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%TCIDATA{<META NAME="Title" CONTENT="Polynomials">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Wed May 14 11:37:59 1997}
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\input{tcilatex}
\begin{document}


\subsection{Polynomials\label{Polynomials}\hrulefill}

Use the commands on the Polynomials and Calculus menus to collect and order
the terms in polynomials, divide polynomials, determine roots, and write a
rational expression as the sum of partial fractions.

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Collect the
terms in \ $3x-7x^{2}+8x-3+x^{5}$}

\begin{enumerate}
\item  Click\FRAME{itbpF}{20.125pt}{20.125pt}{2pt}{}{}{text.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 20.125pt;height 20.125pt;depth
2pt;original-width 16.0625pt;original-height 16.0625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'text.wmf';file-properties
"XNPEU";}}or, from the Insert menu, choose Math to start mathematics.

\item  Type \textbf{3x-7x} and click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{%
}{}{supscrip.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'supscrip.wmf';file-properties
"XNPEU";}}or, from the Insert menu, choose Superscript.

\item  Type \textbf{2} and press spacebar.

\item  Type \textbf{+8x-3+x}.

\item  Click$\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{supscrip.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 19.5625pt;height 18.8125pt;depth 2pt;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'supscrip.wmf';file-properties "XNPEU";}}$, type 
\textbf{5}, and press spacebar.

\item  From the Maple menu, choose Polynomials and then choose Collect.

\item  Type \textbf{x }in the Need Polynomial Variable dialog box. Maple
collects the terms and displays this result: \ \medskip 
\[
\fbox{%
\begin{tabular}{lllll}
\rule[-0.1in]{0in}{0.3in} &  & $3x-7x^{2}+8x-3+x^{5}=\allowbreak
11x-7x^{2}-3+x^{5}$ & \rule[-0.1in]{0in}{0.3in} & 
\end{tabular}
}
\]
\medskip 
\end{enumerate}

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Order the
terms in \ $3x-7x^{2}+8x-3+x^{5}$}

\begin{enumerate}
\item  Follow the steps in the preceding example to enter the expression.

\item  From the Maple menu, choose Polynomials and then choose Sort. Maple
orders the terms and displays this result: \ \medskip 
\[
\fbox{%
\begin{tabular}{lllll}
\rule[-0.1in]{0in}{0.3in} &  & $3x-7x^{2}+8x-3+x^{5}=\allowbreak
x^{5}-7x^{2}+11x-3$ &  & \rule[-0.1in]{0in}{0.3in}%
\end{tabular}
}
\]
\medskip 
\end{enumerate}

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Write $%
\dfrac{x^{6}-5x^{4}+3x+4}{\left( x^{2}-2\right) \left( x+3\right) ^{3}}$ as
the sum of partial fractions}

\begin{enumerate}
\item  Click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{fraction.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 19.5625pt;height 18.8125pt;depth 2pt;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'fraction.wmf';file-properties "XNPEU";}}or,
from the Insert menu, choose Fraction.

\item  Type \textbf{x}, click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{%
supscrip.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'supscrip.wmf';file-properties
"XNPEU";}}or, from the Insert menu, choose Superscript. Type \textbf{6},\
and press spacebar.

\item  Type \textbf{-5x, }click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{%
supscrip.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'supscrip.wmf';file-properties
"XNPEU";}}, type \textbf{4},\ and press spacebar.

\item  Type \textbf{+3x+4 }and press Tab.

\item  Click\FRAME{itbpF}{19.8125pt}{18.9375pt}{2pt}{}{}{parens.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 19.8125pt;height 18.9375pt;depth
2pt;original-width 20.3125pt;original-height 20.3125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'parens.wmf';file-properties
"XNPEU";}}, type \textbf{x}, click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{%
}{supscrip.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'supscrip.wmf';file-properties
"XNPEU";}}, type \textbf{2}, press spacebar, type \textbf{-2},\textbf{\ }and
press spacebar.

\item  Click\FRAME{itbpF}{19.8125pt}{18.9375pt}{2pt}{}{}{parens.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 19.8125pt;height 18.9375pt;depth
2pt;original-width 20.3125pt;original-height 20.3125pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'parens.wmf';file-properties
"XNPEU";}}, type \textbf{x+3},\ and press spacebar.

