1. 2.5 complex 0’s & the fundamental theorem of algebra

2. Complex Conjugate Zeros
Suppose that f(x) is a polynomial function with real coefficients. If a and b are real numbers with b ≠ 0 and a + bi is a zero of f(x), then its complex conjugate a - bi is also a zero of f(x).
Example: find the complex conjugate
4 – 3i
8 + 2i
3 + √3 i

3. Factoring a complex polynomial function
Use the Rational Root theorem
Examples: find all zeros
f(x) = 3x4 + 8x3 + 6x2 + 3x – 2
f(x) = x3 – 10x2 + 44x – 69

4. Fundamental theorem of algebra
A polynomial of degree n has n complex roots (real and non-real). Some of these may be repeated. In other words: if you can solve the polynomial the polynomial has 0’s For example: if k is a zero of a polynomial if and only if x – k is a factor of the polynomial

5. How to use the fundamental theorem of algebra
2 parts: 1) find all roots, how you do this depends on if you are given a real root or a complex root
a) given a real root: do synthetic division, rewrite your answer, find the remaining zeros by doing quadratic formula (these will be complex numbers),write your answer in set notation.)
b) given a complex root: find the conjugate, write both as
(x – (a + bi))(x – (a – bi)) and foil to get rid of the i’s, take that quadratic and the original problem and find the remaining roots by doing long division. If need be factor or use quadratic formula to find the remaining roots or set equal
to 0 and solve, write your answer in set notation.

6. Cont’d 2) given the roots find the polynomial function
write your roots as (x - #)(x - #),etc. If given a complex root or a radical, DO NOT forget about it’s conjugate. Multiply the parenthesis together either by FOIL or distributing. Write your final answer as f(x) = …

7. Use the fundamental theorem of algebra
Use the fundamental theorem of algebra to find all zeros
Examples:
f(x) = x3 – 3x2 + 9x + 13; 2 – 3i
f(x) = x3 + 6x2 + 21x + 26; -2

8. Finding a polynomial given zeros
**if there is an i or radical in 1 parenthesis you must have the conjugate in another
Examples: write a polynomial with minimal degree in standard form with real coefficients whose zeros include those listed. Use the fundamental theorem of algebra
-4, 1 – i, and 1 + i
-2, 1- i
-2- i, 1 + 3i

9. Examples cont’d
4. -1 (multiplicity 3), 3 (multiplicity 1) (x – 1)3(x – 3) = (x – 1)(x – 1)(x – 1)(x – 3) (multiplicity 2), -2 – i (multiplicity 1)

10. Fundamental Polynomial connections in the complex case
The following statements about a polynomial function f are equivalent if k is a complex number:
1. x = k is a solution (or root) of the equation
f(x) = 0
2. k is a zero of the function f
3. x – k is a factor of f(x)

11. Write the function in std
Write the function in std. form & identify the 0’s and x-intercepts of the graph
Examples:
f(x) = (x – 2i)(x + 2i)
f(x) = (x – 5)(x - √2 i)(x + √2 i)
f(x) = (x – 1)(x – 1)(x + 2i)(x – 2i)