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Mathematics of the DFT
    Complex Numbers
       Fundamental Theorem of Algebra

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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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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