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%TCIDATA{<META NAME="Title" CONTENT="Polynomial Functions and their Graphs">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 23 16:21:35 1997}
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\section{Polynomial Functions and their Graphs\label%
{Polynomial Functions and their Graphs}}

You have previously studied constant, linear and quadratic functions having
the forms $f\left( x\right) =c$, $f\left( x\right) =mx+b$, and $f\left(
x\right) =ax^{2}+bx+c$ respectively. All these are special cases of perhaps
the most important class of functions in mathematics, \textbf{polynomials}.
If you add functions obtained by dividing two polynomials, you get the class
of \textbf{rational functions}.

\subsection{Definition of Polynomial Functions\label%
{Definition of Polynomial Functions}}

A function of the form 
\[
P\left( x\right) =a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}\text{, with }%
a_{n}\neq 0 
\]
is a \textbf{polynomial of degree n}. The numbers $a_{0}$, $a_{1}$, $a_{2}$,
...$a_{n-1}$, $a_{n}$ are called the \textbf{coefficients} of the
polynomial. The number $a_{0}$ is called the \textbf{constant coefficient}
and the number $a_{n}$, the coefficient of the highest power, is the \textbf{%
leading coefficient}.

\subsection{\ Local Extrema of Polynomials\label%
{Local Extrema of Polynomials}}

If $P\left( x\right) =a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}$ is a
polynomial of degree $n$, then the graph of $P$ has at most $n-1$ local
extrema.

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