Complex Zeros

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 $a_{n}\neq 0$ 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 $\geq 1$ has at least one complex zero.
 
 Theorem
 Every complex function f of degree n $\geq$ 1 can be factored into n linear factors (not necessarily distinct) of the form
 $f(x)=a_{n}(x-r_{1})(x-r_{2})\ldots (x-r_{n})$ where $a_{n},r_{1},\ldots ,r_{n}$ 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 $b\neq 0$, the complex conjugate ${\overline {r}}=a-bi$ 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.
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