #!x2 # x " 14" " 6 " 1

4

f !x" ! #x2 " x " 6

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Section 2.6 Rational Functions and Asymptotes

! You should know the following basic facts about rational functions.

(a) A function of the form where are polynomials, is calleda rational function.

(b) The domain of a rational function is the set of all real numbers except those which make thedenominator zero.

(c) If is in reduced form, and is a value such that then the line is avertical asymptote of the graph of

(d) The line is a horizontal asymptote of the graph of if or

(e) Let where have no common

factors.

1. If then the -axis is a horizontal asymptote.

2. If then is a horizontal asymptote.

3. If then there are no horizontal asymptotes.n > m,

y !an

bm

n ! m,

!y ! 0"xn < m,

P!x" and Q!x"f!x" !P!x"Q!x"

!anxn " an#1xn#1 " . . . " a1x " a0

bmxm " bm#1xm#1 " . . . " b1x " b0

x ! #$.f!x" ! b as x ! $fy ! b

f!x"!$ or f!x"!#$ as x!a.f.x ! aQ!a" ! 0,af!x" ! P!x"#Q!x"

P!x" and Q!x"f!x" ! P!x"#Q!x", Q!x" % 0,

71.

Intercepts:

Vertex:

54321!2!3!4!5

21

!8

x

y

!# 512, #169

24 " ! 6!x " 5

12"2# 169

24

! 6!x2 " 56x " 25

144" # 6 # 2524

f !x" ! 6x2 " 5x # 6

!23, 0", !#3

2, 0", !0, #6"

f !x" ! 6x2 " 5x # 6 ! !3x # 2"!2x " 3" 72.

Vertex:

Intercepts:

864!4!6!8

2

!14

x

y

!32, 0", !#2, 0", !0, #12"

f !x" ! !2x # 3"!2x " 4"

!#14, #49

4 " ! 4!x " 1

4"2 # 494

! 4!x2 " 12 x " 1

16" # 12 # 14

f !x" ! 4x2 " 2x # 12

Vocabulary Check1. rational functions 2. vertical asymptote 3. horizontal asymptote

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Section 2.6 Rational Functions and Asymptotes 143

1.

(a) Domain: all

(b)

(c) approaches from the left of 1 and from the right of 1.

$#$f

x % 1

f!x" !1

x # 12.

(a) Domain: all

(b)

(c) approaches from the left of 1 and from the right of 1.

$#$f

x % 1

f!x" !5x

x # 1

3.

(a) Domain: all

(b)

(c) approaches from both the left and the rightof 1.

$f

x % 1

f!x" !3x

$x # 1$ 4.

(a) Domain: all

(b)

(c) approaches from both the left and theright of 1.

$f

x % 1

f !x" !3

$x # 1$

x

0.5

0.9

0.99

0.999 #1000

#100

#10

#2

f !x" x

1.5 2

1.1 10

1.01 100

1.001 1000

f !x"

x

5 0.25

10

100

1000 0.001

0.01

0.1

f !x" x

#0.00099#1000

#0.0099#100

#0.09#10

#0.16#5

f !x"

x

0.5

0.9

0.99

0.999 #4995

#495

#45

#5

f !x" x

1.5 15

1.1 55

1.01 505

1.001 5005

f !x"

x

5

10

100

1000 5.005

5.05

5.55

6.25

f !x" x

4.995#1000

4.950495#100

4.54#10

4.16#5

f !x"

x

0.5 6

0.9 30

0.99 300

0.999 3000

f !x" x

1.5 6

1.1 30

1.01 300

1.001 3000

f !x"

x

5 0.75

10

100

1000 0.003

0.03

0.33

f !x" x

0.5

0.0297

0.003#1000

#100

0.27#10

#5

f !x"

x

0.5 3

0.9 27

0.99 297

0.999 2997

f !x" x

1.5 9

1.1 33

1.01 303

1.001 3003

f !x"

x

5 3.75

10

100

1000 3.003

3.03

3.33

f !x" x

#2.997#1000

#2.970#100

#2.727#10

#2.5#5

f !x"

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5.

(a) Domain: all

(b)

(c) approaches from the left of 1, and from the right of 1. approaches from theleft of and from the right of #1.#$#1,

$f$#$f

x % ±1

f !x" !3x2

x2 # 16.

(a) Domain: all

(b)

(c) approaches from the left of 1, and from the right of 1. approaches from theleft of and from the right of #1.$#1,

#$f$#$f

x % ±1

f!x" !4x

x2 # 1

x

0.5

0.9

0.99

0.999 #1498

#147.8

#12.79

#1

f !x" x

1.5 5.4

1.1 17.29

1.01 152.3

1.001 1502.3

f !x"

x

5 3.125

10

100

1000 3

3.0003

3.03

f !x" x

3.125

3#1000

3.0003#100

3.03#10

#5

f !x"

x

0.5

0.9

0.99

0.999 #1999

#199

#18.95

#2.66

f !x" x

1.5 4.8

1.1 20.95

1.01 201

1.001 2001

f !x"

x

5

10

100

1000 0.004

0.04

0.40

#0.833

f !x" x

#0.004#1000

#0.04#100

#0.40#10

#0.833#5

f !x"

7.

Vertical asymptote:

Horizontal asymptote:

Matches graph (a).

y ! 0

x ! #2

f!x" !2

x " 28.

Vertical asymptote:


Matches graph (d).

y ! 0

x ! 3

f!x" !1

x # 39.

Vertical asymptote:


Matches graph (c).

y ! 4

x ! 0

f!x" !4x " 1

x

10.

Vertical asymptote:


Matches graph (e).

y ! #1

x ! 0

f!x" !1 # x

x11.

Vertical asymptote:


Matches graph (b).

y ! 1

x ! 4

f!x" !x # 2x # 4

12.

Vertical asymptote:


Matches graph (f).

y ! #1

x ! #4

f!x" ! #x " 2x " 4

14.

(a) Vertical asymptote:


(b) Holes: none

y ! 0

x ! 2

f!x" !3

!x # 2"313.



(b) Holes: none

y ! 0

x ! 0

f!x" !1x2

16.



(b) Hole at !#1, 0"x ! #1:

y !12

x !32

x % #1

!x " 12x # 3

,!!x " 1"2

!x " 1"!2x # 3" f !x" !x2 " 2x " 12x2 # x # 3

15.



(b) Hole at !0, 1"x ! 0:

y ! #1

x ! 2

x % 0f !x" !x!2 " x"2x # x2 !

2 " x2 # x

,

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19.

(a) Domain: all real numbers

(b) Vertical asymptote: none


(c)

!10

!7

11

7

y ! 3

f !x" !3x2 " x # 5

x2 " 120.

(a) Domain: All real numbers. The denominator hasno real zeros. [Try the Quadratic Formula onthe denominator.]

(b) Vertical asymptote: none


degree

(c)

!7

!2

8

8

p!x" ! degree q!x"%&

y ! 3

f !x" !3x2 " 1

x2 " x " 9

21.

(a) Domain: all real numbers except

(b) Vertical asymptote:


to the right

(to the left)

(c)

!9

!8

9

4

y ! #1

y ! 1

x ! 0

x ! 0

f !x" !x # 3

$x$ 22.

(a) Domain: all x

(b) No vertical asymptotes.

Horizontal asymptotes

(c)

!6

!4

6

4

y ! ±1

f !x" !x " 1

$x$ " 1

17.



(b) Hole at !#5, 2"x ! #5:

y ! 1

x ! 0

!x # 5

x, x % #5

!!x # 5"!x " 5"

x!x " 5"

f !x" !x2 # 25x2 " 5x

18.



(b) Hole at '#3, #165 (x ! #3:

y ! #52

x ! #12

! #5x # 12x " 1

, x % #3

!#!x " 3"!5x # 1"!x " 3"!2x " 1"

f !x" !#!5x2 " 14x # 3"

2x2 " 7x " 3

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(d)

(e) and differ at where is undefined.fx ! 4,gf

(d)


x 0 1 2 3

0 Undef. Undef. 3

0 Undef. 4 3#2#12

14g!x"

#2#12

14f !x"

#1#2#3

24.

(a) Domain of all real numbers except 3

Domain of all real numbers

(b)

has no vertical asymptotes.

(c) Hole at x ! 3

f

f !x" !!x # 3"!x " 3"

x # 3! x " 3, x % 3

g:

f :

f!x" !x2 # 9x # 3

, g!x" ! x " 3

26.

(a) Domain of all real numbers except 1 and 2

Domain of all real numbers except 1

(b)

has a vertical asymptote at

(c) The graph has a hole at x ! 2.

x ! 1.f

f !x" !!x " 2"!x # 2"!x # 2"!x # 1" !

x " 2x # 1

, x % 2

g:

f :

g!x" !x " 2x # 1

f!x" !x2 # 4

x2 # 3x " 2!

!x " 2"!x # 2"!x # 2"!x # 1",

(d)


x 0 1 2 3 4 5 6

3 4 5 Undef. 7 8 9

3 4 5 6 7 8 9g!x"

f !x"

25.


Domain of all real numbers except 3

(b)

has a vertical asymptote at

(c) The graph has a hole at x ! #1.

x ! 3.f

f !x" !!x # 1"!x " 1"!x " 1"!x # 3" !

x # 1x # 3

, x % #1

g:

#1,f :

g!x" !x # 1x # 3

f!x" !x2 # 1

x2 # 2x # 3!

!x # 1"!x " 1"!x " 1"!x # 3",

(d)

(e) and differ at where is undefined.fx ! #1,gf

x 0 1 2 3 4

Undef. 0 Undef. 3

0 Undef. 3#113

12

35g!x"

#113

35f !x"

#1#2

27.

(a) As

(b) As but is less than 4.

(c) As but is greater than 4.x ! #$, f!x" ! 4

x ! $, f!x" ! 4

x ! ±$, f!x" ! 4

f!x" ! 4 #1x

28.

(a) As

(b) As but is greater than 2.

(c) As but is less than 2.x ! #$, f!x" ! 2

x ! $, f!x" ! 2

x ! ±$, f!x" ! 2.

f!x" ! 2 "1

x # 3

x 1 2 3 4 5 6 7

5 6 7 Undef. 9 10 11

5 6 7 8 9 20 11g!x"

f !x"

23.


Domain of all real numbers

(b)

has no vertical asymptotes.

(c) Hole at x ! 4

f

f !x" !!x # 4"!x " 4"

x # 4! x " 4, x % 4

g:

f :

f!x" !x2 # 16x # 4

, g!x" ! x " 4

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39.

(a)

(c)

(e) as No, it would not bepossible to remove 100% of the pollutants.

x ! 100.C ! $

C!75" !255!75"

100 # 75! 765 million dollars

C!10" !255!10"

100 # 10) 28.33 million dollars

C !255p

100 # p, 0 " p < 100

(b)

(d)

00

100

2000

C!40" !255!40"

100 # 40! 170 million dollars

31.

The zeros of are the zeros of the numerator:x ! ±2

g

g!x" !x2 # 4x " 3

!!x # 2"!x " 2"

x " 332.

The zero of corresponds to the zero of the numerator and is x ! 2.

g

g!x" !x3 # 8x2 " 4

33.

The zero of corresponds to the zero of the numerator and is x ! 7.

f

f!x" ! 1 #2

x # 5!

x # 7x # 5

34.

There are no real zeros.

h!x" ! 5 "3

x2 " 1

35.

Zeros: x ! #1, 3

g!x" ! x2 # 2x # 3

x2 " 1!

!x # 3"!x " 1"x2 " 1

! 0 36.

Zeros: x ! 2, 3

g!x" ! x2 # 5x " 6

x2 " 4!

!x # 3"!x # 2"x2 " 4

! 0

37.

Zero: 'x !12

is not in the domain.(x ! 2

x %12

f !x" ! 2x2 # 5x " 22x2 # 7x " 3

!!2x # 1"!x # 2"!2x # 1"!x # 3" !

x # 2x # 3

, 38.

