Chapter 4 - Imaginary Numbers

Question 1

Definition of Imaginary Numbers

Answer 1

Involves square root of negative number

Question 2

What is “i”?

Answer 2

Defined as square root of –1

Question 3

How do you solve for the powers of “i”?
Ex: i^27

Answer 3

• Divides exponent by 4
• i^0 (remainder of 0) = 1
• i^1 (remainder of 1) = i
• i^2 (remainder of 2) = -1
• i^3 (remainder of 3) = -i

Question 4

Definition of Standard Form (FOR IMAGINARY NUMBERS)

Answer 4

Simplest a + bi form

Question 5

How do you verify -1 + 2i & -1 - 2i are solutions of x^2 + 2x + 5 = 0?
(4 Ways)

Answer 5

Option 1: Plug twice into equation, if answer is 0, correct
Option 2: X = -1 ± 2i, convert to quadratic form
Option 3: S = -b/a and P = c/a, plug a, b, c into quadratic form
Option 4: Solve Quadratic (Complete Square)

Question 6

What do you do if “i” is in the denominator? (4)

Answer 6

• Must always be in standard form
• Rationalize Denominator
• Multiply by i for monomial
• Multiply by conjugate for binomial

Question 7

How do you graph imaginary numbers? (3)

Answer 7

• Make a Vector/Ray
• Start at Origin
• 5 - 3i (5 right, 3 Down)

Question 8

What are they asking you to do when they ask for absolute value of an expression in standard form? (3)

Answer 8

• Asking for Length/Magnitude of Vector
• Basically Solving Pythagorean theorem
• Z = √a² + b²

Question 9

How do you express the square root of a negative number using the imaginary unit i? (4)

Answer 9

• √(-x) = i · √x
• x is positive real number & i if form i² = -1.
• Ex: √(-16) = i · √16 = 4i
• Ex: √(-49) = i · √49 = 7i

Question 10

How do you factor something if there are no real solutions?
Factor x² + 4

Answer 10

1. Solve: x² + 4 = 0 → x² = -4 → x = ±√(-4) = ±2i
2. Factored form: x² + 4 = (x - 2i)(x + 2i)

Question 11

What should your answers be if they ask for linear solutions?
What should your answers be if they ask for all complex zeroes?
What should your answers be if they ask all real zeroes?

Answer 11

• Include both real and imaginary solutions
• Include both real and imaginary solutions
• Only include solutions that are real numbers

Question 12

What must you do if you must find all complex zeroes using synthetic division? (7)

Answer 12

1. Start with the expression:
   x² - 4x + 7
2. Move the constant aside:
   x² - 4x + ___ = -7 + ___
3. Add 4 to both sides:
   x² - 4x + 4 = -7 + 4 → x² - 4x + 4 = -3
4. Rewrite as binomial square:
   (x - 2)² = -3
5. Solve using square roots:
   x - 2 = ±√(-3) → x - 2 = ±i√3
6. Final solutions:
   x = 2 ± i√3
7. Factored form:
   x² - 4x + 7 = (x - (2 + i√3))(x - (2 - i√3))

Question 13

How do you write a polynomial function when given zeroes in terms of i?
{2, 3 + 4i}

Answer 13

1. Start with the given zero: x = 3 + 4i
2. Include its conjugate: x = 3 - 4i
3. (x - 3)² = (4i)²
4. Apply square to both number & i: (x - 3)² = 16 (-1)
5. (x - 2) (x² - 6x + 25)

Question 14

What must you do if given a polynomial function and they ask you to make it go through a certain point? (4)

Answer 14

• Must change the stretch (a)
• Convert Polynomial Function into Factored Form
• Plug coordinate into x & y
• Must be given a new value for a