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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 22:42:02 1997}
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\input{tcilatex}
\begin{document}


\section{Polynomials and Rational Expressions 
\index{Polynomials@Polynomials}\label{Polynomials}}

With \textsl{Scientific Notebook} you can perform the usual operations on
polynomials in a variety of ways. The general procedure is as follows.

\begin{quote}
$\blacktriangleright $ To work with a polynomial expression
\end{quote}

\begin{enumerate}
\item[1.]  Enter the expression in mathematics mode.

\item[2.]  Leave the insertion point in the expression.

\item[3.]  Apply one of the commands from the \textsf{Maple }menu.
\end{enumerate}

These commands include \textsf{Evaluate}, \textsf{Simplify, Factor}, and%
\textsf{\ Expand}; from the \textsf{Combine} submenu, \textsf{Powers}; or
from the \textsf{Polynomials}\ submenu, \textsf{Divide}, \textsf{Partial
Fractions}, \textsf{Roots}, and \textsf{Sort}.

\subsection{\ Sums, Differences, and Quotients of Polynomials\label%
{Adding polynomials}}

\begin{quote}
$\blacktriangleright $ To collect terms, to find a sum or difference of
polynomials, or to change a quotient to rational form
\end{quote}

\begin{enumerate}
\item[1.]  Enter the expression in mathematics mode.

\item[2.]  Leave the insertion point in the expression and apply \textsf{%
Evaluate} (or click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{evaluate.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Evaluate.wmf';file-properties "XNPEU";}}or
press \textsc{ctrl +e}).
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) +\left( 8x^{2}+7\right) =\,11x^{2}+3x+7$

$\left( 3x^{2}+3x\right) -\left( 8x^{2}+7\right) =\,-5x^{2}+3x-7$

$x^{2}+x+5-3x^{3}+5x^{2}+4x-13+2x^{2}=\,8x^{2}+5x-8-3x^{3}$

$\left( 3x^{2}+3x\right) /\left( 8x^{2}+7\right) =\,\,\dfrac{3x^{2}+3x}{%
8x^{2}+7}$
\end{quotation}

\strut

Several of the other commands listed earlier have the same effect on these
particular expressions.

\subsection{\ Summation Notation\label{Summation notation}}

\begin{quote}
$\blacktriangleright $ \textsf{To enter a polynomial as a summation }$%
\sum_{k=0}^{5}a_{k}x^{k}$
\end{quote}

\begin{enumerate}
\item  Choose \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{sigma.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Sigma.wmf';file-properties "XNPEU";}}; or
choose \textsf{Insert + Operators}, and from the panel, choose \FRAME{itbpF}{%
0.3009in}{0.3009in}{0.0701in}{}{}{sigma.wmf}{\special{language "Scientific
Word";type "GRAPHIC";display "PICT";valid_file "F";width 0.3009in;height
0.3009in;depth 0.0701in;original-width 17.8125pt;original-height
16.9375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Sigma.wmf';file-properties "XNPEU";}}.

\item  With the insertion point immediately to the right of $\sum $, choose 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{subscrip.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Subscrip.wmf';file-properties "XNPEU";}} or
choose \textsf{Insert + Subscript}. Type $k=0$ in the input box.

\item  Press \textsc{tab}; or press \textsc{spacebar} and choose \FRAME{itbpF%
}{0.3009in}{0.3009in}{0.0701in}{}{}{supscrip.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Supscrip.wmf';file-properties "XNPEU";}} or
choose \textsf{Insert + Superscript}. Type $5$ in the input box.

\item  Press \textsc{spacebar} or an arrow key to return to the line, and
type $a_{k}x^{k}$.
\end{enumerate}

\subsection{\ Sums and Differences of Rational Expressions \label%
{Rational expressions}}

\begin{quote}
$\blacktriangleright $ To combine rational expressions over a common
denominator
\end{quote}

\begin{enumerate}
\item[1.]  Enter the expression in mathematics mode.

\item[2.]  Leave the insertion point in the expression.