Zero: !x ! #2 is not in the domain."x !12

x ! #2

f !x" ! 2x2 " 3x # 2x2 " x # 2

!!x " 2"!2x # 1"!x " 2"!x " 1" !

2x # 1x " 1

,

40. (a)

The cost would be $4411.76.

(c)

The cost would be $225,000.

(e) No. The model is undefined for p ! 100.

C !25,000!90"100 # 90

! 225,000

C !25,000!15"100 # 15

) 4411.76 (b)

The cost would be $25,000.

(d)

00

100

300,000

C !25,000!50"100 # 50

! 25,000

29.

(a) As .


(c) As but is less than 2.x ! #$, f!x" ! 2

x ! $, f!x" ! 2

x ! ±$, f!x" ! 2

f!x" !2x # 1x # 3

30.

(a) As


(c) As but is less than 0.x ! #$, f!x" ! 0x ! $, f!x" ! 0x ! ±$, f!x" ! 0.

f!x" !2x # 1x2 " 1

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42. (a)

The greater the mass, the more time required per oscillation. The model is a good fit to the actual data.

(b) You can find M corresponding to by finding the point of intersection of

and

If you do this, you obtain grams.M ) 1306

t ! 1.056.t !38M " 16,96510!M " 500"

t ! 1.056

M 200 400 600 800 1000 1200 1400 1600 1800 2000

t 0.472 0.596 0.710 0.817 0.916 1.009 1.096 1.178 1.255 1.328

43.

(a)

(b)

(c) The herd is limited by the horizontal asymptote:

N !60

0.04! 1500 deer

N!25" ! 800 deer

N!10" ! 500 deer

N!5" ) 333 deer

00

50

1200

N !20!5 " 3t"1 " 0.04t

, 0 " t 44. (a)

The model is a good fit.

(b) For 2010, and billion.

For 2015, and billion.

For 2020, and billion.

Answers will vary.

(c) Horizontal asymptote

As time passes, the national defense outlaysapproach $292.7 billion.

y !1.4930.0051

) 292.7

D ) $319.1t ! 30

D ) $332.3t ! 25

D ) $366.8t ! 20

7 150

600

45. False. A rational function can have at most verticalasymptotes, where is the degree of the denominator.n

n 46. False. For example, has no vertical

asymptote.

f !x" !1

x2 " 1

47. There are vertical asymptotes at and zerosat Matches (b).x ! ±2.

x ! ±3, 48. There are vertical asymptotes at and is a zero. Matches (c).

x ! 0x ! ±1,

49. f !x" !x # 1x3 # 8

50. f !x" !x # 2

!x " 1"2

51. f !x" !2!x " 3"!x # 3"!x " 2"!x # 1" !

2x2 # 18x2 " x # 2

52. f !x" !#2!x " 2"!x # 3"

!x " 1"!x # 2" !#2x2 " 2x " 12

x2 # x # 2

41. (a) Use data

.

y !1

0.445 # 0.007x

1y

! #0.007x " 0.445

'60, 1

39.4('50, 1

19.7(,

'16, 13(, '32,

14.7(, '44,

19.8(, (b)

(Answers will vary.)

(c) No, the function is negative for x ! 70.

x 16 32 44 50 60

y 3.0 4.5 7.3 10.5 40

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57.

x2 " 5x " 6x # 4

! x " 9 "42

x # 4

42

9x # 36

9x " 6

x2 # 4x

x # 4 ) x2 " 5x " 6

x " 9

59.

2x4 " x2 # 11x2 " 5

! 2x2 # 9 "34

x2 " 5

34

#9x2 # 45

#9x2 # 11

2x4 " 10x2

x2 " 5 ) 2x4 " 0x3 " x2 " 0x # 11

2x2 # 9

60.

4x5 " 3x3 # 102x " 3

! 2x4 # 3x3 " 6x2 # 9x "272

#101

4x " 6

#1012

27x " 812

27x # 10

#18x2 # 27x

#18x2

12x3 " 18x2

12x3

#6x4 # 9x3

#6x4 " 3x3 4x5 " 6x4

2x " 3 ) 4x5 " 0x4 " 3x3 " 0x2 " 0x # 10

2x4 # 3x3 " 6x2 # 9x " 272

58.

x2 # 10x " 15x # 3

! x # 7 "#6

x # 3

3 1

1

#103

#7

15#21

#6

53.

y # x " 1 ! 0

y ! x # 1

y # 2 !#1 # 20 # 3

!x # 3" ! 1!x # 3" 54.

3x " 5y " 13 ! 0

#10y " 10 ! 6x " 36

y # 1 !1 " 5

#6 # 4!x " 6"

55.

3x # y " 1 ! 0

y ! 3x " 1

y # 7 !10 # 73 # 2

!x # 2" ! 3!x # 2" 56.

4x " 9y ! 0

#9y ! 4x

y # 0 !4 # 0

#9 # 0!x # 0"

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Section 2.7 Graphs of Rational Functions

! You should be able to graph

(a) Find the - and -intercepts. (b) Find any vertical or horizontal asymptotes.

(c) Plot additional points. (d) If the degree of the numerator is one more than the degree ofthe denominator, use long division to find the slant asymptote.

yx

f!x" !p!x"q!x"

.

9.

-intercept:

Vertical asymptote:

Horizontal asymptote: y ! 0

x ! "2

#0, 12$y

f!x" !1

x # 2

5.

Vertical shift twounits downward

!6

!4

6

4

f

g

f

g

g!x" !2x2 " 2 6.

Reflection in the -axisx

!6

!4

6

4

f f

g g

7.

Horizontal shift twounits to the right

!6

!1

6

7

f f g

g

g!x" !2

!x " 2"28.

Each -value is multiplied by

Vertical shrink

14.

y

!6

!2

6

6

f f

g

1.

Vertical shift one unitupward

!6

!4

6

4

f

fgg

g!x" !2x

# 1 2.

Horizontal shift oneunit to the right

!6

!4

6

4

f

fg

g

3.

Reflection in the -axisx

!6

!4

6

4

f

fg

g

g!x" ! "2x

4.

Horizontal shift twounits to the left, andvertical shrink

!6

!4

6

4

f

fg

g

Vocabulary Check

1. slant, asymptote 2. vertical

x 0 1

y 1 13

12"1"1

2

"1"3"4

–3 –1

–2

–1

1

2

x

( )0, 12

y©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.

Section 2.7 Graphs of Rational Functions 151

12.

-intercept:

-intercept:

Vertical asymptote:


!2 !1 2 3 4

4

5

6

x

(0, 1) , 013( )

y

y ! 3

x ! 1

!0, 1"y

#13

, 0$x

P!x" !1 " 3x1 " x

!3x " 1x " 1

x 0 2 3

y 2 1 5 4

"1

11.

-intercept:

-intercept:

Vertical asymptote:


( )

–6 –4 2 4x

6

52! , 0

(0, 5)

y

y ! 2

x ! "1

!0, 5"y

#"52

, 0$x

C!x" !5 # 2x1 # x

!2x # 5x # 1

x 0 1 2

1 5 372"11

2C!x"

"2"3"4

x 1 2

y 0 "32"1"3"5

2

12"1"2

13.

-intercept:

Vertical asymptote:


–2 –1 1 2

–3

–1

t( ), 01

2

y

y ! "2

t ! 0

#12

, 0$t

f !t" !1 " 2t

t! "

2t " 1t

14.

-intercept:

-intercept:

Vertical asymptote:


y

x

520,( (5

2! , 0( (

!1!3!4 1 2 3 4!1

!2

1

2

4

5

6

y ! 2

x ! "2

#"52

, 0$x

#0, 52$y

g!x" !1

x # 2# 2 !

2x # 5x # 2

10.

-intercept:

Vertical asymptote:


x ! 6

#0, "16$y

f !x" !1

x " 6 x 0 2 4 8 10

y 14

12"1

2"14"1

6"17

"1

x 0 2

y 0 3 94

52

32

"1"52"4

y

x!2 2 6 8 10

!2

!4

!6

2

4

6

0, ! 16( (

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18.

-intercept:

Vertical asymptote:


x ! 2

#0, "14$y

f!x" ! "1

!x " 2"2

1 3

!4

!3

!2

!1

x( )0, 1

4!y

x 1 3 4

y "14"4

9"1"4"4"1"49"1

4

72

52

32

120

17.

Intercept:

Vertical asymptotes: and


Origin symmetry

y ! 0

x ! "1x ! 1

!0, 0"

f !x" !x

x2 " 1!

x!x # 1"!x " 1" 3

2

1

!2

!1

!3

32!2!3

(0, 0)

x

y

x 0 2 3 4

y 0 415

38

23"2

323"2

3"38

12"1

2"2"3

15.

Intercept:

Vertical asymptotes:


-axis symmetry

6

4

2

!4

!6

64!4!6 !2 2

(0, 0)

x

y

y

y ! 1

x ! 2, x ! "2

!0, 0"

f !x" !x2

x2 " 416.

Intercepts:



Origin symmetry

–2 4 5

–5–4–3–2–1

12345

x(0, 0)

y

y ! 0

x ! ±3

!0, 0"

g!x" !x

x2 " 9

x 0 2 4 5

y 0 516

47"2

525"4

7" 516

"2"4"5x 0 4

y 0 43"1

3"13

43

"1"1"4

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19.

Intercept:



x ! 0 and x ! 4

!"1, 0"

g!x" !4!x # 1"x!x " 4"

–6 –4 –2 2 6 8 10

–8

–6

–4

2

4

6

8

x(!1, 0)

y

21.

Intercept:



x ! "1, 2

!0, 0"

f !x" !3x

x2 " x " 2!

3x!x # 1"!x " 2"

–2 4x

4

2

(0, 0)

y

x 1 2 3 5 6

y 0 73

245"16

3"3"83"1

3

"1"2

x 0 1 3 4

y 0 65

94"3

2" 910

"3

22.

Intercept:



x ! "2, 1

!0, 0"

f!x" !2x

x2 # x " 2!

2x!x # 2"!x " 1"

–3 –1 2 3

–3

–2

–1

2

1

3

4

x

(0, 0)

y

20.



x ! 0, x ! 3

h!x" !2

x2!x " 3"y

x!3 3 4 5

1

2

3

4

x 0 2 3

y 1 0 1 35"4

5"32"4

5

12"1"3"4

x 0 1 2 3 4

y Undef. Undef. 18"1

2"1" 110

"2

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25.

The graph is a line, with a hole at

y

x!2!4!6!8 2 4 6 8

!4

!6

!8

2

4

6

8

x ! "1.

x $ "1

f !x" !x2 " 1x # 1

!!x # 1"!x " 1"

x # 1! x " 1, 26.

Hole at x ! 4

y

x!2!4!6!8 2 4 6 8

!2

!4

2

4

6

8

10

12

f !x" !x2 " 16x " 4

! x # 4, x $ 4

27.

Vertical asymptote:


Domain: x $ 1 or !"%, 1" ! !1, %"

!5

!4

7

4

y ! "1

x ! 1

f!x" !2 # x1 " x

! "x # 2x " 1

28.

-intercept:

-intercept:

Vertical asymptote:


Domain: all x $ 2

y ! 1

x ! 2

#0, 32$y

!3, 0"x !4

!4

8

4f!x" !3 " x2 " x

!x " 3x " 2

x 0 1 2 3

y 0 Undef. 3"113

12

"1"2

23.

Intercept:

Vertical asymptote:

There is a hole at


y

x!2!4!6!8 2 4 6 8

!2

!4

!6

!8

2

4

6

8

y ! 1

x ! "3."!

x ! 2

!0, 0"

x $ "3

f !x" !x2 # 3x

x2 # x " 6!

x!x # 3"!x " 2"!x # 3" !

xx " 2

, 24.

Vertical asymptote:


Hole at x ! "4

y

x2 4 6 8 10

!2

!4

!6

!8

2

4

6

8

y ! 0

x ! 3

x $ "4

g!x" !5!x # 4"

x2 # x " 12!

5!x # 4"!x # 4"!x " 3" !

5x " 3

,

x 0 1 3 4

y Undef. Undef. 5"52"5

3

"4

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29.

Vertical asymptote:


Domain: t $ 0 or !"%, 0" ! !0, %"

!6

!1

6

7

y ! 3

t ! 0

f!t" !3t # 1

t

31.

Domain: all real numbers OR


!6

!1

6

7

y ! 0

!"%, %"

h!t" !4

t2 # 1

30.

-intercept:

-intercept:

Vertical asymptote:


Domain: all x $ 3

y ! 1

x ! 3

#0, 23$y

!2, 0"x !5

!4

10

6h!x" !

x " 2x " 3

32.

Domain: all real numbers except 2 or

Vertical asymptote:


x ! 2

!"%, 2" ! !2, %"

!4

!7

8

1g!x" ! "x

!x " 2"2

33.

Domain: all real numbers except



!9

!6

9

6

y ! 0

x ! 3, x ! "2

x ! 3, "2

f !x" !x # 1

x2 " x " 6!

x # 1!x " 3"!x # 2" 34.

Domain: all real numbersexcept and 2 or



x ! "3, x ! 2

!"%, "3" ! !"3, 2" ! !2, %""3

!6

!4

6

4f!x" !

x # 4x2 # x " 6

35.

Domain: all real numbers except 0, OR

Vertical asymptote:


x ! 0

!"%, 0" ! !0, %" !15

!10

15

10f!x" !

20xx2 # 1

"1x

!19x2 " 1x!x2 # 1"

36.

Domain: all real numbers except and 4



x ! "2, x ! 4

"2!12

!8

12

8

f!x" ! 5# 1x " 4

"1

x # 2$ !30

!x " 4"!x # 2"

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43.

Vertical asymptote:

Slant asymptote:

Origin symmetry

y = 2x

–6 –4 –2 2 4 6

–6

2

4

6

x

y

y ! 2x

x ! 0

f !x" !2x2 # 1

x! 2x #

1x

44.

Intercepts:

Vertical asymptote:

Slant asymptote:

y

x!2!4!6!8 4 6 8

!2

!4

!6

!8

2

4

6

8

y = !x

(!1, 0) (1, 0)

y ! "x

x ! 0

!1, 0", !"1, 0"

g!x" !1 " x2

x45.

Intercept:

Vertical asymptote:

Slant asymptote:

–4 2 4 6 8

–4

–2

2

4

6

8

x(0, 0)

y = x + 1

y

y ! x # 1

x ! 1

!0, 0"

h!x" !x2

x " 1! x # 1 #

1x " 1

37.

There are two horizontalasymptotes, y ! ±6.

!12

!8

12

8

h!x" !6x

%x2 # 138.

There are two horizontalasymptotes, .y ! ±1

!6

!4

6

4

f !x" !"x

%9 # x239.

There are two horizontalasymptotes,

One vertical asymptote:x ! "1

y ! ±4.

!24

!16

24

16

g!x" !4&x " 2&

x # 1

40.

There are two horizontalasymptotes, and

Vertical asymptote: x ! 2

y ! 8.y ! "8

!30

!25

30

15

f !x" !"8&3 # x&

x " 2!

8&3 # x&2 " x

41.

The graph crosses its horizontalasymptote, y ! 4.

!10

!4

14

12

f!x" !4!x " 1"2

x2 " 4x # 542.

The graph crosses itshorizontal asymptote, y ! 3.

!9

!3

9

9

g !x" !3x4 " 5x # 3

x4 # 1

46.

Intercept:


Slant asymptote:

Origin symmetry

y ! x

x ! ±1

!0, 0"

f!x" !x3

x2 " 1! x #

xx2 " 1

–6 –4 –2 2 4 6x

2(0, 0)

y = x

y

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47.

Intercept:


Slant asymptote:

Origin symmetry

–8 –6 –4 4 6 8

4

6

8

x

y = x

(0, 0)

12

y

y !12

x

x ! ±2

!0, 0"

g!x" !x3

2x2 " 8!

12

x #4x

2x2 " 848.

No vertical asymptotes


Intercepts:

4321!1!2!3!4

4

3

2

!2

!3

!4

( 1, 0)! (1, 0)

1

x

y

!±1, 0", #0, "14$

y ! 1

f !x" !x2 " 1x2 # 4

49.

Intercepts:

Slant asymptote:

54321!2 !1!5

7654

!2!3

(0, 4)

y = 12x + 1

x

y

y !x2

# 1

!"2.594, 0", !0, 4"

f !x" !x3 # 2x2 # 4

2x2 # 1!

x2

# 1 #3 "

x2

2x2 # 150.

-intercept:

Vertical asymptote:

Slant asymptote:

!9 !6 !3 3 6 9 12 15

!9

3

6

9

12

15

x

( )0, 52!

y = 2x ! 1

y

y ! 2x " 1

x ! 2

#0, "52$y

f !x" !2x2 " 5x # 5

x " 2! 2x " 1 #

3x " 2

51.

(a) -intercept:

(b)

"1 ! x

"0 ! x # 1

"0 !x # 1x " 3

!"1, 0"x

y !x # 1x " 3 52.

(a) -intercept:

(b)

0 ! x

0 ! 2x

0 !2x

x " 3

!0, 0"x

y !2x

x " 353.

(a) -intercepts:

(b)

x ! ±1

x2 ! 1

x !1x

0 !1x

" x

!±1, 0"x

y !1x

" x

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57.

Domain: all real numbers except 0

or

Vertical asymptote:

Slant asymptote:

!12

!4

12

12

y ! "x # 3

x ! 0

!"%, 0" ! !0, %"

y !1 # 3x2 " x3

x2 !1x2 # 3 " x ! "x # 3 #

1x2

58.

Domain: all real numbers except or

-intercepts:

-intercept:

Vertical asymptote:

Slant asymptote: y ! "12

x # 1

x ! "4

#0, 32$y

!"4.61, 0", !2.61, 0"x

!"%, "4" ! !"4, %""4

y !12 " 2x " x2

2!4 # x" ! "12

x # 1 #2

4 # x

59.



No slant asymptotes, no holes

y ! 1

x ! "2x ! 2,

f !x" !x2 " 5x # 4

x2 " 4!

!x " 4"!x " 1"!x " 2"!x # 2" 60.



No slant asymptotes, no holes

y ! 1

x ! "3x ! 3,

f !x" !x2 " 2x " 8

x2 " 9!

!x " 4"!x # 2"!x " 3"!x # 3"

54.

(a) -intercepts:

(b)

x ! 1, 2

0 ! !x " 1"!x " 2"

0 ! x2 " 3x # 2

0 ! x " 3 #2x

!1, 0", !2, 0"x

y ! x " 3 #2x

55.

Domain: all real numbers except

Vertical asymptote:

Slant asymptote:

!12

!10

12

6

y ! 2x " 1

x ! "1

x ! "1

y !2x2 # xx # 1

! 2x " 1 #1

x # 1

56.

Domain: all

Vertical asymptote:

Slant asymptote:

!14

!8

10

8

y ! x # 2

x ! "3

x $ "3

y !x2 # 5x # 8

x # 3! x # 2 #

2x # 3

!16

!6

8

10

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63.

Long division gives

Vertical asymptote:

No horizontal asymptote

Slant asymptote:

Hole at !"1, 6"x ! "1,

y ! 2x " 7

x ! "2

f !x" !2x2 " 3x # 1

x # 2! 2x " 7 #

15x # 2

.

!!x " 1"!2x " 1"

x # 2, x $ "1

!!x " 1"!x # 1"!2x " 1"

!x # 1"!x # 2"

f !x" !2x3 " x2 " 2x # 1

x2 # 3x # 264.

Long division gives

Vertical asymptote:

No horizontal asymptote

Slant asymptote:

Hole at !2, 20"x ! 2,

y ! 2x # 7

x ! 1

f !x" ! 2x # 7 #9

x " 1.

!!x # 2"(2x # 1"

x " 1, x $ 2

!!x " 2"!x # 2"!2x # 1"

!x " 2"!x " 1"

f !x" !2x3 # x2 " 8x " 4

x2 " 3x # 2

65.

(a)

-intercept:

(b)

x ! "4

"5x ! 20

"4x " 20 ! x

"4!x # 5" ! x

"4x

!1

x # 5

0 !1

x # 5#

4x

!"4, 0"x

!11

!6

7

6

y !1

x # 5#

4x

66.

(a)

-intercept:

(b)

"3 ! x

2x ! 3x # 3

2

x # 1!

3x

2

x # 1"

3x

! 0

!"3, 0"x

!3 3

!12

20

y !2

x # 1"

3x

67.

(a)

-intercept:

(b)

x ! "83

3x ! "8

x # 4 ! "2x " 4

1

x # 2!

"2x # 4

1

x # 2#

2x # 4

! 0

#"83

, 0$x

!10 8

!6

6

y !1

x # 2#

2x # 4

61.

Vertical asymptote:


No slant asymptotes

Hole at #2, 37$x ! 2,

y ! 1

x ! "32

x $ 2

f !x" !2x2 " 5x # 22x2 " x " 6

!!2x " 1"!x " 2"!2x # 3"!x " 2" !

2x " 12x # 3

, 62.

Vertical asymptote:


No slant asymptotes

Hole at #2, 45$x ! 2,

y !32

x ! "12

x $ 2

f !x" !3x2 " 8x # 42x2 " 3x " 2

!!3x " 2"!x " 2"!2x # 1"!x " 2" !

3x " 22x # 1

,

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68.

(a)

-intercept: !"8, 0"x

!10 8

!6

6

y !2

x # 2"

3x " 1

(b)

"8 ! x

2x " 2 ! 3x # 6

2

x # 2!

3x " 1

70.

(a)

-intercepts:

(b)

±3 ! x

9 ! x2

9x

! x

0 ! x "9x

!"3, 0", !3, 0"x

!9

!6

9

6

y ! x "9x

71.

(a)

-intercepts:

(b)

' "2.618, "0.382

!"32

±%52

x !"3 ± %9 " 4

2

x2 # 3x # 1 ! 0

x2 # 3x # 2 ! 1

x # 2 !1

x # 2

!"0.382, 0"!"2.618, 0",x

!10 8

!6

6

y ! x # 2 "1

x # 1

69.

(a)

-intercept:

(b)

x ! "2, x ! 3

0 ! !x # 2"!x " 3"

0 ! x2 " x " 6

6 ! x!x " 1"

6

x " 1! x

0 ! x "6

x " 1

!"2, 0", !3, 0"x

!9

!6

9

6

y ! x "6

x " 1

72.

(a)

-intercepts:

(b)

x ! 1, 32

!x " 1"!2x " 3" ! 0

2x2 " 5x # 3 ! 0

1 ! "2x2 # 5x " 2

1

x " 2! 1 " 2x

2x " 1 #1

x " 2! 0

#32

, 0$!1, 0",x

!3 6

!6

12

y ! 2x " 1 #1

x " 2

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75.

(a)

-intercepts:

(b)

' 0.766, "3.266

!"5 ± %65

4

x !"5 ± %25 " 4!2"!"5"

4

2x2 # 5x " 5 ! 0

2x2 # 5x " 3 ! 2

x # 3 !2

2x " 1

x # 3 "2

2x " 1! 0

!0.766, 0", !"3.266, 0"x

!5 7

!1

7

y ! x # 3 "2

2x " 176.

