1.
- [latex]\displaystyle \sqrt{-25}-4[/latex]
- [latex]\displaystyle \dfrac{-8+\sqrt{-4}}{2}[/latex]
- [latex]\displaystyle \dfrac{-5-\sqrt{-2}}{6}[/latex]
Chapter 10: Polar Coordinates and Complex Numbers
Exercises: 10.3 Complex Numbers
SKILLS
Practice each skill in the Homework Problems listed.
- Write and simplify complex numbers #1–6
- Perform arithmetic operations on complex numbers #7–34
- Evaluate polynomials at complex numbers, expand polynomials #35–46
- Graph complex numbers #47–58
- Find a polynomial with given zeros #67–74
Suggested Homework
Problems: #2, 6, 10, 16, 22, 28, 40, 44, 52, 56, 70, 72
Exercises for 10.3 Complex Numbers
Exercise Group
For Problems 1–2, write the complex number in the form [latex]a+bi\text{,}[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers.
2.
- [latex]\displaystyle \sqrt{-9}+3[/latex]
- [latex]\displaystyle \dfrac{6-\sqrt{-36}}{2}[/latex]
- [latex]\displaystyle \dfrac{7+\sqrt{-3}}{4}[/latex]
Exercise Group
For Problems 3–6, find the zeros of the quadratic polynomial. Write each zero in the form [latex]a+bi\text{,}[/latex] where [latex]a[/latex] and [latex]b[/latex] are real numbers.
3.
[latex]x^2+6x+13[/latex]
4.
[latex]x^2-2x+10[/latex]
5.
[latex]3x^2-x+1[/latex]
6.
[latex]5x^2+2x+2[/latex]
Exercise Group
For Problems 7–10, add or subtract.
7.
[latex](11-4i)-(-2-8i)[/latex]
8.
[latex](7i-2)+(6-4i)[/latex]
9.
[latex](2.1+5.6i)+(-1.8i-2.9)[/latex]
10.
[latex]\left(\dfrac{1}{5}i-\dfrac{2}{5}\right)-\left(\dfrac{4}{5}-\dfrac{3}{5}i\right)[/latex]
Exercise Group
For Problems 11–20, multiply.
11.
[latex]5i(2-4i)[/latex]
12.
[latex]-7i(-1+4i)[/latex]
13.
[latex](4-i)(-6+7i)[/latex]
14.
[latex](2-3i)(2-3i)[/latex]
15.
[latex](7+i\sqrt{3})^2[/latex]
16.
[latex](5-i\sqrt{2})^2[/latex]
17.
[latex](7+i\sqrt{3})(7-i\sqrt{3})[/latex]
18.
[latex](5-i\sqrt{2})(5+i\sqrt{2})[/latex]
19.
[latex](1-i)^3[/latex]
20.
[latex](2+i)^3[/latex]
Exercise Group
For Problems 21–32, divide.
21.
[latex]\dfrac{12+3i}{-3i}[/latex]
22.
[latex]\dfrac{12+4i}{8i}[/latex]
23.
[latex]\dfrac{10+15i}{2+i}[/latex]
24.
[latex]\dfrac{4-6i}{1-i}[/latex]
25.
[latex]\dfrac{5i}{2-5i}[/latex]
26.
[latex]\dfrac{-2i}{7+2i}[/latex]
27.
[latex]\dfrac{\sqrt{3}}{\sqrt{3}+i}[/latex]
28.
[latex]\dfrac{2\sqrt{2}}{1-i\sqrt{2}}[/latex]
29.
[latex]\dfrac{1+i\sqrt{5}}{1-i\sqrt{5}}[/latex]
30.
[latex]\dfrac{\sqrt{2}-i}{\sqrt{2}+i}[/latex]
31.
[latex]\dfrac{3+2i}{2-3i}[/latex]
32.
[latex]\dfrac{4-6i}{-3-2i}[/latex]
33.
Simplify.
- [latex]\displaystyle i^6[/latex]
- [latex]\displaystyle i^{12}[/latex]
- [latex]\displaystyle i^{15}[/latex]
- [latex]\displaystyle i^{102}[/latex]
34.
Express with a positive exponent and simplify.
- [latex]\displaystyle i^{-1}[/latex]
- [latex]\displaystyle i^{-2}[/latex]
- [latex]\displaystyle i^{-3}[/latex]
- [latex]\displaystyle i^{-6}[/latex]
Exercise Group
For Problems 35-40, evaluate the polynomial for the given values of the variable.
35.
[latex]z^2+9[/latex]
- [latex]\displaystyle z=3i[/latex]
- [latex]\displaystyle z=-3i[/latex]
36.
[latex]2y^2-y-2[/latex]
- [latex]\displaystyle y=2-i[/latex]
- [latex]\displaystyle y=-2-i[/latex]
37.
