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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Algebraic Expressions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Mar 20 22:27:47 1997}
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\section{\label{Algebraic Expressions}Algebraic Expressions}

A \textbf{variable} is a letter that can represent any number in a given set
of numbers, whereas a \textbf{constant} represents a fixed (or specific)
number. The \textbf{domain} of a variable is the set of values that the
variable is permitted to take on; so, for instance, in the expression $\sqrt{%
x}$ the domain of $x$ is $\left\{ x\mid x\geq 0\right\} $, whereas in the
expression $\frac{2}{x-3}$ the domain of $x$ is $\left\{ x\mid x\neq
3\right\} $.

\vspace{1pt}\textbf{Algebraic expressions} are obtained by starting with
variables and constants and combining them using addition, subtraction,
multiplication, division, exponentiation and roots. The simplest types of
algebraic expressions use only addition, subtraction, and multiplication;
these are called \textbf{polynomials\label{polynomials}}. The general form
of a polynomial of degree $n$ in the variable $x$ is 
\[
a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}
\]
where $a_{0}$, $a_{1}$, ..., $a_{n}$ are constants and $a_{n}\neq 0$. The 
\textbf{degree} of a polynomial is the highest power of the variable. Any
polynomial is a sum of \textbf{terms} of the form $ax^{k}$, called \textbf{%
monomials}, where $a$ is a constant and $k$ is a nonnegative integer. A 
\textbf{binomial} is the sum of two monomials, a \textbf{trinomial} is the
sum of three monomials, and so on. Thus $7x^{6}+4x$ is a binomial of degree $%
6$, while $173x^{5}$ is a monomial.\bigskip 

\subsection{\label{Special Product Formulas}Special Product Formulas}

\begin{itemize}
\item  $\left( A-B\right) \left( A+B\right) =A^{2}-B^{2}$

\item  $\left( A+B\right) ^{2}=A^{2}+2AB+B^{2}$

\item  $\left( A-B\right) ^{2}=A^{2}-2AB+B^{2}$

\item  $\left( A+B\right) ^{3}=A^{3}+3A^{2}B+3AB^{2}+B^{3}$

\item  $\left( A-B\right) ^{3}=A^{3}-3A^{2}B+3AB^{2}-B^{3}\bigskip $
\end{itemize}

\subsection{\label{Factoring Formulas}Factoring Formulas}

\begin{itemize}
\item  \label{Difference of squares}\textbf{Difference of squares}: $%
A^{2}-B^{2}=\left( A-B\right) \left( A+B\right) $

\item  \textbf{Perfect square: }$A^{2}+2AB+B^{2}=\left( A+B\right) ^{2}$

\item  \textbf{Perfect square: }$A^{2}-2AB+B^{2}=\left( A-B\right) ^{2}$

\item  \label{Difference of cubes}\textbf{Difference of cubes}: $%
A^{3}-B^{3}=\left( A-B\right) \left( A^{2}+AB+B^{2}\right) $

\item  \label{Sum of cubes}\textbf{Sum of cubes}: $A^{3}+B^{3}=\left(
A+B\right) \left( A^{2}-AB+B^{2}\right) \bigskip $
\end{itemize}

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