\item[3.]  Apply \textsf{Simplify} (or click \FRAME{itbpF}{0.3035in}{0.3035in%
}{0.0701in}{}{}{simplify.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
0.3035in;height 0.3035in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'SIMPLIFY.WMF';file-properties "XNPEU";}}) or 
\textsf{Factor}. 
\index{Simplify@Simplify}\label{Simplify}
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Simplify} (or \textsf{Factor})
\end{quote}

\begin{quotation}
$%
\dfrac{3x^{2}+3x}{8x^{2}+7}+\dfrac{5x^{2}+3}{2x^{2}+x+7}=\,\dfrac{%
46x^{4}+9x^{3}+83x^{2}+21x+21}{\left( 8x^{2}+7\right) \left(
2x^{2}+x+7\right) }$

\ 

$\dfrac{\left( 3x^{2}+3x\right) ^{3}}{2x-5}-\dfrac{5x^{2}+3}{2x^{2}+x+7}$

$\quad \quad =\,\dfrac{%
54x^{8}+189x^{7}+432x^{6}+702x^{5}+594x^{4}+179x^{3}+25x^{2}-6x+15}{\left(
2x-5\right) \left( 2x^{2}+x+7\right) }$

\ 

$\dfrac{3x^{2}+4}{5x^{3}-3x}+\dfrac{4x+1}{x^{2}-1}=\,\dfrac{%
23x^{4}-11x^{2}-4+5x^{3}-3x}{x\left( 5x^{2}-3\right) \left( x^{2}-1\right) }$

$\quad \quad $or $\,=\dfrac{23x^{4}-11x^{2}-4+5x^{3}-3x}{x\left(
5x^{2}-3\right) \left( x-1\right) \left( x+1\right) }$
\end{quotation}

The feature of \hyperref{computing in place}{}{}{DM1-3.tex#Computing in
place (2)} is useful for manipulating polynomials.

\strut 
\index{In place computations@In place computations}\label%
{In-place computations}

\begin{example}
To change a factored denominator to expanded form,

enter the expression 
\[
\frac{x+2}{x+1}+\frac{3x}{x-1} 
\]

Leave the insertion point in the expression and apply \textsf{Factor} to
get\ 
\[
\frac{x+2}{x+1}+\frac{3x}{x-1}=\,2\frac{2x^{2}+2x-1}{\left( x-1\right)
\left( x+1\right) } 
\]

Select the denominator. Hold down the \textsc{ctrl } key and choose \textsf{%
Expand}\ to get $\frac{x+2}{x+1}+\frac{3x}{x-1}=2\frac{2x^{2}+2x-1}{x^{2}-1}$%
.
\end{example}

\strut

\begin{example}
Computing in place, used together with the editing features of \textsl{%
Scientific Notebook}, provides an easy way to display intermediate steps in
a computation.

Choose \textsf{Factor}. 
\[
\frac{x+2}{x+1}+\frac{3x}{x-1}=\,2 
\]

Type an equals sign, select the expression $2\frac{2x^{2}+2x-1}{\left(
x+1\right) \left( x-1\right) }$ and with the \textsc{ctrl } key pressed
down, drag a copy to the right of the equals sign. 
\[
\frac{x+2}{x+1}+\frac{3x}{x-1}=\,2\frac{2x^{2}+2x-1}{\left( x+1\right)
\left( x-1\right) }=\,2\frac{2x^{2}+2x-1}{\left( x+1\right) \left(
x-1\right) } 
\]

Select the denominator and, with the \textsc{ctrl } key pressed down, choose 
\textsf{Expand}. 
\[
\frac{x+2}{x+1}+\frac{3x}{x-1}=\,2\frac{2x^{2}+2x-1}{\left( x+1\right)
\left( x-1\right) }=\,2\frac{2x^{2}+2x-1}{x^{2}-1} 
\]
\end{example}

You can find the standard form for the equation of a circle by ``completing
the square.'' 
\index{Completing a square@Completing a square}\label{Completing a square}%
You can take advantage of computing in place for this computation also.

\strut

\begin{example}
Find the center and radius of the circle $\,x^{2}-6x+18+y^{2}+10y=0$.

