1. 3.8 Complex Zeros; Fundamental Theorem of Algebra

2. A variable in the complex number system is referred to as a complex variable.
A complex polynomial function f degree n is a complex function of the form
A complex number r is called a (complex) zero of a complex function f if f (r) = 0.

3. Fundamental Theorem of Algebra
Every complex polynomial function f (x) of degree n > 1 has at least one complex zero.

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

5. Conjugate Pairs Theorem
Let f (x) be a complex polynomial whose coefficients are real numbers. If r = a + bi is a zero of f, then the complex conjugate
is also a zero of f.
Corollary
A complex polynomial f of odd degree with real coefficients has at least one real zero.

6. Find a polynomial f of degree 4 whose coefficients are real numbers and that has zeros 1, 2, and 2+i.
f(x)

7. Find the complex zeroes of the polynomial function
There are 4 complex zeros.
From Rational Zero Theorem find potential rational zeros

9. Zeros are -2, -1/2, 2 +5i, 2 -5i.