(a)

-intercepts:

(b)

!5 ± %17

4' 0.219, 2.281

x !5 ± %25 " 8

4

2x2 " 5x # 1 ! 0

2x2 " 5x # 3 ! 2

x " 1 !2

2x " 3

x " 1 "2

2x " 3! 0

!2.281, 0"!0.219, 0",x

!5 7

!3

5

y ! x " 1 "2

2x " 3

73.

(a)

No -intercepts

(b)

No real zeros

x2 # 1 ! 0

2 ! "x2 # 1

2

x " 1! "x " 1

x # 1 #2

x " 1! 0

x

!10 8

!4

8

y ! x # 1 #2

x " 174.

(a)

No -intercept

(b)

Because there areno real zeros.

b2 " 4ac ! 16 " 24 < 0,

x2 # 4x # 6 ! 0

2 ! "x2 " 4x " 4

2

x # 2! "x " 2

x # 2 #2

x # 2! 0

x

!11 7

!6

6

y ! x # 2 #2

x # 2

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77. (a)

(b) Domain: and

Thus, .

(c)

As the tank fills, the rate that the concentrationis increasing slows down. It approaches the horizontal asymptote When thetank is full the concentration isC ! 0.725.

!x ! 950",C ! 3

4 ! 0.75.

00

950

1

0 ! x ! 950

x ! 1000 " 50 ! 950x " 0

C !3x # 50

4!x # 50"

50 # 3x200 # 4x

! C

12.5 # 0.75x

50 # x! C

0.25!50" # 0.75!x" ! C!50 # x" 78. (a)

(b) Domain:

(c)

meters.y ! 50030 ! 162

3For x ! 30,

10 20 30 40 50 60

20

40

60

80

100

120

x

y

x > 0

y !500

x

Area ! xy ! 500

79. (a) and

Thus, A ! xy ! x#4x # 22x " 2 $ !

2x!2x # 11"x " 2

.

y ! 4 #30

x " 2!

4x # 22x " 2

y " 4 !30

x " 2

!x " 2"!y " 4" ! 30

A ! xy (b) Domain: Since the margins on the left and right areeach 1 inch, or

(c)

The area is minimum when andy ' 11.75 in.

x ' 5.87 in.

050

20

100

!2, %".x > 2,

80. (a) The line passes through the points andand has a slope of

by the point-slope form

!2!a " x"

a " 3, 0 ! x ! a

y !2!x " a"

3 " a!

"2!a " x""1!a " 3"

y " 0 !2

3 " a!x " a"

m !2 " 03 " a

!2

3 " a.

!3, 2"!a, 0" (b) The area of a triangle is

when so

(c)

Vertical asymptote:

Slant asymptote:

is a minimum when and A ! 12.

a ! 6A

A ! a # 3

010

36

36a ! 3

A !a2

a " 3! a # 3 #

9a " 3

A !12

a# 2aa " 3$ !

a2

a " 3

h !2!a " 0"

a " 3!

2aa " 3

.x ! 0,h ! y

b ! a

A ! 12 bh.

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83.

(a) The horizontal asymptote is the -axis, orThis indicates that the chemical

eventually dissipates.

(b) The maximum occurswhen t ' 4.5.

00

10

1

C ! 0.t

C !3t2 # tt3 # 50

, 0 ! t

(c) Graph together with The graphsintersect at and when hours and when hours.t > 8.320 ! t < 2.65

C < 0.345t ' 8.32.t ' 2.65y ! 0.345.C

x 0.5 1 2 3 4 5 6 7

20.1 15.2 12.9 12.3 12.05 12 12.1'12.0C

82.

The minimum average cost occurs when x ! 5.

5 10 302520x

C

15

5

10

15

20

25

30C !

Cx

!0.2x2 # 10x # 5

x, x > 0

84. (a) Rate Time Distance or



x ! 25

y !25x

x " 25

25x ! y!x " 25"

25x ! xy " 25y

25y # 25x ! xy

25x

#25y

! 1

100x

#100

y!

20050

! 4

DistanceRate

! Time!& (c)

The results in the table are unexpected. You wouldexpect the average speed for the round trip to bethe average of the average speeds for the two partsof the trip.

(d)

(e) No, it is not possible to average 20 miles per hour inone direction and still average 50 miles per hour onthe round trip. At 20 miles per hour you would usemore time in one direction than is required for theround trip at an average speed of 50 miles per hour.

300

60

150

x 30 35 40 45 50 55 60

y 150 87.5 66.7 56.3 50 45.8 42.9

81.

The minimum occurs when x ' 40.4 ' 40.0

0300

300

C ! 100#200x2 #

xx # 30$, 1 ! x

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85. (a)

(b) Using the data we obtain:

(c)

!1 150

11,000

y3 !1

"0.00001855t # 0.000315!

1A

y2 ! "0.00001855t # 0.0003150

#t !1A$,

!1 150

11,000

!t ! 0 corresponds to 1990"y1 ! 583.8t # 2414 86. (a) Domain:

(b) At

(c) elk

elk

elk

(d) Yes, the horizontal aymptote is

the limit.

y !2.70.1

' 27

P!100" ' 25

P!50" ' 24

P!25" ' 22

P ! 10.t ! 0,

t " 0

87. False, you will have to lift your pencil to cross thevertical asymptote.

88. False. crosses its horizontal

asymptote at x ! 0.y ! 0

f !x" !x

x3 # 1

91. has a slant asymptote

and a vertical asymptote

Hence, y ! x # 1 "12

x # 2!

x2 # 3x " 10x # 2

.

a ! "12

a4

! "3

0 ! 3 #a4

0 ! 2 # 1 #a

2 # 2

x ! "2.y ! x # 1

y ! x # 1 #a

x # 292. has slant asymptote

and vertical asymptote at We determine

so that has a zero at

Hence, y ! x " 2 #"7

x # 4!

x2 # 2x " 15x # 4

.

0 ! 3 " 2 #a

3 # 4! 1 #

a7

# a ! "7

x ! 3:y

ax ! "4.

y ! x " 2y ! x " 2 #a

x # 4

89.

Since is not reduced and is a factor ofboth the numerator and the denominator, isnot a horizontal asymptote.

There is a hole in the graph at

!6

!4

6

4

x ! 3.

x ! 3!3 " x"h!x"

x $ 3h!x" !6 " 2x3 " x

!2!3 " x"

3 " x! 2, 90.

Since is not reduced is a factor of boththe numerator and the denominator, is not ahorizontal asymptote.

There is a hole at x ! 1.

!7

!4

5

4

x ! 1!x " 1"g!x"

!!x # 2"!x " 1"

x " 1! x # 2, x $ 1

g!x" !x2 # x " 2

x " 1

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Section 2.8 Quadratic Models 165

1. A quadratic model is better. 2. Linear 3. A linear model is better.

4. Neither 5. Neither linear nor quadratic 6. Quadratic

Section 2.8 Quadratic Models

93. !x8"

!3" !8

x"3

"512x3 94. #4x2$!2 "

1#4x2$2 "

116x4 95.

37%6

31%6 " 36%6 " 3

96.

"1x3

x!2 # x1%2

x!1 # x5%2 "x # x1%2

x2 # x5%2 "x3%2

x9%2 97.

Domain: all

Range: y ! &6

x

!6

!1

6

7 98.

Semicircle

Domain:

Range: 0 " y " 11

!11 " x " 11

!15

!5

15

15

99.

Domain: all

Range: y " 0

x

!20

!11

4

5 100.

Parabola

Domain: all x

Range: y " 9

!15

!10

15

10 101. Answers will vary. (Make a Decision)

You should know how to

! Construct and classify scatter plots.

! Fit a quadratic model to data.

! Choose an appropriate model given a set of data.

Vocabulary Check

1. linear 2. quadratic

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7. (a) (b) Linear model is better.

(d)

(c) linear

quadratic

(e)

'( y " !0.00478x2 $ 0.1887x $ 2.1692,

y " 0.14x $ 2.2,0

010

4

00

10

4

9. (a)

(c)

(e)

y " 5.55x2 ! 277.5x $ 3478

00

60

5100 (b) Quadratic model is better.

(d)

00

60

5100

x 0 5 10 15 20 25 30 35 40 45 50 55

y 3480 2235 1250 565 150 12 145 575 1275 2225 3500 5010

Model 3478 2229 1258 564 148 9 148 564 1258 2229 3478 5004

8. (a)

(c) , linear

quadratic

(e)

y " 0.006x2 ! 0.23x $ 10.5,

y " !0.19x $ 10.5

!3 90

15 (b) Quadratic model is better. Answers will vary.

(d)

!3 90

15

x 0 1 2 3 4 5 6 7 8

y 11 10.7 10.4 10.3 10.1 9.9 9.6 9.4 9.4 9.2 9.0

Model 11 10.7 10.4 10.3 10.1 9.9 9.7 9.5 9.3 9.2 9.0

!1!2

x 0 1 2 3 4 5 6 7 8 9 10

y 2.1 2.4 2.5 2.8 2.9 3.0 3.0 3.2 3.4 3.5 3.6

Model 2.2 2.4 2.5 2.7 2.8 2.9 3.1 3.2 3.4 3.5 3.6

10. (a)

(c)

—CONTINUED—

y " !17.793x2 $ 354.797x $ 6162.9

05200

22

8000 (b) Quadratic

(d)

05200

22

8000

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16. (a)

(b)

(c)

(d) Using a graphics utility, whenor 2009.

(e) No, because the coefficient of is negative.The graph turns downward as time increases.Answers will vary.

t2

t ) 19.4,S > 2000

1571400

2000

#t " 8 # 1998$

S " !3.07t2 $ 131.9t $ 597

1571400

2000 17. (a)

(b)

(c)

(d) According to the model, whenor 2024.

(e) Answers will vary.

t ) 24,y > 10,000

!1 64000

6000

y " !2.630t2 $ 301.74t $ 4270.2

!1 64000

6000

13. (a) , linear

, quadratic

(b) 0.99982 for the linear model

0.99987 for the quadratic model

(c) The quadratic model is slightly better.

y " 0.001x2 ! 0.90x $ 5.3

y " !0.89x $ 5.3

15. (a)

(b) P " 0.1322t2 ! 1.901t $ 6.87

10

12

5 (c)

(d) The model’s minimum is at This corresponds to July.

t " 7.2.H ) 0.03

10

12

5

14. (a)

(b) 0.98235, linear

0.99653, quadratic

(c) Quadratic is better.

y " 0.08x2 ! 9.0x $ 595

y " !8.6x $ 612

11. (a) linear

quadratic

(b) 0.98995 for linear model

0.99519 for quadratic model

(c) Quadratic fits better.

y " 0.071x2 $ 1.69x $ 2.7,

y " 2.48x $ 1.1, 12. (a) linear model

quadratic model

(b) 0.99978 for the linear model

0.99984 for the quadratic model

(c) The quadratic model is slightly better.

y " 0.006x2 $ 2.04x,

y " 2.10x,

10. —CONTINUED—

(e)

(Answers will vary.)

x 0 2 4 6 8 10 12 14 16 18 20 22

y 6140 6815 7335 7710 7915 7590 7975 7700 7325 6820 6125 5325

Model 6163 6801 7297 7651 7863 7932 7858 7643 7285 6784 6142 5357

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18. (a)

(b) , linear model

(c)

6 145400

6400

#0.98921$S " 117.1t $ 4727

6 145400

6400 (d)

(e)

(f) Answers will vary.

(g) for or 2009.

is never greater than 7000.S2

t > 19.4,S1 > 7000

6 145400

6400

#0.99151$S2 " !3.26t2 $ 182.3t $ 4413

24. (a)

(b) #g % f $#x$ " g#2x ! 1$ " #2x ! 1$2 $ 3 " 4x2 ! 4x $ 4

# f % g$#x$ " f #x2 $ 3$ " 2#x2 $ 3$ ! 1 " 2x2 $ 5

19. (a)

(b)

(c) ,quadratic model, #0.99859$y " !1.1357t2 $ 18.999t $ 50.32

!1 640

140

!1 640

140

20. (a)

(b) Answers will vary.