Subtract the constant term $18$ from both sides of the equation. 
\[
x^{2}-6x+18+y^{2}+10y-18=0-18 
\]

Select the left side of the equation, and while holding down the \textsc{%
ctrl } key, apply \textsf{Simplify}. Then do the same to the right side. 
\[
x^{2}-6x+y^{2}+10y=-18 
\]

Select the terms containing $x$ and click 
\FRAME{itbpF}{0.2793in}{0.2992in}{0.0692in}{}{}{parens.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.2793in;height 0.2992in;depth
0.0692in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1.0179";cropright "0.9866";cropbottom "0";filename
'Parens.wmf';file-properties "XNPEU";}}. Then, do the same to the terms
containing $y$. 
\[
\left( x^{2}-6x\right) +\left( y^{2}+10y\right) =-18. 
\]

Add the square of one-half the coefficient of $x$ to both sides. Do the same
for the coefficient of $y$. 
\[
\left( x^{2}-6x+\left( \frac{-6}{2}\right) ^{2}\right) +\left(
y^{2}+10y+\left( \frac{10}{2}\right) ^{2}\right) =-18+\left( \frac{-6}{2}%
\right) ^{2}+\left( \frac{10}{2}\right) ^{2} 
\]

Select the term $\left( x^{2}-6x+\left( \frac{-6}{2}\right) ^{2}\right) $,
and while holding down the \textsc{ctrl } key, apply \textsf{Factor}. Do the
same for the term with $y$. 
\[
\left( x-3\right) ^{2}+\left( y+5\right) ^{2}=-18+\left( \frac{-6}{2}\right)
^{2}+\left( \frac{10}{2}\right) ^{2} 
\]

Select the right side of the equation, and while holding down the \textsc{%
ctrl } key, apply \textsf{Simplify}. 
\[
\left( x-3\right) ^{2}+\left( y+5\right) ^{2}=16 
\]

You can read the solution to this problem from this form of the equation.
The center of the circle is $(3,-5)$ and the radius is $\sqrt{16}=\,4$.
\end{example}

\strut

\subsection{\ Partial Fractions 
\index{Partial fractions@Partial fractions}\label{Partial fractions}}

With the command \textsf{Partial Fractions}, you can write a rational
expression as a sum of simpler fractions---essentially the reverse of the
operation demonstrated in the previous section.

\begin{quote}
$\blacktriangleright $ To write a rational expression as a sum of rational
expressions
\end{quote}

\begin{enumerate}
\item[1.]  Enter the rational expression in mathematics mode.

\item[2.]  Leave the insertion point in the expression.

\item[3.]  From the \textsf{Polynomials} submenu (or from the \textsf{%
Calculus} submenu), choose \textsf{Partial Fractions} . 
\index{Polynomials@Polynomials!partial fractions@partial fractions}
\end{enumerate}

The \textsf{Partial Fractions} command expands a rational expression into a
sum of rational expressions having denominators that are multiples of powers
of linear and irreducible quadratic factors. In this case \emph{irreducible}
means the roots are neither rational nor rational combinations of the
coefficients of the polynomials. This operation does not accept
floating-point numbers, so write the coefficients as integers or quotients
of integers. There is no automatic way to change a decimal to a fraction,
such as $.58$ to $%
\frac{58}{100}$, but here again you can take advantage of the fact that you
are working in a word processor. If more than one variable occurs in the
expression, specify your choice of variable in the dialog box that appears.

\strut

\begin{example}
In each of the following cases, apply \textsf{Partial Fractions} to the
expression on the left to expand the expression.

The variable is recognized automatically, because only one occurs. 
\[
\dfrac{36}{\left( x-2\right) \left( x-1\right) ^{2}\left( x+1\right) ^{2}}=%
\dfrac{4}{x-2}-\dfrac{9}{\left( x-1\right) ^{2}}-\dfrac{3}{\left( x+1\right)
^{2}}-\dfrac{4}{x+1} 
\]

The variable is $x$. The denominators are multiples of $x-y$, $(x-y)^{2}$,
and $x+1$. 
\begin{eqnarray*}
\dfrac{y}{(x-y)^{2}(x+1)}\! &=&\allowbreak -\frac{1}{y^{2}+2y+1}\frac{y}{x-y}%
+\frac{y}{\left( y+1\right) \left( x-y\right) ^{2}}+\frac{1}{y^{2}+2y+1}%
\frac{y}{x+1} \\
&=&\,\!\!-\dfrac{y}{\left( y^{2}+2y+1\right) \!\left( x-y\right) }+\dfrac{y}{%
\left( y+1\right) \!\left( x-y\right) ^{2}}+\dfrac{y}{\left(
y^{2}+2y+1\right) \!\left( x+1\right) }
\end{eqnarray*}

