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==Introduction== Every polynomial that we has been mentioned so far have been polynomials with real numbers as coefficients and real numb..."

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==Introduction==

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.

==Complex Polynomials==

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 <math>a_n \neq 0</math> 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 <math>\ge 1</math> has at least one complex zero.

'''Theorem'''
Every complex function f of degree n <math>\ge </math> 1 can be factored into n linear factors (not necessarily distinct) of the form
<math>f(x) = a_n(x - r_1)(x-r_2)\ldots (x - r_n)</math> where <math>a_n, r_1, \ldots , r_n</math> are complex numbers. That is, every complex
polynomial completely factors into linear terms, some of which may be repeated.

==Complex Roots==

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 <math> b \neq 0</math>, the complex conjugate <math> \overline{r} = a - bi</math> 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.
[[Math_5|'''Return to Topics Page]]

Complex Zeros - Revision history

MathAdmin: Created page with "__TOC__ ==Introduction== Every polynomial that we has been mentioned so far have been polynomials with real numbers as coefficients and real numb..."

MathAdmin: Created page with "
TOC
==Introduction== Every polynomial that we has been mentioned so far have been polynomials with real numbers as coefficients and real numb..."