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D. Division of Complex Numbers

D. Division of Complex Numbers

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Section 3.1 Complex Numbers



289



3.1 EXERCISES





CONCEPTS AND VOCABULARY



Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.



1. Given the complex number 3 ϩ 2i, its complex

conjugate is

.

4 ϩ 6i12

is written in the standard

2

form a ϩ bi, then a ϭ

and b ϭ

.



3. If the expression



5. Discuss/Explain which is correct:

a. 1Ϫ4 # 1Ϫ9 ϭ 11Ϫ42 1Ϫ92 ϭ 136 ϭ 6

b. 1Ϫ4 # 1Ϫ9 ϭ 2i # 3i ϭ 6i 2 ϭ Ϫ6





2. The product 13 ϩ 2i2 13 Ϫ 2i2 gives the real

number

.

4. For i ϭ 1Ϫ1, i 2 ϭ

, i4 ϭ

, i6 ϭ

, and

8

3

5

7

i ϭ

,i ϭ

,i ϭ

,i ϭ

, and i 9 ϭ .

6. Compare/Contrast the product 11 ϩ 12211 Ϫ 132

with the product 11 ϩ i12211 Ϫ i132. What is the

same? What is different?



DEVELOPING YOUR SKILLS



Simplify each radical (if possible). If imaginary, rewrite

in terms of i and simplify.



7. a. 1Ϫ144

c. 127



b. 1Ϫ49

d. 172



8. a. 1Ϫ100

c. 164



Write each complex number in the standard form

a ؉ bi and clearly identify the values of a and b.



17. a. 5



b. 3i



18. a. Ϫ2



b. Ϫ4i



b. 1Ϫ169

d. 198



19. a. 2 1Ϫ81



b.



1Ϫ32

8



9. a. Ϫ 1Ϫ18

c. 3 1Ϫ25



b. Ϫ 1Ϫ50

d. 2 1Ϫ9



20. a. Ϫ31Ϫ36



b.



1Ϫ75

15



10. a. Ϫ 1Ϫ32

c. 3 1Ϫ144



b. Ϫ 1Ϫ75

d. 2 1Ϫ81



21. a. 4 ϩ 1Ϫ50



b. Ϫ5 ϩ 1Ϫ27



11. a. 1Ϫ19

Ϫ12

c.

A 25



b. 1Ϫ31

Ϫ9

d.

A 32



22. a. Ϫ2 ϩ 1Ϫ48



b. 7 ϩ 1Ϫ75



23. a.



14 ϩ 1Ϫ98

8



b.



5 ϩ 1Ϫ250

10



12. a. 1Ϫ17

Ϫ45

c.

A 36



b. 1Ϫ53

Ϫ49

d.

A 75



24. a.



21 ϩ 1Ϫ63

12



b.



8 ϩ 1Ϫ27

6



Simplify each expression, writing the result in terms of i.



13. a.

14. a.

15. a.

16. a.



2 ϩ 1Ϫ4

2

16 Ϫ 1Ϫ8

2

8 ϩ 1Ϫ16

2

6 Ϫ 1Ϫ72

4



b.

b.

b.

b.



6 ϩ 1Ϫ27

3

4 ϩ 31Ϫ20

2

10 Ϫ 1Ϫ50

5

12 ϩ 1Ϫ200

8



Perform the addition or subtraction. Write the result in

a ؉ bi form. Check your answers using a calculator.