The variable is $y$. The denominators are multiples of $\left( x-y\right)
^{2}$ and $x-y$. 
\begin{eqnarray*}
\frac{y}{(x-y)^{2}(x+1)} &=&\allowbreak \frac{1}{x+1}\frac{x}{\left(
x-y\right) ^{2}}-\frac{1}{\left( x+1\right) \left( x-y\right) } \\
&=&\frac{x}{\left( x+1\right) \left( x-y\right) ^{2}}-\frac{1}{\left(
x+1\right) \left( x-y\right) }
\end{eqnarray*}
\end{example}

\strut

\subsection{\ Products and Powers of Polynomials \label{Polynomial products}%
\label{Expand (2)}}

The command \textsf{Expand} can be used to expand products or powers of
polynomials.

\begin{quote}
$\blacktriangleright $ \textsf{Expand} 
\index{Expand@Expand!polynomial@polynomial} (or click 
\FRAME{itbpF}{0.3001in}{0.3001in}{0.0701in}{}{}{cxpand.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 0.3001in;height 0.3001in;depth
0.0701in;original-width 17.8125pt;original-height 16.9375pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";filename
'Cxpand.wmf';file-properties "XNPEU";}})
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) \left( 8x^{2}+7\right)
=\,24x^{4}+21x^{2}+24x^{3}+21x$

$\left( 3x^{2}+3x\right) ^{3}=\,27x^{6}+81x^{5}+81x^{4}+27x^{3}$
\end{quotation}

\strut

\subsection{\ Division by Polynomials \label{Polynomials, divide}}

\begin{quote}
$\blacktriangleright $ To convert a quotient of polynomials $\dfrac{f(x)}{%
g(x)}$ with rational coefficients to the form $q(x)+\dfrac{r(x)}{g(x)}$,
where $r(x)$ and $q(x)$ are polynomials and $\deg r(x)<\deg g(x)$
\end{quote}

\begin{enumerate}
\item[1.]  Enter a quotient of polynomials.

\item[2.]  Leave the insertion point in the expression.

\item[3.]  From the \textsf{Polynomials} submenu, 
\index{Polynomials@Polynomials!long division@long division}\label%
{Polynomials, long division}choose 
\index{Divide@Divide!polynomials@polynomials}\label{Divide (2)} \textsf{%
Divide} 
\index{Polynomials@Polynomials!divide@divide}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Divide}
\end{quote}

\begin{quotation}
$\left( 3x^{2}+3x\right) /\left( 8x^{2}+7\right) =%
\dfrac{3}{8}+\dfrac{3x-\frac{21}{8}}{8x^{2}+7}$

$\dfrac{3x^{5}+3x^{3}-4x^{2}+5}{8x^{2}+7}=\dfrac{3}{8}x^{3}+\dfrac{3}{64}x-%
\dfrac{1}{2}+\dfrac{\frac{17}{2}-\frac{21}{64}x}{8x^{2}+7}\medskip $

$\dfrac{128x^{6}+128x^{5}+128x^{4}+768x^{3}-128x^{2}+\allowbreak 128x-128}{%
16x^{4}+8x^{3}+70x^{2}+7x+49}=8x^{2}+4x-29+\dfrac{%
664x^{3}+1482x^{2}+135x+1293}{16x^{4}+8x^{3}+70x^{2}+7x+49}$
\end{quotation}

\strut

\begin{quotation}
\emph{Note\quad }This algorithm is the familiar long division algorithm 
\index{Long division@Long division!polynomials@polynomials}\label%
{Long division (2)}for polynomials.
\end{quotation}

\strut

\subsection{\ Collecting and Ordering Terms 
\index{Polynomials@Polynomials!sort@sort}\label{Polynomials, sort}\label%
{Polynomials, collect}}

The 
\index{Sort@Sort}\label{Sort} \textsf{Sort }command on the \textsf{%
Polynomials} submenu collects numeric coefficients of terms of a polynomial
expression and returns the terms in order of decreasing degree. The \textsf{%
Collect} command on the \textsf{Polynomials} submenu collects all
coefficients of terms of a polynomial expression, but does not necessarily
sort the terms by degree. Specify your choice of polynomial variable in the
dialog box that appears.

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Sort}
\end{quote}

\begin{itemize}
\item  
\begin{quotation}
$5t^{2}+2t-16t^{5}+t^{3}-2t^{2}+9+at^{2}=\allowbreak
-16t^{5}+t^{3}+3t^{2}+at^{2}+2t+9$ (The variable is $t$.)

