Algebra 2 and Trigonometry

Ann Xavier Gantert

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS - all with Video Answers

Educators

Section 5

Operations with Complex Numbers

Problem 1

Tim said that the binomial $x^{2}+16$ can be written as $x^{2}-16 i^{2}$ and factored over the set of
complex numbers. Do you agree with Tim? Explain why or why not.

Heather Z.

Heather Z.

Numerade Educator

Problem 2

Joshua said that the product of a complex number and its conjugate is always a real number. Do you agree with Joshua? Explain why or why not.

Jeremy S.

Numerade Educator

Problem 3

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(6+7 i)+(1+2 i)
$$

Heather Z.

Numerade Educator

Problem 4

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(3-5 i)+(2+i)
$$

Jeremy S.

Numerade Educator

Problem 5

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(5-6 i)+(4+2 i)
$$

Heather Z.

Numerade Educator

Problem 6

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(-3+3 i)-(1+5 i)
$$

Jeremy S.

Numerade Educator

Problem 7

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(2-8 i)-(2+8 i)
$$

Heather Z.

Numerade Educator

Problem 8

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(4+12 i)+(-4-2 i)
$$

Jeremy S.

Numerade Educator

Problem 9

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(1+9 i)-(1+2 i)
$$

Heather Z.

Numerade Educator

Problem 10

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(10-12 i)-(12+7 i)
$$

Jeremy S.

Numerade Educator

Problem 11

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(0+3 i)+(0-3 i)
$$

Heather Z.

Numerade Educator

Problem 12

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
\left(\frac{1}{2}+\frac{1}{2} i\right)+\left(\frac{1}{4}-\frac{3}{4} i\right)
$$

Jeremy S.

Numerade Educator

Problem 13

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
\left(\frac{2}{3}-\frac{1}{6} i\right)+(2-i)
$$

Heather Z.

Numerade Educator

Problem 14

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
(1+0 i)-\left(\frac{1}{5}-\frac{2}{5} i\right)
$$

Jeremy S.

Numerade Educator

Problem 15

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
\left(-\frac{3}{2}+\frac{5}{3} i\right)-\left(\frac{9}{4}-\frac{1}{5} i\right)
$$

Heather Z.

Numerade Educator

Problem 16

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
\left(\frac{1}{4}+12 i\right)+\left(7+\frac{i}{10}\right)
$$

Jeremy S.

Numerade Educator

Problem 17

In $3-17,$ find each sum or difference of the complex numbers in $a+b i$ form.
$$
\left(\frac{3}{7}+\frac{1}{3} i\right)+\left(\frac{1}{6}-\frac{7}{8} i\right)
$$

Heather Z.

Numerade Educator

Problem 18

In $18-25,$ write the complex conjugate of each number.
$$
3+4 i
$$

Jeremy S.

Numerade Educator

Problem 19

In $18-25,$ write the complex conjugate of each number.
$$
2-5 i
$$

Heather Z.

Numerade Educator

Problem 20

In $18-25,$ write the complex conjugate of each number.
$$
-8+i
$$

Jeremy S.

Numerade Educator

Problem 21

In $18-25,$ write the complex conjugate of each number.
$$
-6-9 i
$$

Heather Z.

Numerade Educator

Problem 22

In $18-25,$ write the complex conjugate of each number.
$$
\frac{1}{2}-3 i
$$

Jeremy S.

Numerade Educator

Problem 23

In $18-25,$ write the complex conjugate of each number.
$$
-4+\frac{1}{3} i
$$

Heather Z.

Numerade Educator

Problem 24

In $18-25,$ write the complex conjugate of each number.
$$
\frac{5}{3}-\frac{2}{3} i
$$

Jeremy S.

Numerade Educator

Problem 25

In $18-25,$ write the complex conjugate of each number.
$$
\pi+2 i
$$

Heather Z.

Numerade Educator

Problem 26

In $26-37,$ find each product.
$$
(1+5 i)(1+2 i)
$$

Jeremy S.

Numerade Educator

Problem 27

In $26-37,$ find each product.
$$
(3+2 i)(3+3 i)
$$

Heather Z.

Numerade Educator

Problem 28

In $26-37,$ find each product.
$$
(2-3 i)^{2}
$$

Jeremy S.

Numerade Educator

Problem 29

In $26-37,$ find each product.
$$
(4-5 i)(3-2 i)
$$

Heather Z.

Numerade Educator

Problem 30

In $26-37,$ find each product.
$$
(7+i)(2+3 i)
$$

Jeremy S.

