%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Complex Roots and the Fundamental Theorem of Algebra">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 23 16:31:53 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=Reference.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{<LINK REL="contents" HREF="..\..\Reference\ReferenceIndex.tex#Reference Index">}
%TCIDATA{<LINK REL="next" HREF="al04_08.tex">}
%TCIDATA{<LINK REL="parent" HREF="Al_contents.tex">}
%TCIDATA{<LINK REL="previous" HREF="Al04_04.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{Complex Roots and the Fundamental Theorem of Algebra\label%
{Complex Roots and the Fundamental Theorem of Algebra}}

Remarkably, simply adding $\sqrt{-1}$ and its real multiples to the real
number system is sufficient to provide a number system in which \emph{every}
polynomial equation has a root.

\subsection{Fundamental Theorem of Algebra\label%
{Fundamental Theorem of Algebra}}

\begin{theorem}
Every polynomial 
\[
P\left( x\right) =a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\text{, }n\geq 1%
\text{, }a_{n}\neq 0
\]
with complex coefficients has at least one complex zero.
\end{theorem}

Because every real number is also a complex number, the theorem applies to
polynomials with real coefficients as well. This theorem then promises that
there is a complex number $c_{1}$ such that $P\left( c_{1}\right) =0$. The 
\hyperref{Factor Theorem}{}{}{Al04_03.tex#Factor Theorem for Polynomials}
then implies that 
\[
P\left( x\right) =\left( x-c_{1}\right) Q_{1}\left( x\right) \text{, some
complex polynomial }Q_{1}\left( x\right) 
\]
Note that the degree of $Q_{1}\left( x\right) $ is now $n-1$. But then the
Fundamental Theorem applies also to $Q_{1}\left( x\right) $, so there is a
root $c_{2}$ of $Q_{1}\left( x\right) $. Together with the Factor Theorem,
this leads to 
\[
Q_{1}\left( x\right) =\left( x-c_{2}\right) Q_{2}\left( x\right) \text{,
some complex polynomial }Q_{2}\left( x\right) 
\]
Now the degree of $Q_{2}\left( x\right) $ is $n-2$. By continuing this
process, you must finally reach a quotient $Q_{n}\left( x\right) $ whose
degree is $0$; that is, $Q_{n}\left( x\right) $ is a constant complex number.

\subsection{Complete Factorization Theorem\label%
{Complete Factorization Theorem}}

\begin{theorem}
If $P\left( x\right) $ is a polynomial of degree $n>0$, then there exist
complex numbers $a$, $c_{1}$, $c_{2}$, ...$c_{n}$ (with $a\neq 0$) such that 
\[
P\left( x\right) =a\left( x-c_{1}\right) \left( x-c_{2}\right) ...\left(
x-c_{n}\right) 
\]
\end{theorem}

Note that the $c_{k}$ that appear in the factorization need not be
different. However, counting those which appear according to their \hyperref{%
multiplicity}{}{}{Al04_03.tex#Definition of multiplicity of a root} leads to
the Zeros Theorem.

\subsection{Zeros Theorem\label{Zeros Theorem for polynomials}}

\begin{theorem}
Every polynomial of degree $n\geq 1$ has exactly $n$ complex zeros, provided
that a zero with \hyperref{multiplicity}{}{}{Al04_03.tex#Definition of
multiplicity of a root} $k$ is counted $k$ times.
\end{theorem}

\subsection{Conjugate Roots Theorem\label%
{Conjugate Roots Theorem for Polynomials}}

\begin{theorem}
If the polynomial $P\left( x\right) $ of degree $n>0$ has real coefficients,
and if the complex number $z$ is a root of the equation $P\left( x\right) =0$%
, then so is its complex conjugate $\bar{z}$.
\end{theorem}

\noindent \hrulefill

\begin{center}
\begin{tabular}{ccc}
{\small \hyperref{\FRAME{itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Computing Techniques}{}{}{%
../../Help/DMindex.tex}} & {\small \hyperref{\FRAME{itbpF}{13.9375pt}{%
14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
13.9375pt;height 14.4375pt;depth 3pt;original-width
24.0625pt;original-height 24.8125pt;cropleft "0";croptop "0.9959";cropright
"0.9937";cropbottom "0";filename 'Nblogo.wmf';file-properties "XNPEU";}}%
General Information}{}{}{../../Help/Index.tex}} & {\small \hyperref{\FRAME{%
itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Reference Library}{}{}{%
../ReferenceIndex.tex}}
\end{tabular}
\end{center}

\noindent \hrulefill

\end{document}
