Every polynomial that we has been mentioned so far have been polynomials with real numbers as coefficients and real numbers as zeros. In this section we introduce the notion of a polynomial with complex numbers as coefficients and complex numbers as zeros.
We define the complex polynomial function the same way we defined a polynomial function. The only difference is the coefficients are complex numbers instead of real numbers. We still want and say a complex number r is a complex zero if f(r) = 0.
Fundamental Theorem of Algebra
This section would not be complete without mentioning the Fundamental Theorem of Algebra and an important consequence.
Fundamental Theorem of Algebra Every complex polynomial function f of degree n has at least one complex zero. Theorem Every complex function f of degree n 1 can be factored into n linear factors (not necessarily distinct) of the form where are complex numbers. That is, every complex polynomial completely factors into linear terms, some of which may be repeated.
If a root is a complex number that is not a real number, it has a non-zero imaginary part, we have some useful theorems to provide us with additional information.
Conjugate Pairs Theorem Let f be a polynomial function whose coefficients are real numbers. If r = a + bi is a zero of f, with , the complex conjugate is also a zero of f. A polynomial function f of odd degree with real coefficients has at least one real zero.
Every polynomial function with real coefficients can be uniquely factored over the real numbers into a product of linear factors and/or irreducible quadratic factors. Return to Topics Page