Warm-up Divide the following using Long Division: (6x3 - 16x2 + 17x - 6)  (3x –2 ) Divide the following with Synthetic Division (5x3 – 6x2 + 8) (x – 4) Given the following polynomial and one of its factors, Find the remaining factors (3x3 + 2x2 –19x + 6) : (x + 3) is a factor

Warm-up Divide the following using Long Division: (6x3 - 16x2 + 17x - 6)  (3x –2 ) 2x2 – 4x + 3

Warm-up Divide the following with Synthetic Division (5x3 – 6x2 + 8) (x – 4)

Warm-up Given the following polynomial and one of its factors, Find the remaining factors (3x3 + 2x2 –19x + 6) : (x + 3) is a factor (x – 2)(3x – 1)

Complex Numbers Section 2-4 Digital Lesson Complex Numbers Section 2-4

Objectives I can use “i” to write complex numbers I can add, subtract, and multiply complex numbers I can simplify Negative Square Roots

Applications Impedance readings for electronics and electrical circuits are all measured in complex units

Complex Numbers Real Numbers Imaginary Numbers Rational Irrational

Complex Numbers The set of all numbers that can be written in the format: a + bi ; “a” is the real number part “bi’ is the imaginary part

The Imaginary Unit

Negative Radicals

Negative Radicals

Add or Subtract Complex Numbers To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers. (a + bi ) + (c + di ) = (a + c) + (b + d )i (a + bi ) – (c + di ) = (a – c) + (b – d )i Add or Subtract Complex Numbers

Adding Complex Numbers Example: Add (11 + 5i) + (8 – 2i ) = (11 + 8) + (5i – 2i ) Group real and imaginary terms. = 19 + 3i a + bi form Adding Complex Numbers

Subtracting Complex Numbers Examples: Subtract: (– 21 + 3i ) – (7 – 9i) = (– 21 – 7) + [(3 – (– 9)]i Group real and imaginary terms. = (– 21 – 7) + (3i + 9i) = –28 + 12i a + bi form Subtracting Complex Numbers

Product of Complex Numbers The product of two complex numbers is defined as: (a + bi)(c + di ) = (ac – bd ) + (ad + bc)i 1. Use the FOIL method to find the product. 2. Replace i2 by – 1. 3. Write the answer in the form a + bi. Product of Complex Numbers

1. 7i (11– 5i) = 77i – 35i2 = 77i – 35 (– 1) = 35 + 77i 2. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2 = 12 + 4i – 21i2 = 12 + 4i – 21(–1) = 12 + 4i + 21 = 33 + 4i Examples

Homework WS 3-7