Numerade Educator

Problem 31

In $26-37,$ find each product.
$$
(-2-i)(1+2 i)
$$

Heather Z.

Numerade Educator

Problem 32

In $26-37,$ find each product.
$$
(-4-i)(-4+i)
$$

Jeremy S.

Numerade Educator

Problem 33

In $26-37,$ find each product.
$$
(-12-2 i)(12-2 i)
$$

Heather Z.

Numerade Educator

Problem 34

In $26-37,$ find each product.
$$
(5-3 i)(5+3 i)
$$

Jeremy S.

Numerade Educator

Problem 35

In $26-37,$ find each product.
$$
(3-i)\left(\frac{3}{10}+\frac{1}{10} i\right)
$$

Heather Z.

Numerade Educator

Problem 36

In $26-37,$ find each product.
$$
\left(\frac{1}{5}-\frac{1}{10} i\right)(4+2 i)
$$

Jeremy S.

Numerade Educator

Problem 37

In $26-37,$ find each product.
$$
\left(-\frac{1}{8}+\frac{1}{8} i\right)(4+4 i)
$$

Heather Z.

Numerade Educator

Problem 38

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
1+i
$$

Jeremy S.

Numerade Educator

Problem 39

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
2+4 i
$$

Heather Z.

Numerade Educator

Problem 40

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
-1+2 i
$$

Jeremy S.

Numerade Educator

Problem 41

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
3-3 i
$$

Heather Z.

Numerade Educator

Problem 42

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
\frac{1}{2}-\frac{1}{4} i
$$

Jeremy S.

Numerade Educator

Problem 43

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
2-\frac{1}{2} i
$$

Heather Z.

Numerade Educator

Problem 44

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
\frac{5}{6}+3 i
$$

Jeremy S.

Numerade Educator

Problem 45

In $38-45,$ find the multiplicative inverse of each of the following in $a+b i$ form.
$$
9+\pi i
$$

Heather Z.

Numerade Educator

Problem 46

In $46-60,$ write each quotient in $a+b i$ form.
$$
(8+4 i) \div(1+i)
$$

Jeremy S.

Numerade Educator

Problem 47

In $46-60,$ write each quotient in $a+b i$ form.
$$
(2+4 i) \div(1-i)
$$

Heather Z.

Numerade Educator

Problem 48

In $46-60,$ write each quotient in $a+b i$ form.
$$
(10+5 i) \div(1+2 i)
$$

Jeremy S.

Numerade Educator

Problem 49

In $46-60,$ write each quotient in $a+b i$ form.
$$
(5-15 i) \div(3-i)
$$

Heather Z.

Numerade Educator

Problem 50

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{2+2 i}{4-2 i}
$$

Jeremy S.

Numerade Educator

Problem 51

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{10-5 i}{2+6 i}
$$

Heather Z.

Numerade Educator

Problem 52

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{8+2 i}{1+3 i}
$$

Jeremy S.

Numerade Educator

Problem 53

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{3+i}{3-i}
$$

Heather Z.

Numerade Educator

Problem 54

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{5-2 i}{5+2 i}
$$

Jeremy S.

Numerade Educator

Problem 55

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{12+3 i}{3 i}
$$

Heather Z.

Numerade Educator

Problem 56

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{8+6 i}{2 i}
$$

Jeremy S.

Numerade Educator

Problem 57

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{\frac{1}{5}-\frac{1}{5} i}{\frac{5}{3}-4 i}
$$

Heather Z.

Numerade Educator

Problem 58

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{1+i}{\frac{1}{2}-\frac{1}{2} i}
$$

Jeremy S.

Numerade Educator

Problem 59

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{3 i}{\pi+\frac{1}{2} i}
$$

Heather Z.

Numerade Educator

Problem 60

In $46-60,$ write each quotient in $a+b i$ form.
$$
\frac{\frac{1}{7}+i}{\frac{5}{i}}
$$

Jeremy S.

Numerade Educator

Problem 61

Impedance is the resistance to the flow of current in an electric circuit measured in ohms.
The impedance, $Z,$ in a circuit is found by using the formula $Z=\frac{V}{I}$ where $V$ is the voltage
(measured in volts) and $I$ is the current (measured in amperes). Find the impedance when
$V=1.8-0.4 i$ volts and $I=-0.3 i$ amperes.

Heather Z.

Numerade Educator

Problem 62

Find the current that will flow when $V=1.6-0.3 i$ volts and $Z=1.5+8 i$ ohms using the
formula $Z=\frac{V}{I} .$

Jeremy S.

Numerade Educator