1. Warm Up Factor: 6x4 – 18x3 + 12x2 Factor: 16x4 – 81
Solve for x: 4x2 + 13x + 9

2. Complex Numbers Objective
SWBAT use complex numbers in polynomial identities and equations
SWBAT perform arithmetic operations with complex numbers

3. Complex numbers We can never take the square root of a negative number
Only take the square root of positive numbers
If we want to take the square root of a negative number we use complex numbers

4. Complex numbers
We use the letter i which stands for ‘imaginary’ number to represent complex numbers
The letter i is the simplest complex number and it replaces the square root of negative 1

5. Complex numbers
So if , what happens if we solve ?

6. Example

7. Example

8. Example

9. Example

10. Your turn Simplify the sqrt(-81) Simplify the sqrt(-49)

11. Complex number arithmetic
We can combine like terms and solve the same way we did with other variables
For example:

12. Your turn: Simplify each of the following
2i + 7i – 5i
13i -8i i
(6i)(5i)
(3i)(4i)(5i)
4(3i – 6) + 3i - 7

13. Pattern of powers We know that and we know that
If we keep increasing the exponent we will notice a pattern

14. Pattern of powers

15. Pattern of power trick To calculate any high power of I For example:
convert it to a lower power by taking the closest multiple of 4 that's no bigger than the exponent
The subtract this multiple from the exponent
For example:

16. Example

17. Example

18. Example

19. Your turn Simplify i16 Simplify i45 Simplify i98 Simplify i3456

20. Example
(3i – 6) (3i – 6)
(2i2 – 5) (3i – 6i)
i2(i + 2i6 – i14)

21. Practice
(6i – 2) (7i – 5)
(8i2 – 5i) (i – 4)
i4(i + 4i5 – i8) + 12i