\item  $x^{2}+3x+5-3x^{3}+5x^{2}+4x^{3}+13+2x^{4}=\allowbreak
2x^{4}+x^{3}+6x^{2}+3x+18$

\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak
t^{2}x+5t^{2}-16t^{5}+y^{3}+9$ (The variable is $x$.)

\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak
-16t^{5}+xt^{2}+5t^{2}+y^{3}+9$ (The variable is $t$.)
\end{quotation}
\end{itemize}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Collect}
\end{quote}

\begin{itemize}
\item  
\begin{quotation}
$5t^{2}+2t-16t^{5}+t^{3}-2t^{2}+9+at^{2}=\allowbreak -16t^{5}+t^{3}+\left(
3+a\right) t^{2}+2t+9$ (The variable is $t$.)

\item  $x^{2}+3x+5-3x^{3}+5x^{2}+4x^{3}+13+2x^{4}=\allowbreak
6x^{2}+3x+18+x^{3}+2x^{4}$ (The variable is $x$.)

\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak
5t^{2}+xt^{2}-16t^{5}+y^{3}+9$ (The variable is $x$.)

\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak -16t^{5}+\left(
x+5\right) t^{2}+9+y^{3}$ (The variable is $t$.)\medskip
\end{quotation}
\end{itemize}

For some expressions, you may want to apply both commands.

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Collect, Polynomials + Sort}
\end{quote}

\begin{quotation}
(The variable is $x$)
\end{quotation}

\begin{itemize}
\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak
5t^{2}+xt^{2}-16t^{5}+y^{3}+9=\allowbreak t^{2}x+5t^{2}-16t^{5}+y^{3}+9$
\end{itemize}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Sort, Polynomials + Collect}
\end{quote}

\begin{quotation}
(The variable is $x$)
\end{quotation}

\begin{itemize}
\item  $5t^{2}+3xt^{2}-16t^{5}+y^{3}-2xt^{2}+9=\allowbreak
t^{2}x+5t^{2}-16t^{5}+y^{3}+9=\allowbreak t^{2}x+5t^{2}-16t^{5}+y^{3}+9$

\item  $x^{3}b+e+x^{2}c+x^{2}+x^{4}k+x-x^{3}d+a=\allowbreak
kx^{4}-dx^{3}+bx^{3}+x^{2}+cx^{2}+x+e+a=\allowbreak kx^{4}+\left(
-d+b\right) x^{3}+\left( c+1\right) x^{2}+x+e+a$
\end{itemize}

\strut

\subsection{\ Factoring Polynomials \label{Polynomials, factor}}

The ability to factor polynomials 
\index{Polynomials@Polynomials!factor@factor}is an important algebraic tool.
You will find that the factoring capabilities of \textsl{Scientific Notebook}
are powerful and useful. You can factor polynomials with integer or rational
roots and with other roots directly related to the coefficients of the
expanded polynomial.

To factor a polynomial, you must type it without using decimal notation.
Numbers such as $1.5$ are interpreted as floating-point 
\index{Floating point@Floating point}\label{Floating point}numbers, and 
\textsf{Factor} does not handle polynomials with floating-point
coefficients. Replace decimal numbers with fractions (such as $1.5=%
\frac{15}{10}$), and then apply \textsf{Factor}.

\strut

\begin{example}
Apply \textsf{Factor } 
\index{Factor@Factor!polynomial@polynomial}\label{Factor, polynomial}and
then \textsf{Expand } 
\index{Expand@Expand!polynomial@polynomial}\label{Expand, polynomial}to get
the following results.

$5x^{5}+5x^{4}-10x^{3}-10x^{2}+5x+5=\allowbreak 5\left( x-1\right)
^{2}\left( x+1\right) ^{3}=\allowbreak 5x^{5}+5x^{4}-10x^{3}-10x^{2}+5x+5%
\vspace{14pt}$

$\frac{1}{16}x^{2}-\frac{37}{30}x-\frac{56}{15}=\allowbreak \frac{1}{240}%
\left( 3x+8\right) \left( 5x-112\right) =\allowbreak \frac{1}{16}x^{2}-\frac{%
37}{30}x-\frac{56}{15}$ \vspace{14pt}

Apply \textsf{Expand} and then \textsf{Factor} to produce the following.