25. a. 112 Ϫ 1Ϫ42 ϩ 17 ϩ 1Ϫ92

b. 13 ϩ 1Ϫ252 ϩ 1Ϫ1 Ϫ 1Ϫ812

c. 111 ϩ 1Ϫ1082 Ϫ 12 Ϫ 1Ϫ482



26. a. 1Ϫ7 Ϫ 1Ϫ722 ϩ 18 ϩ 1Ϫ502

b. 1 13 ϩ 1Ϫ22 Ϫ 1 112 ϩ 1Ϫ82

c. 1 120 Ϫ 1Ϫ32 ϩ 1 15 Ϫ 1Ϫ122

27. a. 12 ϩ 3i2 ϩ 1Ϫ5 Ϫ i2

b. 15 Ϫ 2i2 ϩ 13 ϩ 2i2

c. 16 Ϫ 5i2 Ϫ 14 ϩ 3i2



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28. a. 1Ϫ2 ϩ 5i2 ϩ 13 Ϫ i2

b. 17 Ϫ 4i2 Ϫ 12 Ϫ 3i2

c. 12.5 Ϫ 3.1i2 ϩ 14.3 ϩ 2.4i2

29. a. 13.7 ϩ 6.1i2 Ϫ 11 ϩ 5.9i2

3

2

b. a8 ϩ ib Ϫ aϪ7 ϩ ib

4

3

5

1

c. aϪ6 Ϫ ib ϩ a4 ϩ ib

8

2



1

2



Ϫ 23i



48. a. Ϫ5i



b.



3

4



ϩ 15i



49. x2 ϩ 36 ϭ 0; x ϭ Ϫ6

50. x2 ϩ 16 ϭ 0; x ϭ Ϫ4

51. x2 ϩ 49 ϭ 0; x ϭ Ϫ7i

52. x2 ϩ 25 ϭ 0; x ϭ Ϫ5i



53. 1x Ϫ 32 2 ϭ Ϫ9; x ϭ 3 Ϫ 3i



54. 1x ϩ 12 2 ϭ Ϫ4; x ϭ Ϫ1 ϩ 2i



55. x2 Ϫ 2x Ϫ 5 ϭ 0; x ϭ 1 Ϫ 2i



Multiply and write your answer in a ؉ bi form.



b. 14i21Ϫ4i2



32. a. 312 Ϫ 3i2



b.



Use substitution to determine if the value shown is a

solution to the given equation. Use a calculator check

for Exercises 57 to 60.



30. a. 19.4 Ϫ 8.7i2 Ϫ 16.5 ϩ 4.1i2

3

7

b. a3 ϩ ib Ϫ aϪ11 ϩ ib

5

15

3

5

c. aϪ4 Ϫ ib ϩ a13 ϩ ib

6

8

31. a. 5i # 1Ϫ3i2



47. a. 7i



56. x2 ϩ 6x ϩ 11 ϭ 0; x ϭ Ϫ1 Ϫ 3i

57. x2 Ϫ 4x ϩ 9 ϭ 0; x ϭ 2 ϩ i 15



b. Ϫ713 ϩ 5i2



58. x2 Ϫ 2x ϩ 4 ϭ 0; x ϭ 1 Ϫ i13



b. 6i1Ϫ3 ϩ 7i2



34. a. 1Ϫ4 Ϫ 2i213 ϩ 2i2 b. 12 Ϫ 3i21Ϫ5 ϩ i2



59. Verify that x ϭ 1 ϩ 4i is a solution to

x2 Ϫ 2x ϩ 17 ϭ 0. Then show its complex

conjugate 1 Ϫ 4i is also a solution.



36. a. 15 ϩ 2i21Ϫ7 ϩ 3i2 b. 14 Ϫ i217 ϩ 2i2



60. Verify that x ϭ 2 Ϫ 3i 12 is a solution to

x2 Ϫ 4x ϩ 22 ϭ 0. Then show its complex

conjugate 2 ϩ 3i 12 is also a solution.



33. a. Ϫ7i15 Ϫ 3i2



35. a. 1Ϫ3 ϩ 2i212 ϩ 3i2 b. 13 ϩ 2i211 ϩ i2



Compute the special products.



37. a. 14 Ϫ 5i214 ϩ 5i2



Simplify using powers of i.



b. 17 Ϫ 5i217 ϩ 5i2



38. a. 1Ϫ2 Ϫ 7i21Ϫ2 ϩ 7i2

b. 12 ϩ i212 Ϫ i2



61. a. i48



b. i26



c. i39



d. i53



62. a. i36



b. i50



c. i19



d. i65



Divide and write your answer in a ؉ bi form. Check

your answer using multiplication.



