Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


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.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA