D. Division of Complex Numbers
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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 ⍀).
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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.