Transcript Complex Numbers

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
1
Warm-up
• Divide the following using Long Division:
• (6x3 - 16x2 + 17x - 6)  (3x –2 )
• 2x2 – 4x + 3
2
Warm-up
• Divide the following with Synthetic Division
• (5x3 – 6x2 + 8) (x – 4)
232
5 x  14 x  56 
x4
2
3
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)
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
6
Applications
• Impedance readings for electronics and
electrical circuits are all measured in
complex units
7
Complex Numbers
Real Numbers
Rational
Imaginary Numbers
Irrational
8
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
9
The Imaginary Unit
i   1 where i  1
2
10
Negative Radicals
25  25 i  5i
20  20 i  2 5 i
11
Negative Radicals
18  32
18  4 2i
12
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
13
Example: Add (11 + 5i) + (8 – 2i )
= (11 + 8) + (5i – 2i )
Group real and imaginary terms.
= 19 + 3i
a + bi form
14
Examples: Subtract: (– 21 + 3i ) – (7 – 9i)
= (– 21 – 7) + [(3 – (– 9)]i
= (– 21 – 7) + (3i + 9i)
= –28 + 12i
Group real and
imaginary terms.
a + bi form
15
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.
16
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
17
(3  4i)
2
(3  4i )(3  4i )
9  12i  12i  16i
2
7  24i
18
Homework
• WS 3-7
19