\item  Click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{supscrip.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 19.5625pt;height 18.8125pt;depth 2pt;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'supscrip.wmf';file-properties "XNPEU";}}and
type \textbf{3}.

\item  From the Maple menu, choose Calculus and then choose Partial
Fractions. Maple computes the partial fraction and returns this result: \ 
\end{enumerate}

\noindent 
\[
\fbox{%
\begin{tabular}{lll}
\rule[-0.2in]{0in}{0.5in} & $\dfrac{x^{6}-5x^{4}+3x+4}{\left( x^{2}-2\right)
\left( x+3\right) ^{3}}=\allowbreak x-9+\dfrac{319}{7\left( x+3\right) ^{3}}-%
\dfrac{4491}{49\left( x+3\right) ^{2}}+\dfrac{17126}{343\left( x+3\right) }%
+\allowbreak \dfrac{1}{343}\dfrac{-534+367x}{x^{2}-2}$ & 
\rule[-0.2in]{0in}{0.5in}%
\end{tabular}
}
\]

\subparagraph{\FRAME{itbpF}{50.75pt}{21.75pt}{6pt}{}{}{ex3.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 50.75pt;height 21.75pt;depth 6pt;original-width
48.875pt;original-height 19.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'ex3.WMF';file-properties "XNPEU";}}Find the
roots of the expression $x^{2}+4$}

\begin{enumerate}
\item  Click\FRAME{itbpF}{20.125pt}{20.125pt}{2pt}{}{}{text.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 20.125pt;height 20.125pt;depth
2pt;original-width 16.0625pt;original-height 16.0625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'text.wmf';file-properties
"XNPEU";}}or, from the Insert menu, choose Math to start mathematics.

\item  Type \textbf{x}, click\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{%
supscrip.wmf}{\special{language "Scientific Word";type "GRAPHIC";display
"PICT";valid_file "F";width 19.5625pt;height 18.8125pt;depth
2pt;original-width 17.8125pt;original-height 16.9375pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'supscrip.wmf';file-properties
"XNPEU";}}, type \textbf{2},\textbf{\ }press spacebar, type \textbf{+4}.

\item  From the Maple menu, choose Polynomials and then choose Roots. Your
screen shows this result: \medskip 
\[
\fbox{%
\begin{tabular}{lll}
\rule[-0.2in]{0in}{0.45in} & $x^{2}+4$, roots: $
\begin{array}{c}
2i \\ 
-2i
\end{array}
$ & \rule[-0.2in]{0in}{0.45in}%
\end{tabular}
}
\]
\medskip 
\end{enumerate}

\subparagraph{Try these exercises with polynomials\textit{\label{exercises}}}

\begin{quotation}
\rule[-0.05in]{0in}{0.3in}\quad Collect the terms in these expressions:

\[
5t^{2}+2t-16t^{5}+t^{3}-2t^{2}+9\text{\hspace{0.5in}}%
x^{2}+y+5-3x^{3}y+5x^{2}+4y^{3}+13+2x^{4}\text{ (Use the variable }x\text{%
)\medskip } 
\]

\quad Order the terms in these expressions:

\[
3x-7x^{2}+8x-3+x^{5}\text{\hspace{0.5in}}%
x^{2}+y+5-3x^{3}y+5x^{2}+4y^{3}+13+2x^{4}\text{ (Use the variable }x\text{)}%
\medskip 
\]

\quad Divide these polynomials:

\[
\dfrac{x^{5}-5x^{4}+3x+4}{x^{2}-2}\text{\hspace{0.5in}}\dfrac{\left(
3x^{2}+3x\right) }{\left( 8x^{2}+7\right) }\text{\medskip }
\]

\quad Find the roots of these expressions:

\[
x^{3}-2x-2x^{2}+4\text{\hspace{0.5in}}x^{3}-\dfrac{13}{5}ix^{2}-8x^{2}+%
\dfrac{29}{5}ix+\dfrac{81}{5}x+6i-\dfrac{18}{5}
\]
\end{quotation}

\end{document}