39. a. 13 Ϫ i122 13 ϩ i 122

b. 1 16 ϩ 23i21 16 Ϫ 23i2



63. a.



b. 1 12 ϩ 34i21 12 Ϫ 34i2



64. a.



40. a. 15 ϩ i132 15 Ϫ i 132

41. a. 12 ϩ 3i2 2



b. 13 Ϫ 4i2 2



43. a. 1Ϫ2 ϩ 5i2 2



b. 13 ϩ i122 2



42. a. 12 Ϫ i2 2



44. a. 1Ϫ2 Ϫ 5i2



2



65. a.



b. 13 Ϫ i2 2



b. 12 Ϫ i132



66. a.



2



For each complex number given, name the complex

conjugate and compute the product.



45. a. 4 ϩ 5i



b. 3 Ϫ i12



46. a. 2 Ϫ i



b. Ϫ1 ϩ i15



67. a.

68. a.



Ϫ2

1Ϫ49

2

1 Ϫ 1Ϫ4

7

3 ϩ 2i

6

1 ϩ 3i

3 ϩ 4i

4i

Ϫ4 ϩ 8i

2 Ϫ 4i



b.

b.

b.

b.

b.

b.



4

1Ϫ25

3

2 ϩ 1Ϫ9

Ϫ5

2 Ϫ 3i

7

7 Ϫ 2i

2 Ϫ 3i

3i

3 Ϫ 2i

Ϫ6 ϩ 4i



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Section 3.1 Complex Numbers



WORKING WITH FORMULAS



69. Absolute value of a complex number:

ͦa ؉ biͦ ‫ ؍‬2a2 ؉ b2

The absolute value of any complex number a ϩ bi

(sometimes called the modulus of the number) is

computed by taking the square root of the sum of

the squares of a and b. Find the absolute value of

the given complex numbers.

a. | 2 ϩ 3i|

b. | 4 Ϫ 3i |

c. | 3 ϩ i12|





70. Binomial cubes:

1A ϩ B2 3 ϭ A3 ϩ 3A2B ϩ 3AB2 ϩ B3

The cube of any binomial can be found using the

formula shown, where A and B are the terms of the

binomial. Use the formula to compute 11 Ϫ 2i2 3 (note

A ϭ 1 and B ϭ Ϫ2i2.



APPLICATIONS



71. Dawn of imaginary numbers: In a day when

imaginary numbers were imperfectly understood,

Girolamo Cardano (1501–1576) once posed the

problem, “Find two numbers that have a sum of 10

and whose product is 40.” In other words,

A ϩ B ϭ 10 and AB ϭ 40. Although the solution

is routine today, at the time the problem posed an

enormous challenge. Verify that A ϭ 5 ϩ i 115

and B ϭ 5 Ϫ i 115 satisfy these conditions.

72. Verifying calculations using i: Suppose Cardano

had said, “Find two numbers that have a sum of 4

and a product of 7” (see Exercise 71). Verify that

A ϭ 2 ϩ i 13 and B ϭ 2 Ϫ i13 satisfy these

conditions.

Although it may seem odd, complex numbers have

several applications in the real world. Many of these

involve a study of electrical circuits, in particular

alternating current or AC circuits. Briefly, the

components of an AC circuit are current I (in amperes),

voltage V (in volts), and the impedance Z (in ohms). The

impedance of an electrical circuit is a measure of the

total opposition to the flow of current through the

circuit and is calculated as Z ‫ ؍‬R ؉ iXL ؊ iXC where R

represents a pure resistance, XC represents the

capacitance, and XL represents the inductance. Each of

these is also measured in ohms (symbolized by ⍀).





291



73. Find the impedance Z if R ϭ 7 ⍀, XL ϭ 6 ⍀, and

XC ϭ 11 ⍀.