(c)quadratic model, #0.99474$y " 2.36t2 $ 53.7t $ 3489

!1 63450

3850

!1 63450

3850

21. True 22. True 23. The model is above all data points.

(d)

(e) The cubic model is a better fit.

(f )

!1 640

140

Year 2006 2007 2008

127.76 140.15 154.29

Cubic 129.91 145.13 164.96

Quadratic 123.40 127.64 129.60

A*

#t " 8$#t " 7$#t " 6$


(e) Cubic model is better.

(f)

!1 63450

3850

Year 2006 2007 2008

3890 3949 4059

Cubic 3858 3878 3862

Quadratic 3896 3981 4070

A*

#t " 8$#t " 7$#t " 6$©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.


28. is one-to-one.

y "#x ! 5$

2 $ f !1#x$ "

x ! 52

2y " x ! 5

x " 2y $ 5

y " 2x $ 5

f 29. f is one-to-one.

5x $ 4 " y $ f!1#x$ " 5x $ 4

x "y ! 4

5

y "x ! 4

5

25. (a)

(b) g# f #x$$ " g#5x $ 8$ " 2#5x $ 8$2 ! 1 " 50x2 $ 160x $ 127

f #g#x$$ " f #2x2 ! 1$ " 5#2x2 ! 1$ $ 8 " 10x2 $ 3

26. (a)

(b) #g % f $#x$ " g#x3 ! 1$ " 3&x3 ! 1 $ 1 " x

# f % g$#x$ " f # 3&x $ 1 $ " x $ 1 ! 1 " x 27. (a)

(b) g# f #x$$ " g# 3&x $ 5$ " (3&x $ 5'3 ! 5 " x

f #g#x$$ " f #x3 ! 5$ " 3&x3 ! 5 $ 5 " x

32.

54321!1!2!3!4

5

34

21

!2!3!4!5

Realaxis

Imaginaryaxis

33.

54321!1!2!3!4

5

34

21

!2!3!4!5

Realaxis

Imaginaryaxis

34.

54321!1!2!3!4

5

34

21

!2!3!4!5

Realaxis

Imaginaryaxis

35.

Realaxis8642!2!4!6

8

6

4

2

!4

!6

!8

Imaginaryaxis

30. is one-to-one on .

y " &x ! 5 $ f !1#x$ " &x ! 5, x ! 5

y2 " x ! 5

y ! 0 x " y2 $ 5,

x ! 0 y " x2 $ 5,

(0, &$f 31. f is one-to-one.

"&2x $ 6

2, x ! !3

y "&x $ 32

$ f !1#x$ "&x $ 32

y2 "#x $ 3$

2

x " 2y2 ! 3, y ! 0

y " 2x2 ! 3, x ! 0

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1.

(a) is a vertical stretch.

(b) is a vertical stretch and reflection inthe -axis.

(c) is a vertical shift two units upward.

(d) is a horizontal shift five units to the left.y ! !x " 5"2

y ! x2 " 2

xy ! #2x2

y ! 2x2

!9

!6

9

6

a

b

d

c

Review Exercises for Chapter 2

2.

(a) Vertical shift three units downward

(b) Reflection in the x-axis and vertical shift three units upward

(c) Horizontal shift four units to the right

(d) Vertical shrink followed by vertical shift four units upward

!9 9

!4

8

a

b

c

d

5.

!13#x "

52$

2

#4112

!13%#x "

52$

2

#414 &

!13 #x2 " 5x "

254

#254

# 4$

f!x" !13

!x2 " 5x # 4" Vertex:

-intercept:

-intercepts:

Use the Quadratic Formula.

##5 ± '412

, 0$

x !#5 ± '41

2

0 ! x2 " 5x # 4

0 !13

!x2 " 5x # 4"x

#0, #43$y

–10 –8 –6 –2 2 4 6

–4

2

4

6

8

10

12

x

y##52

, #4112$

3.

Vertex:

-intercept:

No -intercepts

–4 –3 –2 –1 1 2

1

3

4

5

6

x

y

x

!0, 134 "y

!#32, 1"

f!x" ! !x " 32"2

" 1 4.

Vertex:

-intercept:

-intercepts:

–6 –4 –2 4 8 10 12

–4

2

4

6

8

10

12

14

x

y

!2, 0", !6, 0"x

!0, 12"y

!4, #4"

f!x" ! !x # 4"2 # 4

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Review Exercises for Chapter 2 171

12. Vertex:

Point:

y ! #2449 !x " 1

4"2" 3

2

! 4916 a " 3

2 ! a ! #2449

!#2, 0" ! 0 ! a!#2 " 14"2

" 32

!#14, 32" ! y ! a!x " 1

4"2" 3

211. Vertex:

Point:

Thus, f !x" ! 2!x " 2"2 # 2.

a ! 2

!#1, 0" ! 0 ! a!#1 " 2"2 # 2

!#2, #2" ! f !x" ! a!x " 2"2 # 2

7.

Vertex:

Intercepts: !#2 ± '7, 0"!0, 3",

!#2, 7"

! 7 # !x " 2"2

! 3 # !x2 " 4x " 4" " 4

!1!2!3!4!6!7 2 3

!2

1234

678

x

yf(x) ! 3 # x2 # 4x

8.

Vertex:

Intercepts: !0, 30"!#53, 0",!#6, 0",

x ! #6, #53

x " 236 ! ±'169

36 ! ±136

3!x " 236 "2 ! 169

12

!#236 , #169

12 " ! 3!x " 23

6 "2# 169

12!1!3!4!5!7!8!9 1

!4!6!8

!10!12!14!16!18

2x

y

f(x) ! 3!x2 " 233 x " 529

36 " " 30 # 52912

9. Vertex:

Point:

Thus, f !x" ! !x # 1"2 # 4.

1 ! a

!2, #3" ! #3 ! a!2 # 1"2 # 4

!1, #4" ! f !x" ! a!x # 1"2 # 4 10. Vertex:

Point:

y ! #14!x # 2"2 " 3

! 4a " 3 ! a ! #14

!0, 2" ! 2 ! a!0 # 2"2 " 3

!2, 3" ! y ! a!x # 2"2 " 3

6.

Vertex:

-intercept:

-intercepts:

#2 "13'3, 0$, #2 #

13'3, 0$

! 2 ±13'3x !

12 ± '126

x

!0, 11"y

!2, #1"

! 3!x # 2"2 # 1

! 3%!x # 2"2 #13&

! 3#x2 # 4x " 4 # 4 "113 $

–6 –4 –2 4 6 8 10

2

4

6

8

10

12

14

x

yf!x" ! 3x2 # 12x " 11

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13. (a) since

Since the figure is in the first quadrant and and must be positive, the domain of

(b)

The dimensions that will produce a maximumarea seem to be and

(d)

The maximum area of 8 occurs when x ! 4 and y !8 # 4

2! 2.

! #12

!x # 4"2 " 8

! #12

(!x # 4"2 # 16)

! #12

!x2 # 8x " 16 # 16"

! #12

!x2 # 8x"

!12

!8x # x2"

A ! x#8 # x2 $

y ! 2.x ! 4

A ! x#8 # x2

$ is 0 < x < 8.

yx

x " 2y # 8 ! 0 ! y !8 # x

2.A ! xy ! x#8 # x

2 $,

x y Area

1

2

3

4

5

6 !6"(4 # 12 !6") ! 64 # 1

2 !6"!5"(4 # 1

2 !5") ! 1524 # 1

2 !5"!4"(4 # 1

2 !4") ! 84 # 12 !4"

!3"(4 # 12 !3") ! 15

24 # 12 !3"

!2"(4 # 12 !2") ! 64 # 1

2 !2"!1"(4 # 1

2 !1") ! 724 # 1

2 !1"

(c)

The maximum area of 8 occurs at the vertex

when

(e) The answers are the same.

x ! 4 and y !8 # 4

2! 2.

00

8

9

14.

The minimum is 122 units.

C ! 10,000 # 110x " 0.45x2

x C

100 3500

120 3280

124 3279.2

130 3305

x C

121 3278.5

122 3277.8

122.5 3277.8

123 3278.1

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18. (a) (b) (c) (d)

!2!4!10!12 2

!6

!4

x

y

f (x) = ! (x + 7)6 + 2

y = x6

!2!4!6!8 2 4 6 8

!4

!2

x

y

y = x6

f (x) = ! x6 ! 512!2!3 2

!3

!2

!1

1

2

3

x

y

y = x6

f (x) = ! x614

!2!3 2 3

!3

!1

1

2

3

x

y

y = x6

f (x) = x6 ! 2

15.

Total amount of fencing

Area enclosed

Because

The vertex is at Thus feet,

and the dimensions are 375 feet by 187.5 feet.

y !14

!1500 # 6!125"" ! 187.5x ! 125x !#b2a

!#1125

2!#9*2" ! 125.

! #92

x2 " 1125x.

A ! 3x #14$(1500 # 6x)

y !14

!1500 # 6x",

A ! 3xy

6x " 4y ! 1500

x x x

y

16. (a)

(b) The maximum is

To maximize profit, sell 1750 songs.

!1750, 1031.25".

P ! 1.75x # !0.0005x2 " 500"

0 35000

1250

17. (a) (b) (c) (d)

!2!3!5!6 1

!4

!3

!2

2

1

3

4

x

y

y = x5

f (x) = 2(x + 3)5

!2!3!4!5 1 2 3

!3

!2

2

1

4

5

x

y

f (x) = 3 ! x512

y = x5

!2!3!4!5 1 2 3

!4

!3

!2

2

3

4

x

y

y = x5f (x) = x5 + 1

!6 !2!8!10!12 2 4

2

4

6

8

x

y

y = x5

f (x) = (x + 4)5

(c)opens downward. The vertex is at

(d) The maximum profit is P!1750" ! $1031.25.

x !#b2a

!#1.75#0.001

! 1750.

P ! #0.0005x2 " 1.75x # 500

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25. (a)

Zeros:

(b)

(c) Zeros: ; the samex ! #1, 0, 2

!6

!4

6

4

x ! #1, 0, 2

! x2!x # 2"!x " 1" ! 0

x4 # x3 # 2x2 ! x2!x2 # x # 2" 26. (a)

Zeros:

(b)

(c) Zeros: ; the same#1, 0, 0.5

!6

!4

6

4

x ! #1, 0, 12

! #x!2x # 1"!x " 1" ! 0

#2x3 # x2 " x ! #x!2x2 " x # 1"

19.

!18

!12

18

12

fg

f!x" ! 12x3 # 2x " 1; g!x" ! 1

2x3 20.

!12

!12

12

4

f

g

f!x" ! #x4 " 2x3; g!x" ! #x4 21.

The degree is even and theleading coefficient is negative.The graph falls to the leftand right.

f!x" ! #x2 " 6x " 9

22.

The degree is odd and theleading coefficient is positive.The graph falls to the left andrises to the right.

f!x" ! 12x3 " 2x 24.

The degree is odd and theleading coefficient is negative.The graph rises to the left andfalls to the right.

h!x" ! #x5 # 7x2 " 10x23.

The degree is even and theleading coefficient is positive.The graph rises to the leftand right.

f!x" ! 34!x4 " 3x2 " 2"

27. (a)

Zeros:

(b)

(c) Zeros: the samet ! 0, ±1.732,

!6

!4

6

4

t ! 0, ±'3

t3 # 3t ! t!t2 # 3" ! t!t " '3"!t # '3" ! 0 28. (a)

For the quadratic,

Zeros:

(b)

(c) Real zero: x ! #8

!16

!14

8

2

#8, #5 ± '3i

! #5 ± '3i.

x !#10 ± '100 # 112

2

!x " 8"!x2 " 10x " 28"

x3 " 18x2 " 108x " 224 ! 0

#!x " 6"3 # 8 ! 0

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36. (a) Degree is odd and leading coefficient is Rises to the left and falls to the right.

