1. 1 Using the Fundamental Theorem of Algebra

2.  Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x 2 + 27 you end up with 2 imaginary solutions!  Directions only asked for REAL Solutions 2

3. Remember??? Complex numbers : a real number and an imaginary number a + bi Complex Conjugate Theorem: If a + bi is a zero/solution of the function, then a – bi is also a zero.

4. The Fundamental Theorem of Algebra  Recall:  the degree of the polynomial = the number of roots/solutions the polynomial has  Every polynomial of degree “n” with complex coefficients has “n” roots in the complex numbers.  This does NOT say n distinct roots  So, there may be repeated roots 4

5. Find all the zeros of the polynomial. (real and complex) 5 List all possible rational zeros. Narrow down list using A graphing calculator. -4, 3

6. 6 Use synthetic division to test these zeros and factor the polynomial. The quadratic will not factor, so use the quadratic formula to solve it.

7. Assignment  p.369 # 25-34 all 7

8. 8 Using the Fundamental Theorem of Algebra

9.  Going backwards!  Last week --- given the function and ask to find zero’s  Now --- given zero’s and asked to find the function 9

10. Examples  Write a polynomial function of least degree that has real coefficients and a leading coefficient of 1 with the following zeros. 1. 3, -2, 5 2. 4, i, -i 3. 1 – 2i, 3, -3 10

11. 11 1. 3, -2, 5 3 as a zero is (x - 3) as a factor -2 as a zero is (x + 2) as a factor 5 as a zero os (x – 5) as a factor FOIL the first two binomials Distribute each term. Combine like terms

12. 12 2. 4, i, -i 4 as a zero is (x - 4) as a factor i as a zero is (x - i) as a factor -i as a zero os (x + i) as a factor FOIL the last two binomials (multiply the complex parts first!) FOIL Combine like terms & replace i 2 with -1 Rearrange terms

13. 13 3. 1 – 2i, 3, -3 All imaginary factors come in conjugate pairs, so 1 + 2i is also a zero. Distribute the negative through each complex # Distribute the first two polynomials Combine like terms & replace i 2 with -1 Now distribute and FOIL

14.  Zero’s: 3, 6i, 2-3i  Factors = (x - 3) (x + 6i)(x - 6i) (x –(2-3i)) (x – (2+3i)) 14

15. Assignment  p.369 36-46 evens 15