1. COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

2. Fundamental Theorem of Algebra  Every complex polynomial function f(x) of degree n ≥ 1 has at least one complex zero. Explanation from the expert

3. Theorem  Every complex polynomial function f(x) of degree n ≥ 1 can be factored into n linear factors (not necessarily distinct) of the form

4. Theorem  The fundamental theorem of algebra was proved by Karl Friedrich Gauss (1777 – 1855) when he was 22 years old.

5. Conjugate Pairs Theorem

6. Finding a Polynomial Whose Zeros are Given  Extra Examples Extra Examples  Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 1, 1, and -4 + i. (x – 1) (x – 1) (x – (-4 + i)) (x – (-4 – i)) Do FOIL on the conjugate pairs first. (x^2 –(-4 – i)x – (-4 + i)x +(-4 + i)(-4 – i))= x^2 +4x – ix +4x +ix +16 + 4i – 4i – i^2= x^2 + 8x + 17 (Remember i^2 = -1)

7. Finding a Polynomial Whose Zeros are Given  Now multiply the other two zeros together using FOIL: (x – 1) (x – 1) = x^2 – 2x + 1  Finally multiply the two polynomials together using the distributive property (x^2 – 2x + 1) (x^2 + 8x + 17) X^4 + 8x^3 + 17x^2 -2x^3 – 16x^2 – 34x x^2 + 8x + 17= X^4 + 6x^3 + 2x^2 – 26x + 17