74. Find the impedance Z if R ϭ 9.2 ⍀, XL ϭ 5.6 ⍀,

and XC ϭ 8.3 ⍀.

The voltage V (in volts) across any element in an AC

circuit is calculated as a product of the current I and the

impedance Z: V ‫ ؍‬IZ.



75. Find the voltage in a circuit with a current

I ϭ 3 Ϫ 2i amperes and an impedance of

Z ϭ 5 ϩ 5i ⍀.

76. Find the voltage in a circuit with a current

I ϭ 2 Ϫ 3i amperes and an impedance of

Z ϭ 4 ϩ 2i ⍀.

In an AC circuit, the total impedance (in ohms) is given

Z1Z2

by Z ‫؍‬

, where Z represents the total impedance

Z1 ؉ Z2

of a circuit that has Z1 and Z2 wired in parallel.



77. Find the total impedance Z if Z1 ϭ 1 ϩ 2i and

Z2 ϭ 3 Ϫ 2i.

78. Find the total impedance Z if Z1 ϭ 3 Ϫ i and

Z2 ϭ 2 ϩ i.



EXTENDING THE CONCEPT



79. Up to this point, we’ve said that expressions like

x2 Ϫ 9 and p2 Ϫ 7 are factorable:

x2 Ϫ 9 ϭ 1x ϩ 32 1x Ϫ 32 and



p2 Ϫ 7 ϭ 1p ϩ 1721p Ϫ 172,



while x2 ϩ 9 and p2 ϩ 7 are prime. More correctly,

we should state that x2 ϩ 9 and p2 ϩ 7 are

nonfactorable using real numbers, since they



actually can be factored if complex numbers are

used. Specifically,

1x ϩ 3i21x Ϫ 3i2 ϭ x2 ϩ 9 and



1p ϩ i 172 1p Ϫ i 172 ϭ p2 ϩ 7.

a. Verify that in general,

1a ϩ bi21a Ϫ bi2 ϭ a2 ϩ b2.



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81. Simplify the expression

i17 13 Ϫ 4i2 Ϫ 3i3 11 ϩ 2i2 2.



Use this idea to factor the following.



b. x ϩ 36



c. m ϩ 3



d. n2 ϩ 12



e. 4x2 ϩ 49



2



2



80. In this section, we noted that the product property of

radicals 1AB ϭ 1A1B, can still be applied when

at most one of the factors is negative. So what

happens if both are negative? First consider the

expression 1Ϫ4 # Ϫ25. What happens if you first

multiply in the radicand, then compute the square

root? Next consider the product 1Ϫ4 # 1Ϫ25.

Rewrite each factor using the i notation, then

compute the product. Do you get the same result as

before? What can you say about 1Ϫ4 # Ϫ25 and

1Ϫ4 # 1Ϫ25?





3–12



CHAPTER 3 Quadratic Functions and Operations on Functions



82. While it is a simple concept for real numbers, the

square root of a complex number is much more

involved due to the interplay between its real and

imaginary parts. For z ϭ a ϩ bi the square root of

z can be found using the formula:

12

11ͿzͿ ϩ a Ϯ i 1ͿzͿ Ϫ a2, where the sign

1z ϭ

2

is chosen to match the sign of b (see Exercise 69).

Use the formula to find the square root of each

complex number, then check by squaring.

a. z ϭ Ϫ7 ϩ 24i



b. z ϭ 5 Ϫ 12i



c. z ϭ 4 ϩ 3i



MAINTAINING YOUR SKILLS



83. (1.4) Two boats leave Nawiliwili (Kauai) at the

same time, traveling in opposite directions. One

travels at 15 knots (nautical miles per hour) and the

other at 20 knots. How long until they are 196 mi

apart?

84. (2.4) State the domain of the following functions

using interval notation.

a. f 1x2 ϭ x3

2



c. f 1x2 ϭ



3.2



x

xϪ3



A. Establish a relationship



D.



