Exercises: 10.3 Complex Numbers

Bimal Kunwor; Donna Densmore; Jared Eusea; Yi Zhen

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.

1.

[latex]\displaystyle \sqrt{-25}-4[/latex]
[latex]\displaystyle \dfrac{-8+\sqrt{-4}}{2}[/latex]
[latex]\displaystyle \dfrac{-5-\sqrt{-2}}{6}[/latex]

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.