(b)

Zeros:

(c) and (d)

!2!4!5 1 2 4 5

!24!32

2416

324048

x

y

x ! ±3, #23

#3x3 # 2x2 " 27x " 18 ! #!x # 3"!x " 3"!3x " 2"

#3 < 0.

29. (a)

Zeros:

(b)

(c) Zeros: the samex ! #3, 0,

!8

!5

4

3

x ! 0, #3

x!x " 3"2 ! 0 30. (a)

Zeros:

(b)

(c) Zeros: the samet ! 0, ±2,

!7

!5

8

5

t ! 0, ±2

t2!t " 2"!t # 2" ! 0

t2!t2 # 4" ! 0

t4 # 4t2 ! 0

31.

33.

! x3 # 7x2 " 13x # 3

f!x" ! !x # 3"!x # 2 " '3"!x # 2 # '3" ! x4 # 5x3 # 3x2 " 17x # 10

f!x" ! !x " 2"!x # 1"2!x # 5" 32.

34.

! x3 # x2 # 46x " 70

f!x" ! !x " 7"!x # 4 " '6"!x # 4 # '6"

! x4 # 2x3 # 11x2 " 12x

f!x" ! !x " 3"x!x # 1"!x # 4"

35. (a) Degree is even and leading coefficient is Rises to the left and rises to the right.

(b)

Zeros:

(c) and (d)

!2!4!5 1 2 3 4 5

!24!32!40

243240

x

y

±3, #1

x4 # 2x3 # 12x2 " 18x " 27 ! !x # 3"2!x " 1"!x " 3"

1 > 0.

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37.

(a)

(b) Zeros: #2.247, #0.555, 0.802

f !0" < 0, f !1" > 0 ! zero in (0, 1)

f !#1" > 0, f !0" < 0 ! zero in (#1, 0)

f !#3" < 0, f !#2" > 0 ! zero in (#3, #2)

f !x" ! x3 " 2x2 # x # 1

!6

!4

6

4

38. (a)

(b) Zeros: #2.979, #0.554, 3.533

f !3" < 0, f !4" > 0 ! zero in (3, 4)

f !#1" > 0, f !0" < 0 ! zero in (#1, 0)

f !#3" < 0, f !#2" > 0 ! zero in (#3, #2)

f!x" ! 0.24x3 # 2.6x # 1.4

40.

(a)

(b) Zeros: #1.897, 0.738

f !0" < 0, f !1" > 0 ! zero in (0, 1)

f !#2" > 0, f !#1" < 0 ! zero in (#2, #1)

f!x" ! 2x4 " 72 x3 # 2

39.

(a)

(b) Zeros: ±2.570

f !2" < 0, f !3" > 0 ! zero in (2, 3)

f !#3" > 0, f !#2" < 0 ! zero in (#3, #2)

f !x" ! x4 # 6x2 # 4!5

!14

5

4

41.

!16

!10

20

14

!x2

x # 2! y1

!x2 # 4x # 2

"4

x # 2

!!x " 2"!x # 2"

x # 2"

4x # 2

y2 ! x " 2 "4

x # 2

y1 !x2

x # 242.

!11 7!14

12

!x2 " 2x # 1

x " 3! y1

!!x # 1"!x " 3" " 2

x " 3

y2 ! x # 1 "2

x " 3

y2 ! x # 1 "2

x " 3y1 !

x2 " 2x # 1x " 3

,

!6 6

!5

3

!5 4

!4

2

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49.

Thus,

x $12

!3 ±'5 ".

5x3 # 13x2 # x " 2x2 # 3x " 1

! 5x " 2,

0

2x2 # 6x " 2

2x2 # 6x " 2

5x3 # 15x2 " 5x

x2 # 3x " 1 ) 5x3 # 13x2 # x " 2

5x " 2 50.

Thus,x4 " x3 # x2 " 2x

x2 " 2x! x2 # x " 1, !x $ 0, #2".

0

x2 " 2x

x2 " 2x

#x3 # 2x2

#x3 # x2

x4 " 2x3

x2 " 2x ) x4 " x3 # x2 " 2x

x2 # x " 1

47.

Thus,x4 # 3x2 " 2

x2 # 1! x2 # 2, !x $ ±1".

0

# 2x2 " 2

# 2x2 " 2

x4 # x2

x2 # 1 ) x4 # 3x2 " 2

x2 # 2 48.

Thus,3x4 " x2 # 1

x2 # 1! 3x2 " 4 "

3x2 # 1

.

3

4x2 # 4

4x2 # 1

3x4 # 3x2

x2 # 1 ) 3x4 " x2 # 1

3x2 " 4

43.

!12

!4

12

12

!x4 " 1x2 " 2

! y1

!x4 " 2x2 # 2x2 # 4 " 5

x2 " 2

!x2!x2 " 2"

x2 " 2#

2!x2 " 2"x2 " 2

"5

x2 " 2

y2 ! x2 # 2 "5

x2 " 2

y1 !x4 " 1x2 " 2

44.

!6 6

!2

6

!x4 " x2 # 1

x2 " 1! y1

!x2!x2 " 1" # 1

x2 " 1

y2 ! x2 #1

x2 " 1

y1 !x4 " x2 # 1

x2 " 1

45.

Thus,24x2 # x # 8

3x # 2! 8x " 5 "

23x # 2

.

2

15x # 10

15x # 8

24x2 # 16x

3x # 2 ) 24x2 # x # 8

8x " 5 46.

4x2 " 73x # 2

!43

x "89

"799

3x # 2!

43

x "89

"79

27x # 18

799

83 x # 169

83x " 7

4x2 # 83 x

3x # 2 ) 4x2 " 0x " 7

43 x " 89

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51.

6x4 " 10x3 " 13x2 # 5x " 22x2 # 1

! 3x2 " 5x " 8 "10

2x2 # 1

10

16x2 " 0 # 8

16x2 # 0 " 2

10x3 " 0x2 # 5x

10x3 " 16x2 # 5x

6x4 " 0x3 # 3x2

2x2 " 0x # 1 ) 6x4 " 10x3 " 13 x2 # 5x " 2

3x2 " 5x " 8

52.

x4 # 3x3 " 4x2 # 6x " 3x2 " 2

! x2 # 3x " 2 "#1

x2 " 2

# 1

2x2 " 4

2x2 " 3

# 3x3 # 6x

# 3x3 " 2x2 # 6x

x4 " 2x2

x2 " 2 ) x4 # 3x3 " 4x2 # 6x " 3

x2 # 3x " 2

53.

Hence,

0.25x4 # 4x3

x " 2!

14

x3 #92

x2 " 9x # 18 "36

x " 2.

#2 0.25

14

#4#1

2

#92

09

9

0#18

#18

036

36

54.

0.1x3 " 0.3x2 # 0.5x # 5

! 0.1x2 " 0.8x " 4 "19.5

x # 5

5 0.1

0.1

0.30.5

0.8

04

4

#0.520

19.5

55.

Thus,6x4 # 4x3 # 27x2 " 18x

x # !2%3" ! 6x3 # 27x, x $23

.

23 6

6

#44

0

#270

#27

18#18

0

00

0

57.

Thus,

3x3 # 10x2 " 12x # 22x # 4

! 3x2 " 2x " 20 "58

x # 4.

4 3

3

#1012

2

128

20

#2280

58

56.

2x3 " 2x2 # x " 2x # !1*2" ! 2x2 " 3x "

12

"9*4

x # !1*2"

12 2

2

21

3

#13212

21494

58.

2x3 " 6x2 # 14x " 9x # 1

! 2x2 " 8x # 6 "3

x # 1

1 2

2

62

8

#148

#6

9#6

3

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63.

(a)

is a factor.

is a factor.!x # 3"

3 1

1

#63

#3

5#9

#4

12#12

0

!x " 2"

#2 1

1

#4#2

#6

#712

5

22#10

12

24#24

0

f !x" ! x4 # 4x3 # 7x2 " 22x " 24

(b)

Remaining factors:

(c)

(d) Zeros:

!4

!10

5

40

#2, 3, 4, #1

f !x" ! !x " 2"!x # 3"!x # 4"!x " 1"

!x # 4", !x " 1"

x2 # 3x # 4 ! !x # 4"!x " 1"

64. (a)

5 1

1

#95

#4

23#20

3

#1515

0

2 1

1

#112

#9

41#18

23

#6146

#15

30#30

0

f!x" ! x4 # 11x3 " 41x2 # 61x " 30 (b) Remaining factors of are .

(c)

(d) Zeros: 1, 2, 3, 5

f !x" ! !x # 2"!x # 5"!x # 3"!x # 1"

!x # 3", !x # 1"x2 # 4x " 3

62. (a)

#6 2

2

11#12

#1

#216

#15

#9090

0

f!x" ! 2x3 " 11x2 # 21x # 90 (b) Remaining factors of are

(c)

(d) Zeros: #6, #52, 3

f !x" ! !x " 6"!2x " 5"!x # 3"

!2x " 5", !x # 3".2x2 # x # 15

59. (a)

! f !#3"

#3 1

1

10#3

7

#24#21

#45

20135

155

44#465

#421

(b)

! f !#2"

#2 1

1

10#2

8

#24#16

#40

2080

100

44#200

#156

60.

(a)

(b)

! g!'2 "

'2 2

2

#52'2

2'2 # 5

04 # 5'2

4 # 5'2

04'2 # 10

4'2 # 10

#88 # 10'2

#10'2

20#20

0

! g!#4"

#4 2

2

#5#8

#13

052

52

0#208

#208

#8832

824

20#3296

#3276

g!t" ! 2t5 # 5t4 # 8t " 20

61.

(a)

is a factor.

(b)

Remaining factors: !x " 1", !x " 7"

x2 " 8x " 7 ! !x " 1"!x " 7"

!x # 4"

4 1

1

44

8

#2532

7

#2828

0

f !x" ! x3 " 4x2 # 25x # 28

(c)

(d) Zeros:

!10

!60

10

80

4, #1, #7

f !x" ! !x # 4"!x " 1"!x " 7"

©H

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65.


Zeros: 1, 1, 34

±3, ±32, ±3

4, ±1, ±12, ±1

4

f!x" ! 4x3 # 11x2 " 10x # 3 66.


Actual rational zeros: #2, 12, #35

± 310, ±2

5, ±35, ±6

5, ±32±1, ±2, ±3, ±6, ±1

5, ± 110, ±1

2,

f!x" ! 10x3 " 21x2 # x # 6

67.

Real zero: 56

! !6x # 5"!x2 " 4"

f!x" ! 6x3 # 5x2 " 24x # 20 68.

Zeros: #1, 2, 310

! 110!x # 2"!x " 1"!10x # 3"

f!x" ! x3 # 1.3x3 # 1.7x " 0.6

69.


Use a graphing utility to see that and are probably zeros.

Thus, the zeros of f are x ! #1, x ! 3, x ! 23, and x ! 3

2.

! !x " 1"!x # 3"!3x # 2"!2x # 3"

6x4 # 25x3 " 14x2 " 27x # 18 ! !x " 1"!x # 3"!6x2 # 13x " 6"

3 6

6

#3118

#13

45#39

6

#1818

0

#1 6

6

#25#6

#31

1431

45

27#45

#18

#1818

0

x ! 3x ! #1

±1, ±2, ±3, ±6, ±9, ±18, ±12, ±3

2, ±92, ±1

3, ±23, ±1

6

f!x" ! 6x4 # 25x3 " 14x2 " 27x # 18

70.

No real zeros

! !5x2 " 1"!x2 " 25"

f!x" ! 5x4 " 126x2 " 25

71. has two variations in sign 0 or 2 positive real zeros.

has one variation in sign 1 negative real zero.!g!#x" ! #5x3 " 6x " 9

!g!x" ! 5x3 # 6x " 9

72. has three variations in sign 1 or 3 positive real zeros.

has no variations in sign 0 negative real zeros.! f!#x" ! #2x5 # 3x2 # 2x # 1

! f!x" ! 2x5 # 3x2 " 2x # 1

73.

