1. Sullivan Algebra and Trigonometry: Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Objectives Utilize the Conjugate Pairs Theorem to Find the Complex Zeros of a Polynomial Find a Polynomial Function with Specified Zeros Find the Complex Zeros of a Polynomial

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. Note that real numbers are also complex numbers in the form a + bi where b = 0. So, this definition of a complex polynomial function is a generalization of what was previously introduced.

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

5. Find a polynomial f of degree 4 whose coefficients are real and has zeros 0, -2 and 1 - 3i. Graph f to verify the solution.

6. Find the complex zeros of the polynomial There are 4 complex zeros. Using Descartes’ Rule of Sign, there are two, or no, positive real zeros. Using Descartes’ Rule of Sign, there are two, or no, negative real zeros.

7. Now, list all possible rational zeros p/q by factoring the first and last coefficients of the function. Now, begin testing each potential zero using synthetic division. If a potential zero k is in fact a zero, then x - k divides into f (remainder will be zero) and is a factor of f.

8. Test k = -2 Thus, -2 is a zero of f and x + 2 is a factor of f. Test k = -1/2 Thus, -1/2 is a zero of f and x + 1/2 is a factor of f.

9. Now, find the complex zeros of the quadratic factor.