1

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

2

Warm-up• Divide the following using Long Division:

• (6x3 - 16x2 + 17x - 6) (3x –2 )

• 2x2 – 4x + 3

3

Warm-up• Divide the following with Synthetic Division

• (5x3 – 6x2 + 8) (x – 4)

4

23256145 2

xxx

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 NumbersSection 2-4

Digital Lesson

6

Objectives

• I can use “i” to write complex numbers

• I can add, subtract, and multiply complex numbers

• I can simplify Negative Square Roots

7

Applications

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

8

Complex Numbers

Real Numbers Imaginary Numbers

Rational Irrational

9

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

10

The Imaginary Unit

11 2 iwherei

11

Negative Radicals

25 25 5i i

20 20 2 5i i

12

Negative Radicals

18 32

18 4 2i

13

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

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Example: Add (11 + 5i) + (8 – 2i )

= 19 + 3i

Group real and imaginary terms.

a + bi form

= (11 + 8) + (5i – 2i )

15

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

16

The product of two complex numbers is defined as:

1. Use the FOIL method to find the product.

2. Replace i2 by – 1.

3. Write the answer in the form a + bi.

(a + bi)(c + di ) = (ac – bd ) + (ad + bc)i

17

1. 7i (11– 5i) = 77i – 35i2

= 35 + 77i

2. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i2

= 12 + 4i – 21i2

= 12 + 4i – 21(–1)

= 12 + 4i + 21

= 33 + 4i

= 77i – 35 (– 1)

18

2(3 4 )i

(3 4 )(3 4 )i i

29 12 12 16i i i

7 24i

19

Homework

• WS 3-7