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Chapter 2: Polynomial and Rational Functions Topic 1: Complex Numbers What is an imaginary number? What is a complex number? The imaginary unit is defined as 풊 = √−ퟏ A complex number is defined as the set of all numbers in the form of 푎 + 푏푖, where 푎 is the real component and 푏 is the coefficient of the imaginary component. An imaginary number is when the real component (푎) is zero.

Checkpoint: Since 풊 = √−ퟏ Then 풊ퟐ =

Operations with Complex Numbers Adding & Subtracting: Combine like terms (풂 + 풃풊) + (풄 + 풅풊) = (풂 + 풄) + (풃 + 풅)풊 Examples: 1. (5 − 11푖) + (7 + 4푖) 2. (−5 + 7푖) − (−11 − 6푖)

3. (5 − 2푖) + (3 + 3푖) 4. (2 + 6푖) − (12 − 4푖)

Multiplying: Just like polynomials, use the distributive property. Then, combine like terms and simplify powers of 푖. Remember! Multiplication does not require like terms. Every term gets distributed to every term. Examples: 1. 4푖(3 − 5푖) 2. (7 − 3푖)(−2 − 5푖)

3. 7푖(2 − 9푖) 4. (5 + 4푖)(6 − 7푖)

5. (3 + 5푖)(3 − 5푖)

A note about conjugates: Recall that when multiplying conjugates, the middle terms will cancel out. With complex numbers, this becomes even simpler: (풂 + 풃풊)(풂 − 풃풊) = 풂ퟐ + 풃ퟐ Try again with the shortcut: (3 + 5푖)(3 − 5푖)

Dividing: Just like polynomials and rational expressions, the denominator must be a rational number. Since complex numbers include imaginary components, these are not rational numbers. To remove a complex number from the denominator, we multiply numerator and denominator by the conjugate of the Remember! You can simplify first IF factors can be canceled. NO breaking up terms. Examples: 7+4푖 5+4푖 1. 2. 2−5푖 4−2푖

Operations with Square Roots of Negative Numbers Begin by expressing all square roots of negatives in terms of 푖, then proceed with the operation Examples:

1. √−18 − √−8 −14+√−12 2. 2

2 2 3. (−1 + √−5) 4. (−2 + √−3)

5. √−27 + √−48 −25+√−50 6. 15

Quadratic Formula with Complex/Imaginary Solutions Remember that the discriminant of the quadratic formula tells us ABOUT the roots of the equation. (How would the discriminant behave if it were under the square roots in the full quadratic formula?) When the discriminant is negative, it tells us that the quadratic equation has two complex, conjugate roots.

Examples: Using the quadratic formula, find the roots of the following 1. 3푥2 − 2푥 + 4 = 0 2. 푥2 − 2푥 + 2 = 0