All entries positive; is upper bound.

Alternating signs; is lower bound.x ! #14

#14 4

4

#3#1

#4

41

5

#3#5

4

#174

x ! 1

1 4

4

#34

1

41

5

#35

2

74.

All positive is upper bound.

Alternating signs is lower bound. ! x ! #4

#4 2

2

#5#8

#13

#1452

38

8#152

#144

! x ! 8

8 2

2

#516

11

#1488

74

8592

600

75. 6 " '#25 ! 6 " 5i 76. #'#12 " 3 ! #2'3 i " 3 ! 3 # 2'3 i

77. #2i2 " 7i ! 2 " 7i 78. #i2 # 4i ! 1 # 4i

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79.

! 3 " 7i

!7 " 5i" " !#4 " 2i" ! !7 # 4" " !5i " 2i" 80. #'22

#'22

i$ # #'22

"'22

i$ ! #'2 i

81. 5i!13 # 8i" ! 65i # 40i2 ! 40 " 65i 82. !1 " 6i"!5 # 2i" ! 5 # 2i " 30i " 12 ! 17 " 28i

83.

! #26 " 7i

! #20 # 8i " 15i # 6

!'#16 " 3"!'#25 # 2" ! !4i " 3"!5i # 2" 84.

! 29

! 25 " 4

!5 # '#4"!5 " '#4" ! !5 # 2i"!5 " 2i"

85.

! 3 " 9i

'#9 " 3 " '#36 ! 3i " 3 " 6i 86.

! 7 # 2i

7 # '#81 " '#49 ! 7 # 9i " 7i

89.

! #80

!3 " 7i"2 " !3 # 7i"2 ! !9 " 42i # 49" " !9 # 42i # 49"

87. ! #4 # 46i

!10 # 8i"!2 # 3i" ! 20 # 30i # 16i " 24i2 88.

! i!20 # 9i" ! 9 " 20i

i!6 " i"!3 # 2i" ! i!18 " 3i # 12i " 2"

90.

! #16i

!4 # i"2 # !4 " i"2 ! !16 # 8i # 1" # !16 " 8i # 1"

91.

!#6i " 1

1! 1 # 6i

6 " ii

!6 " i

i&

#i#i

!#6i # i2

#i292.

4#3i

!#43i &

#i#i

!4i3

!43

i

93.

!1726

"726

i

3 " 2i5 " i &

5 # i5 # i

!15 " 10i # 3i " 2

25 " 194.

!#1913

"#1713

i

1 # 7i2 " 3i &

2 # 3i2 # 3i

!2 # 21 # 17i

4 " 9

95. #3 # 2i 96. 2 # i

97.

4321

!2!3!4!5!6

54321!1!2!3!4!5

Realaxis

Imaginaryaxis

2 # 5i 98.

54321!1!2!3!4!5

5

321

!2!3!4!5

Realaxis

Imaginaryaxis

#1 " 4i 99.

321

!2!3

!7

!4!5!6

54321!1!2!3!4!5

Realaxis

Imaginaryaxis

#6i

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106.

Quadratic:

Zeros:

g!x" ! !x " 1"!3x # 1"!x # 1 "'3i"!x # 1 #'3i"

#1, 13

, 1 ± '3i

x !2 ± '4 # 16

2! 1 ± '3i

! !x " 1"!3x # 1"!x2 # 2x " 4"

g!x" ! 3x4 # 4x3 " 7x2 " 10x # 4

107.

is a zero. Applying the Quadratic Formulaon

Zeros:

h!x" ! !x # 4"#x #3 " '15i

2 $#x #3 # '15i

2 $

4, 32

"'15

2i,

32

#'15

2i

x !3 ±'9 # 4!6"

2!

32

±'15

2i.

x2 # 3x " 6,x ! 4

4 1

1

#74

#3

18#12

6

#2424

0

h!x" ! x3 # 7x2 " 18x # 24 108.

Quadratic:

Zeros:

f !x" ! !2x " 5"#x #52

"'72

i$#x #52

#'72

i$#

52

, 52

±'72

i

x !5 ± '25 # 32

2!

5 ± '7i2

! !2x " 5"!x2 # 5x " 8"

f !x" ! 2x3 # 5x2 # 9x " 40

105.

is a zero.

is a zero.

By the Quadratic Formula, applied to

Zeros:

f !x" ! !x # 2"!2x " 3"!x # 1 " i"!x # 1 # i"

2, #32

, 1 ± i

x !2 ±'4 # 4!2"

2! 1 ± i.

x2 # 2x " 2,

! !x # 2"!2x " 3"!x2 # 2x " 2"

f !x" ! !x # 2"#x "32$!2x2 # 4x " 4"

x ! #32

#32 2

2

#1#3

#4

#26

4

6#6

0

x ! 2

2 2

2

#54

#1

0#2

#2

10#4

6

#1212

0

f !x" ! 2x4 # 5x3 " 10x # 12

103.

Zeros: 0, 2, 2

f !x" ! 3x!x # 2"2 104.

Zeros: 4, #9, #9

f !x" ! !x # 4"!x " 9"2

100.

54321!1!2!3!4!5

5

34

6

87

2

!2

1Realaxis

Imaginaryaxis

7i 101. 3

Realaxis4321!1!2!3!4

4

3

2

1

!2

!3

!4

Imaginaryaxis

102.

54321!1!2!3!4!5

5

34

21

!2!3!4!5

Realaxis

Imaginaryaxis

#2

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111.

(a)

By the Quadratic Formula, for

Zeros:

(b)

(c) -intercept:

!5

!6

7

2

!2, 0"x

f!x" ! !x # 2"!x # 1 # i"!x # 1 " i"

2, 1 " i, 1 # i

x !2 ±'!#2"2 # 4!2"

2! 1 ± i.

x2 # 2x " 2,

x3 # 4x2 " 6x # 4 ! !x # 2"!x2 # 2x " 2"

f!x" ! x3 # 4x2 " 6x # 4 112. (a)

Zeros:

(b)

(c) x-intercept: !#3, 0"

f !x" ! !x " 3"!x # 4 # i"!x # 4 " i"

#3, 4 " i, 4 # i

! 4 ± i

x !8 ± '!#8"2 # 4!17"

2!

8 ± '#42

! !x " 3"!x2 # 8x " 17"

f !x" ! x3 # 5x2 # 7x " 51

113. (a)

(b)

(c) -intercepts: !#1, 0", !#6, 0", #23

, 0$x

! #!x " 1"!3x # 2"!x " 6"

f!x" ! #!x " 1"!3x2 " 16x # 12"

#1 #3

#3

#193

#16

#416

12

12#12

0

f!x" ! #3x3 # 19x2 # 4x " 12 114. (a)

Zeros:

(b)

(c) x-intercept: #52

, 0$ f !x" ! !2x # 5"!x # 1 # '5i"!x # 1 " '5i"

52

, 1 " '5i, 1 # '5i

x !2 ± '!#2"2 # 4!6"

2! 1 ± '5i

! !2x # 5"!x2 # 2x " 6"

f !x" ! 2x3 # 9x2 " 22x # 30

115.

(a)

Zeros:

(b)

(c) No x-intercepts

(d)

!5

!100

5

800

!x " 3i"!x # 3i"!x " 5i"!x # 5i"

±3i, ±5i

x4 " 34x2 " 225 ! !x2 " 9"!x2 " 25"

f !x" ! x4 " 34x2 " 225

109.

Zeros: 0, 0, #1, ±'5i

! x2!x " 1"!x " '5i"!x # '5i" ! x2!x " 1"!x2 " 5"

! x2 (x2!x " 1" " 5!x " 1")

! x2!x3 " x2 " 5x " 5"

f!x" ! x5 " x4 " 5x3 " 5x2 110.

Zeros: 0, ±1, ±2

f!x" ! x!x # 2"!x " 2"!x # 1"!x " 1"

! x!x2 # 4"!x2 # 1"

! x!x4 # 5x2 " 4"

f!x" ! x5 # 5x3 " 4x

116. (a), (b)

Zeros:

(c) x-intercept: !#5, 0"

± i, #5, #5

! !x2 " 1"!x " 5"2 ! !x " i"!x # i"!x " 5"2

! !x2 " 1"!x2 " 10x " 25"

f !x" ! x4 " 10x3 " 26x2 " 10x " 25

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121.

(a)

For the quadratic

(b)

(c) f !x" ! !x " 3i"!x # 3i"!x # 1 " '2"!x # 1 # '2" f !x" ! !x2 " 9"!x # 1 " '2 "!x # 1 # '2 "

x2 # 2x # 1, x !2 ±'!#2"2 # 4!#1"

2! 1 ±'2.

f !x" ! !x2 " 9"!x2 # 2x # 1"

f !x" ! x4 # 2x3 " 8x2 # 18x # 9

122.

(a)

(b)

(c)

f !x" ! #x #12

#'17

2 $#x #12

"'17

2 $#x #32

"'72

i$#x #32

#'72

i$

x !3 ± '!#3"2 # 4!4"

2!

32

±'72

i

f !x" ! #x #12

#'17

2 $#x #12

"'17

2 $!x2 # 3x " 4"

x !1 ± '!#1"2 # 4!#4"

2!

12

±'17

2

f !x" ! !x2 # x # 4"!x2 # 3x " 4"

f !x" ! x4 # 4x3 " 3x2 " 8x # 16

123. Zeros:

is a factor.

Zeros: ±2i, #3

f!x" ! !x2 " 4"!x " 3"

!x " 2i"!x # 2i" ! x2 " 4

#2i, 2i

124. Zeros:

is a factor.

Zeros: x ! #12, 2 ± '5i

f!x" ! !x2 # 4x " 9"!2x " 1"

!x # 2 # '5i"!x # 2 " '5i" ! !x # 2"2 " 5 ! x2 # 4x " 9

2 " '5i, 2 # '5i

125. (a) Domain: all

(b) Horizontal asymptote:

Vertical asymptote: x ! #3

y ! #1

x $ #3 126. (a) Domain: all



y ! 4

x $ 8

120.

! x4 " 6x3 " 4x2 " 64

! !x2 " 8x " 16"!x2 # 2x " 4"

! !x2 " 8x " 16"!!x # 1"2 " 3"

f !x" ! !x " 4"!x " 4"!x # 1 # '3i"!x # 1 " '3i"


! x4 # 2x3 " 17x2 # 50x # 200

! !x2 # 2x # 8"!x2 " 25"

f!x" ! !x # 4"!x " 2"!x # 5i"!x " 5i"

#5i.5i 118. Since is a zero, so is

! x4 # 16

! !x2 # 4"!x2 " 4"

f!x" ! !x # 2"!x " 2"!x # 2i"!x " 2i"

#2i.2i

119.

! x4 " 9x3 " 48x2 " 78x # 136

! !x2 " 3x # 4"!x2 " 6x " 34"

! !x2 " 3x # 4"!!x " 3"2 " 25"

f !x" ! !x # 1"!x " 4"!x " 3 # 5i"!x " 3 " 5i"

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131.

(a) Domain: all


Vertical asymptotes: x ! ±'32

! ±'62

y ! 2

x $ ±'32

! ±'62

f !x" !4x2

2x2 # 3132.

(a) Domain: all x


No vertical asymptote

y ! 3

f !x" !3x2 # 11x # 4

x2 " 2

133.

(a) Domain: all


There is a hole at


x ! 5."!x ! #3

x $ 5, #3

x $ 5

f !x" !2x # 10

x2 # 2x # 15!

2!x # 5"!x # 5"!x " 3" !

2x " 3

, 134.