E.

F.



3



Solving Quadratic Equations and Inequalities



In Section 3.2 you will see how we can:



C.



86. (R.4) Factor the following expressions completely.

a. x4 Ϫ 16

b. n3 Ϫ 27

c. x3 Ϫ x2 Ϫ x ϩ 1

d. 4n2m Ϫ 12nm2 ϩ 9m3



b. f 1x2 ϭ x2



LEARNING OBJECTIVES



B.



85. (1.5) John can run 10 m/sec, while Rick can only

run 9 m/sec. If Rick gets a 2-sec head start, who

will reach the 200-m finish line first?



between zeroes of a

quadratic function and the

x-intercepts of its graph

Solve quadratic equations

using the square root

property of equality

Solve quadratic equations

by completing the square

Solve quadratic equations

using the quadratic formula

and the discriminant

Solve quadratic inequalities

Solve applications of

quadratic functions and

inequalities



In Section R.4 we reviewed how to solve polynomials by factoring and applying the

zero factor property. While this is an extremely valuable skill, many polynomials are

unfactorable, and a more general method for finding solutions is necessary. We begin

our search here, with the family of quadratic polynomials (degree 2).



A. Zeroes of Quadratic Functions and x-Intercepts

of Quadratic Graphs

Understanding quadratic equations and functions is an important step towards a more

general study of polynomial functions. Due to their importance, we begin by restating

their definition:

Quadratic Equations

A quadratic equation is one that can be written in the form

ax2 ϩ bx ϩ c ϭ 0,

where a, b, and c are real numbers and a



0.



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Section 3.2 Solving Quadratic Equations and Inequalities



As shown, the equation is in standard form, meaning the terms are in decreasing

order of degree and the equation is set equal to zero. The family of quadratic functions

is similarly defined.

Quadratic Functions

A quadratic function is one that can be written in the form

f 1x2 ϭ ax2 ϩ bx ϩ c,



where a, b, and c are real numbers and a



0.



In Section R.4, we noted that some quadratic equations have two real solutions, others

have only one, and still others have none. When these possibilities are explored graphically, we note a clear connection between the zeroes of a quadratic function and the

x-intercepts of its graph.

EXAMPLE 1







Solution







Noting Relationships between Zeroes and x-Intercepts



Consider the functions f 1x2 ϭ x2 Ϫ 2x Ϫ 3 and g1x2 ϭ x2 Ϫ 4x ϩ 4.

a. Find the zeroes of each function algebraically.

b. Find the x-intercepts of each function graphically.

c. Comment on how the zeroes and x-intercepts are related.

a. To find the zeroes algebraically, replace f (x) and g(x) with 0, then solve. In

each case, the solutions can be found by factoring.

For f (x):



x2 Ϫ 2x Ϫ 3 ϭ 0

1x Ϫ 321x ϩ 12 ϭ 0

x ϭ 3 or x ϭ Ϫ1



For g(x):



x2 Ϫ 4x ϩ 4 ϭ 0

1x Ϫ 221x Ϫ 22 ϭ 0

x ϭ 2 or x ϭ 2



There are two real solutions,

There is only one real solution,

x ϭ 3 and x ϭ Ϫ1.

but it is repeated twice.

b. To find the x-intercepts, go to the Y= screen and enter the first function as Y1

with Y2 ϭ 0 (the x-axis), then graph them in the standard window. Locate the

x-intercepts using the 2nd TRACE (CALC) 5:Intersect feature.

• For f 1x2 ϭ x2 Ϫ 2x Ϫ 3

Figure 3.9



Figure 3.10



10



Ϫ10



10



10



Ϫ10



Ϫ10



10



Ϫ10



The result shows there are two x-intercepts, which occur at x ϭ Ϫ1 (Figure 3.9)

and x ϭ 3 (Figure 3.10). These were also the real zeroes of f 1x2 ϭ x2 Ϫ 2x Ϫ 3.



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D. Division of Complex Numbers

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