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Fundamental Theorem of Algebra

\fbox{\emph{Every $n$th-order polynomial possesses exactly $n$\ complex roots.}}
This is a very powerful algebraic tool.2.4 It says that given any polynomial

\begin{eqnarray*}
p(x) &=& a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots
+ a_2 x^2 + a_1 x + a_0 \\
&\isdef & \sum_{i=0}^n a_i x^i
\end{eqnarray*}

we can always rewrite it as

\begin{eqnarray*}
p(x) &=& a_n (x - z_n) (x - z_{n-1}) (x - z_{n-2}) \cdots (x - z_2) (x - z_1) \\
&\isdef & a_n \prod_{i=1}^n (x-z_i)
\end{eqnarray*}

where the points $ z_i$ are the polynomial roots, and they may be real or complex.


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA