Section+2.5

__2.5 Complex Zeros and the Fundamental Theorem of Algebra__ **Fundamental Theorem of Algebra**: A polynomial function of degree n, has n complex zeros (real and imaginary). Ex. x^3 = 0 x = 0, mult. 3 Example: Write in Standard Form, Identify Zeros, and Identify X-intercepts f(x) = (x-2i)(x+2i) x^2 - 4i --> x^2 + 4 Zeros: -2i and 2i There are no x-intercepts. Example 2: Find the Zeros and the x-intercepts. f(x) = (x-5)(x-i Ö 2)(x+i Ö 2) =(x-5)(x^2+2) =x^3-2x-5x^2-10 =x^3-5x^2-2x-10 Zeros: 5, i Ö2, - Ö2i ) X-intercepts: 5

If a+bi is a zero, then a-bi is also a zero. Ex. Write a polynomial that has the zeros: -3, 4, 2, -i f(x) = (x+3)(x-4)(x-(2-i))(x-(2+i)) =(x^2-x-12)(x-2+i)(x-2-i) =(x^2-x-12)(x^2-2x-ix-2x+4+2i+ix-2i-i^2) =(x^2-x-12)(x^2-4x+4+1) =(x^2-x-12)(x^2-4x+5) =x^4-4x^3+5x^2-x^3+4x^2-5x-12x^2+48x-60 f(x) = x^4 - 5x^3 - 3x^2 + 43x - 60
 * Complex Conjugate Zeros**:

Example 3: Find all the zeros: f(x) = x^5-3x^4-5x^3+5x^2-6x+8 p: 1,2,4,8 q: 1 p/q: ± 1,2,4,8 2 û 1 -3 -5 5 -6 8 __-2 10 -10 10 -8__ 1 -5 5 -5 4 0 <-- now a degree 4 polynomial

4 û 1 -5 5 -8 4 __4 -4 4 -4__ 1 -1 1 -1 0 <-- now a degree 3 polynomial

1 û 1 -1 1 -1 __1 1 1__ 1 0 1 0

x^2 + 1 x^2 = 1 x = ±1 Zeros: -2,4,1,+i,-i

Ex. Write as a product of linear factors, or irreducible quadratic factors with real coefficients. f(x) = 3x^5-2x^4+6x^3-4x^2-24x+16 p: 1,2,4,8,16 q: 1,3 p/q: ±1,2,4,8,16,1/3,2/3,4/3,8/3,16/3 2/3 û 3 -2 6 -4 -24 16 __2 0 4 0 -16__ 3 0 6 0 -24 0
 * Factoring a Polynomial**:

3x^4+6x^2-24 = 0 3(x^4 + 2x^2 - 8) = 0 3(x^2 + 4)(x^2 - 2) = 0 factored form of the original: 3(x-2/3)(x^2+4)(x- Ö 2)(x+ Ö 2) --> cannot factor any further without using i.

Homework: page 234, 1-19 odd and page 234 21-41 odd.