(a) Domain: all


No horizontal asymptotes

x ! #2, x ! #1

x $ #1, #2

f !x" !x3 # 4x2

x2 " 3x " 2!

x2!x # 4"!x " 2"!x " 1"

135.

(a) Domain: all real numbers

(b) No vertical asymptotes

Horizontal asymptotes: y ! 1, y ! #1

f !x" !x # 2

+x+ " 2136.

(a) Domain: all


Horizontal asymptotes: (to the right)(to the left)y ! #1

y ! 1

x !12

x $12

f !x" !2x

+2x # 1+

137.

(a) When million.

When million.

When million.C !528!75"

100 # 75! 1584p ! 75,

C !528!50"

100 # 50! 528p ! 50,

C !528!25"

100 # 25! 176p ! 25,

C !528p

100 # p, 0 " p < 100

(b)

(c) No. As tends to infinity.Cp # 100,

00

100

5000

129.

(a) Domain: all



y ! #1

x $ 7

f !x" !7 " x7 # x

130.

(a) Domain: all


Vertical asymptotes: x ! ±1

y ! 0

x $ ±1

f !x" !6x

x2 # 1!

6x!x " 1"!x # 1"

127.

(a) Domain: all


Vertical asymptotes: x ! 6, x ! #3

y ! 0

x $ 6, #3

f !x" !2

x2 # 3x # 18!

2!x # 6"!x " 3" 128. The denominator has no zeros.

Domain: all


Vertical asymptotes: none

y ! 2

x

x2 " x " 3

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139.



No slant asymptotes

Hole at #1, #32$x ! 1:

y ! 1

x ! #1

!x # 4x " 1

, x $ 1

!!x # 4"!x # 1"!x # 1"!x " 1"

f!x" !x2 # 5x " 4

x2 # 1140.



No slant asymptotes

Holes: none

y ! 1

x ! ±2

!x2 # 3x # 8

!x # 2"!x " 2"

f!x" !x2 # 3x # 8

x2 # 4

141.

Vertical asymptote:


No slant asymptotes

Hole at #3, 59$x ! 3:

y ! 1

x ! #32

!2x # 12x " 3

, x $ 3

!!x # 3"!2x # 1"!x # 3"!2x " 3"

f!x" !2x2 # 7x " 32x2 # 3x # 9

142.

Vertical asymptote:


No slant asymptotes

Hole at ##5, 179 $x ! #5:

y !32

x ! #12

!3x # 22x " 1

, x $ #5

!!x " 5"!3x # 2"!x " 5"!2x " 1"

f!x" !3x2 " 13x # 102x2 " 11x " 5

143.

Vertical asymptote:


Slant asymptote:

Hole at !#2, #28"x ! #2:

y ! 3x # 10

x ! #1

! 3x # 10 "12

x " 1, x $ #2

!!x # 2"!3x # 1"

x " 1, x $ #2

!!x # 2"!x " 2"!3x # 1"

!x " 1"!x " 2"

f!x" !3x3 # x2 # 12x " 4

x2 " 3x " 2144.

Vertical asymptote:


Slant asymptote:

Hole at !1, #10"x ! 1:

y ! 2x " 9

x ! 2

! 2x " 9 "21

x # 2, x $ 1

!!x " 1"!2x " 3"

x # 2, x $ 1

!!x # 1"!x " 1"!2x " 3"

!x # 1"!x # 2"

f !x" !2x3 " 3x2 # 2x # 3

x2 # 3x " 2

138.

The moth will be satiated at the horizontal asymptote, mg.y !1.5686.360

, 0.247

y !1.568x # 0.001

6.360x " 1, x > 0

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146.

x-intercept:

y-intercept:

Vertical asymptote:


x

65432

!2!3!4!5!6

8765431!1!2!3

0, 32 )(

(3, 0)

y

y ! 1

x ! 2

#0, 32$

!3, 0"

f !x" !x # 3x # 2

x 0 1 3 4 5

y 2 0 23

12

32

43

#1

147.

Intercept:

Origin symmetry


3

2

1

!1

!2

!3

321

(0, 0)

x

y

y ! 0

!0, 0"

f!x" !2x

x2 " 4

x 0 1 2

y 0 12

25#2

5#12

#1#2

148.

Intercept:

y-axis symmetry



–6 –4 4 6

4

6

x(0, 0)

y

y ! 2

x ! #2

x ! 2,

!0, 0"

f !x" !2x2

x2 # 4

x 0

y 0#23

185

83

5021

±1±3±4±5

149.

Intercept:

y-axis symmetry


(0, 0)–3 –2 –1 2 3

–2

2

3

4

x

y

y ! 1

!0, 0"

f !x" !x2

x2 " 1

x 0

y 012

45

910

±1±2±3

145.

Intercepts:

Vertical asymptote:


!!!

( )

10864

!4

!6!8

!10

141210864246

2

0, 15

12

)(

, 0

x

y

y ! 2

x ! 5

#0, 15$, #1

2, 0$

f !x" !2x # 1x # 5

150.

Intercept:

Symmetry: origin


No vertical asymptotes

y ! 0

!0, 0"

x

54321

!3!4!5

54321!1!2!3!4

(0, 0)

y

f !x" !5x

x2 " 1

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151.

Intercept:


Vertical asymptote:

x4321!2!3!4!5!6

34

21

!1

(0, 2)

y

x ! #1

y ! 0

!0, 2"

f !x" !2

!x " 1"2 152.

-intercept:

Vertical asymptote:


–3 –2 –1 2 3 4 5

1

3

5

6

7

x

(0, 4)

y

y ! 0

x ! 1

!0, 4"y

h!x" !4

!x # 1"2

x 0 2 3 4

y 1 4 4 1 49

49

#1#2

153.

Intercept:

Origin symmetry

Slant asymptote:

1 2 3 4!2!3!4

!3

!4

1

2

3

4

(0, 0)

y x= 2

x

y

y ! 2x

!0, 0"

f!x" !2x3

x2 " 1! 2x #

2xx2 " 1

x 0 1 2

y 0 1 165#1#16

5

#1#2

154.

Intercepts:


Slant asymptote:

x

54321

!3!2

!4!5

5432!1!5 !4

(0, 0)

y = 13

x

y

y !13

x

x ! ±'2

!0, 0"

f!x" !x3

3x2 # 6!

13

x "2x

3x2 # 6!

13%x "

2xx2 # 2&

155.

Intercept:

Vertical asymptote:

Slant asymptote:

4 8 12 16 20

16

12

8

!4

!8

0, ! 13)(

x

y

y x= + 2

y ! x " 2

x ! 3

#0, #13$

f !x" !x2 # x " 1

x # 3! x " 2 "

7x # 3

156.

Intercepts:

Vertical asymptote:

Slant asymptote:

x321!2!4!5!6!7

567

(0, 3)

y x= 2 + 5

y

y ! 2x " 5

x ! #1

!0, 3", !#3, 0", ##12

, 0$

f !x" !2x2 " 7x " 3

x " 1! 2x " 5 #

2x " 1

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157.

(a)

(b) fish

fish

fish

(c) The limit is

fish, the horizontal asymptote.60

0.05! 1,200,000

N!25" , 702,222

N!10" , 453,333

N!5"1 ! 304,000

00

25

800

N !20!4 " 3t"1 " 0.05t

, t $ 0 158. (a)

(b)

(c) Domain:

(d)

9.48 by 9.48

40

16

200

x > 4

!2x!2x " 7"

x # 4

! x%4x # 16 " 30x # 4 &

Area ! A ! xy ! x%4 "30

x # 4&

!x # 4"!y # 4" ! 30 ! y ! 4 "30

x # 4

x

y30 in.2

2 in.

2 in.

163. (a)

(b)

(c)

Yes, the model is a good fit.

(d) From the model, when or2005.

(e) Answers will vary.

t , 15.1,y $ 500

5 15250

450

y ! 8.03t2 # 157.1t " 1041; 0.98348

5 15250

450

159. Quadratic model 160. Neither 161. Linear model 162. Quadratic

164. (a)

(b)

Yes, the model is an excellent fit.

(c)

(d)

The quadratic model is not a good fit.

(e) The cubic model is better.

(f ) Answers will vary.

8 145

6.3

y ! 0.129t2 # 2.99t " 22.8

8 145

6.3

8 145

6.3

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165. False. The degree of the numerator is two morethan the degree of the denominator.

166. False. A fourth degree polynomial with realcoefficients can have at most four zeros. Since

and are zeros, so are and .#4i8i4i#8i

168. It means that the divisor is a factor of the dividend.

169. Not every rational function has a vertical asymptote. For example,

y !x

x2 " 1.

167. False. a real number!1 " i" " !1 # i" ! 2,

170.

In fact, '#6'#6 ! '6i '6i ! #6.

'#6'#6 $ '!#6"!#6"

171. The error is In fact,

#i!'#4 # 1" ! #i!2i # 1" ! 2 " i.

'#4 $ 4i. 172. (a)

(b)

(c)

(d) i67 ! i3!i64" ! #i!1" ! #i

i50 ! i2!i48" ! !#1"!1" ! #1

i25 ! i!i24" ! i!1" ! i

i40 ! !i4"10 ! 110 ! 1

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Practice Test for Chapter 2 191

Chapter 2 Practice Test

1. Sketch the graph of byhand and identify the vertex and the intercepts.

f!x" ! x2 " 6x # 5 2. Find the number of units that produce a minimum cost if C ! 0.01x2 " 90x # 15,000.C

x

3. Find the quadratic function that has a maximumat and passes through the point !2, 5".!1, 7"

4. Find two quadratic functions that have-intercepts and !4

3, 0".!2, 0"x

5. Use the leading Coefficient Test to determine the right-hand and left-handbehavior of the graph of the polynomial function f!x" ! "3x5 # 2x3 " 17.

6. Find all the real zeros of Verify your answer with a graphing utility.f!x" ! x5 " 5x3 # 4x.

7. Find a polynomial function with 0, 3,and as zeros."2

8. Sketch by hand.f!x" ! x3 " 12x

9. Divide using long division.

3x4 " 7x2 # 2x " 10 by x " 3 10. Divide x3 " 11 by x2 # 2x " 1.

11. Use synthetic division to divide3x5 # 13x4 # 12x " 1 by x # 5.

12. Use synthetic division to find f!"6" when f !x" ! 7x3 # 40x2 " 12x # 15.

13. Find the real zeros of f!x" ! x3 " 19x " 30.

14. Find the real zeros of f!x" ! x4 # x3 " 8x2 " 9x " 9.

15. List all possible rational zeros of the function f!x" ! 6x3 " 5x2 # 4x " 15.

20. Find a polynomial with real coefficients that has 2, and as zeros.3 " 2i3 # i,

21. Use synthetic division to show that isa zero of f!x" ! x3 # 4x2 # 9x # 36.

3i

25. Find all the asymptotes of f!x" !8x2 " 9x2 # 1

.

26. Find all the asymptotes of f!x" !4x2 " 2x # 7

x " 1.

22. Find a mathematical model for the statement, “varies directly as the square of and inversely asthe square root of ”.y

xz 23. Sketch the graph of and label all

intercepts and asymptotes.

f!x" !x " 1

2x

24. Sketch the graph of and label all

intercepts and asymptotes.

f!x" !3x2 " 4

x

27. Sketch the graph of f!x" !x " 5

!x " 5"2 .

16. Find the rational zeros of the polynomial f!x" ! x3 " 20

3 x2 # 9x " 103 .

17. Write as a product of linear factors.

f !x" ! x4 # x3 # 3x2 # 5x " 10

18. Write in standard form.2

1 # i19. Write in standard form.

3 # i2

"i # 1

4

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Documents

CHAPTER 2 Polynomial and Rational Functionscrunchymath.weebly.com/uploads/8/2/4/0/8240213/ch2.1-2.3_solutions.pdf92 Chapter 2 Polynomial and Rational Functions 32. is the vertex. Since