1. Warm-Up

2. TEST Our Ch. 9 Test will be on 5/29/14

3. Complex Number Operations

4. Learning Targets Adding Complex Numbers Multiplying Complex Numbers Rules for Adding and Multiplying Conjugates

5. Addition of Complex Numbers

6. Multiplication of Complex Numbers

7. Multiplying Notes

8. You Try

10. Conjugate Operations Complex Conjugate operations are needed in order to factor quadratics and determine their complex roots. There are two main operations that we need to know about

11. Sum of Complex Conjugates

12. Multiplication of Complex Conjugates

13. Why is the Conjugate Important Our pairs of complex numbers will always be conjugates

14. Conjugate cont.

15. Now we can begin to divide polynomials In order to divide polynomials we have to be able to determine one of its factors Once a factor is known we can begin to divide it throughout the standard form of the polynomial and simplify it If the factor used is indeed a root our remainder will be zero

16. Division Cont. We can then repeat the process until we are only left with all of the roots of the polynomial This process allows us to transform a polynomial from Standard Form to Factored Form

17. Types of Division There are two methods that we can use to divide polynomials ▫Long Division ▫Synthetic Division (preferred method)

18. First divide 3 into 6 or x into x 2 Now divide 3 into 5 or x into 11x Long Division If the divisor has more than one term, perform long division. You do the same steps with polynomial division as with integers. Let's do two problems, one with integers you know how to do and one with polynomials and copy the steps. 32 698x - 3 x 2 + 8x - 5 2 x 64 x 2 – 3x Now multiply by the divisor and put the answer below. Subtract (which changes the sign of each term in the polynomial) 5 11x Bring down the next number or term 8 - 5 1 + 11 Multiply and put below 32 11x - 33 subtract 26 28 This is the remainder Remainder added here over divisor So we found the answer to the problem x 2 + 8x – 5  x – 3 or the problem written another way:

19. List all coefficients (numbers in front of x's) and the constant along the top. If a term is missing, put in a 0. You want to divide the factor into the polynomial so set divisor = 0 and solve for first number. Let's try a problem where we factor the polynomial completely given one of its factors. - 24 8 -25 -50 4 Bring first number down below line Multiply these and put answer above line in next column - 8 Add these up 0 Multiply these and put answer above line in next column 0 Add these up - 25 50 0 Multiply these and put answer above line in next column Add these up No remainder so x + 2 IS a factor because it divided in evenly Put variables back in (one x was divided out in process so first number is one less power than original problem). x 2 + x So the answer is the divisor times the quotient: You could check this by multiplying them out and getting original polynomial

20. Comparison Between Synthetic and Long Division Why Synthetic Division Works

21. Example:

22. You try:

25. For Tonight Worksheet