Do Now

• What is the exact square root of 50?

• What is the exact square root of -50?

Chapter 9: Polynomial Functions

Lesson 6: Complex Numbers

Mrs. Parziale

Complex Numbers:

• is an imaginary number (i). • A complex number has the form (a+bi) where

a is the real part, b = imaginary part. Both a and b are real numbers.

• The expressions (a+bi) and (a-bi) are complex conjugates. When you multiply them you get a real number.

1

: (3 2 )(3 2 )Example i i

Example 1:

• Simplify (exact values):

16 27 2(2 )i

Example 2:

• Multiply: (3+5i)(2-i)

Example 3:

• Multiply (3+i)(3-i)

Example 4:

• Rewrite in a + bi form. Let z = 2+3i and w = 1-5i

(a) 4z (b) 2z – 3w

Example 5:

• Write in a+bi form (rationalize the denominator):

2 3

5

i

i

Example 6:

• Factor x2 – 5 over the set of polynomials with real coefficients (not only integers):

Example 7:

• Factor over the set of complex numbers:

(a) x2 + 9 (b) 4x2 + 16

Closure

• Factor over the set of complex numbers.

2 2 2c c