1. COMPLEX NUMBERS Unit 4Radicals

2. Complex/imaginary numbers WHAT IS? WHY? There is no real number whose square is -25 so we have to use an imaginary number “i” is an imaginary number. “i” is equal to the square root of -1 BASICALLY: any time you see a negative under a SQUARE ROOT an “i” gets pulled out.

4. i Since “i” raised to a power follows a pattern you can easily find the answer by dividing the exponent by 4 and using the remainder to simplify. What about higher exponents? 4-7? 4 goes into 12, 3 times with a remainder of zero. 4 goes into 22, 5 times with a remainder of 2 4 goes into 33, 8 times with a remainder of 1

5. ALWAYS pull the “i” out first before multiplying together.

6. Practice

7. Answers

8. OPERATIONS WITH COMPLEX NUMBERS Unit 4 radicals

9. Adding & subtracting complex numbers A complex number is a number with “i” in it. Complex numbers can be written in the form : Real part Imaginary part To add or subtract complex numbers combine the real parts and combine the imaginary parts separately.

10. Adding & subtracting complex numbers

11. Multiplying complex numbers You multiply complex numbers like you would binomials. (Double Distribute, Box, FOIL…etc)

12. Dividing complex numbers Remember that we don’t want to leave a radical in the denominator. To simplify a quotient, multiply by the conjugate of the denominator. Conjugate – change only the middle sign CONJUGATE =

13. Rationalize the denominator Simplify Imaginary # song

14. Practice Pg 253 #8-12, 19-31 odd

15. COMPLEX SOLUTIONS Unit 4 Radicals

16. Finding complex solutions Find the solution/roots/zeros. I squared

17. Find all Solutions/Roots/zeros Graph, find x-intercepts Use Syn. Div. and quadratic formula to find any missing solutions The graph gives us one solutions. (Use syn. Div. to find the missing two) Result is: Now solve this quadratic:

18. Write a polynomial function with roots 3i & -3i

19. Practice Pg 253 #39-44 Pg322 #16-19, 24