[latex]x^2-2x+2[/latex]
- [latex]\displaystyle x=1-i[/latex]
- [latex]\displaystyle x=1+i[/latex]
38.
[latex]3w^2+5[/latex]
- [latex]\displaystyle w=2i[/latex]
- [latex]\displaystyle w=-2i[/latex]
39.
[latex]q^2+4q+13[/latex]
- [latex]\displaystyle q=-2+3i[/latex]
- [latex]\displaystyle q=-2-3i[/latex]
40.
[latex]v^2+2v+3[/latex]
- [latex]\displaystyle v=1+i[/latex]
- [latex]\displaystyle v=-1+i[/latex]
Exercise Group
For Problems 41–46, expand the product of polynomials.
41.
[latex](2z+7i)(2z-7i)[/latex]
42.
[latex](5w+3i)(5w-3i)[/latex]
43.
[latex][x+(3+i)][x+(3-i)][/latex]
44.
[latex][s-(1+2i)][s-(1-2i)][/latex]
45.
[latex][v-(4+i)][v-(4-i)][/latex]
46.
[latex][Z+(2+i)][Z+(2-i)][/latex]
Exercise Group
For Problems 47–50, plot the number and its complex conjugate in the complex plane. What is the geometric relationship between complex conjugates?
47.
[latex]z=-3+2i[/latex]
48.
[latex]z=4-3i[/latex]
49.
[latex]z=\dfrac{\sqrt{3}}{2}-\dfrac{1}{2}i[/latex]
50.
[latex]z=-\dfrac{\sqrt{2}}{2}-\dfrac{\sqrt{2}}{2}i[/latex]
Exercise Group
For Problems 51–54, sketch the set of points in the complex plane.
51.
[latex]{|z|} \le 2[/latex]
52.
[latex]1 \lt {|z|} \lt 3[/latex]
53.
The set of all [latex]z=a+bi[/latex] for which [latex]a \ge b[/latex]
54.
The set of all [latex]z=a+bi[/latex] for which [latex]a+b \lt 2[/latex]
Exercise Group
For Problems 55–58, illustrate addition using the parallelogram rule in the complex plane.
55.
[latex](1-4i)+(-3+2i)[/latex]
56.
[latex](-4+2i)+(-2+i)[/latex]
57.
[latex](2+6i)-(3+3i)[/latex]
58.
[latex](-5-2i)-(-3+2)[/latex]
59.
Prove that the product of two complex numbers [latex]z_1=a+bi[/latex] and [latex]z_2=c+di[/latex] is
[latex]z_1z_2=(a+bi)(c+di) = (ac-bd)+(ad+bc)i[/latex]
60.
Prove that the quotient of two complex numbers [latex]z_1=a+bi[/latex] and [latex]z_2=c+di[/latex] is
[latex]\dfrac{z_1}{z_2} = \dfrac{a+bi}{c+di} = \dfrac{a+bi}{c+di} \cdot \dfrac{c-di}{c-di} = \dfrac{ac+bd}{c^2+d^2} + \dfrac{bc-ad}{c^2+d^2}i[/latex]
61.
Prove the commutative laws for addition and multiplication of complex numbers:
[latex]z_1+z_2=z_2+z_1[/latex]
[latex]z_1z_2=z_2z_1[/latex]
62.
Prove the distributive law for complex numbers:
[latex]z_1(z_2+z_3)=z_1z_2+z_1z_3[/latex]
63.
- Show that [latex]z+\bar{z}[/latex] is a real number, and [latex]z-\bar{z}[/latex] is an imaginary number.
- Show that [latex]z\bar{z}={|z|}^2[/latex]
64.
Show that [latex]\overline{zw}=\bar{z}\cdot \bar{w}[/latex]
65.
Suppose [latex]t,~ w[/latex] and [latex]z[/latex] are complex numbers. If [latex]w=t+z\text{,}[/latex] is it necessarily true that [latex]{|w|}={|t|}+{|z|}\text{?}[/latex] Provide examples to support your conclusion.
66.
Prove the triangle inequality for complex numbers:
[latex]{|w+z|} \le {|w|}+{|z|}[/latex]
Exercise Group
In Problems 67–70,
- Given one solution of a quadratic equation with rational coefficients, find the other solution.
- Write a quadratic equation that has those solutions.
67.
[latex]2+\sqrt{5}[/latex]
68.
[latex]3-\sqrt{2}[/latex]
69.
[latex]4-3i[/latex]
70.
[latex]5+i[/latex]
Exercise Group
For Problems 71–74, find a fourth-degree polynomial with real coefficients that has the given complex numbers as two of its zeros.
71.
[latex]1-3i,~2-i[/latex]
72.
[latex]5-4i,~-i[/latex]
73.
[latex]\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i,~ 3+2i[/latex]
74.
[latex]-\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}i,~ 4-i[/latex]