$5\left( x+\frac{1}{\sqrt{3}}\right) ^{3}(x-\frac{1}{4})^{2}=\allowbreak
5x^{5}-\frac{5}{2}x^{4}+\frac{85}{16}x^{3}+5x^{4}\sqrt{3}-\frac{5}{2}x^{3}%
\sqrt{3}+\frac{125}{144}x^{2}\sqrt{3}-\allowbreak \frac{5}{2}x^{2}+\frac{5}{%
16}x-\frac{5}{18}x\sqrt{3}+\frac{5}{144}\sqrt{3}=\allowbreak \frac{5}{432}%
\left( 4x-1\right) ^{2}\left( 3x+\sqrt{3}\right) ^{3}\vspace{14pt}$

$5\left( x+3i\right) ^{2}(x-\frac{1}{4}+\frac{2}{3}i)^{2}=\allowbreak 5x^{4}-%
\frac{5}{2}x^{3}+\frac{110}{3}ix^{3}-\frac{12515}{144}x^{2}-\frac{50}{3}%
ix^{2}-\frac{1715}{24}ix+\allowbreak \frac{65}{2}x+\frac{275}{16}%
+15i=\allowbreak \frac{5}{144}\left( x+3i\right) ^{2}\left( 12x-3+8i\right)
^{2}\vspace{14pt}$
\end{example}

\strut

Note that factoring polynomials generally returns factors with integer
coefficients. \textsf{Factor} is effective primarily for polynomials with
integer or rational coefficients, although it also factors polynomials whose
roots are closely related to the coefficients, as demonstrated in the two
preceding examples. Technically, the polynomial is factored over the field
generated by its coefficients. If all the coefficients are rational, then
the polynomial is factored over the rationals. Apply \textsf{Factor} to get
the following product.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$\frac{1}{2}x^{2}+3x-\frac{20}{9}=\,\frac{1}{18}\left( 3x+20\right) \left(
3x-2\right) $
\end{quotation}

You can factor not only the difference of two squares and the sum and
difference of two cubes, but also the difference of any two equal powers
with \textsl{Scientific Notebook}.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$x^{2}-y^{2}=\left( x-y\right) \left( x+y\right) $

$x^{3}-y^{3}=\left( x-y\right) \left( x^{2}+xy+y^{2}\right) $

$x^{4}-y^{4}=\left( x-y\right) \left( x+y\right) \left( x^{2}+y^{2}\right) $
\end{quotation}

You can also factor the sum of any two equal odd powers.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$x^{3}+y^{3}=\left( x+y\right) \left( x^{2}-xy+y^{2}\right) $

$x^{5}+y^{5}=\left( x+y\right) \left(
x^{4}-x^{3}y+x^{2}y^{2}-xy^{3}+y^{4}\right) $

$x^{7}+y^{7}=\left( x+y\right) \left(
y^{6}-xy^{5}+x^{2}y^{4}-x^{3}y^{3}+x^{4}y^{2}-x^{5}y+x^{6}\right) $
\end{quotation}

\strut

\subsection{\ Greatest Common Divisor of Two Polynomials \label%
{Polynomials, gcd}}

You find the greatest common divisor of two polynomials in the same way as
you found the \hyperref{greatest common divisor}{}{}{DM2-2.tex#Greatest
common divisor} of two integers.

\begin{quote}
$\blacktriangleright $ To find the greatest common divisor of two or more
polynomials
\end{quote}

\begin{enumerate}
\item[1.]  Type $gcd$ in mathematics mode, or choose it from the functions
menu.

\item[2.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{parens.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Parens.wmf';file-properties "XNPEU";}} and type
the polynomials in mathematics mode, separated by red commas.

\item[3.]  Choose \textsf{Evaluate}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\gcd (5x^{2}-5x,10x-10)=5x-5$

$\gcd \left( yx+3x-5y-15,\,xz-53x-5z+265\right) =\,x-5$

$\gcd \left( x^{2}+3x+yx+3y,\,x^{2}-4yx-5y^{2},3x^{2}+2yx-y^{2}\right) =x+y$
\end{quotation}

You can check these results by factoring the polynomials and comparing the
factors.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$5x^{2}-5x=5x\left( x-1\right) $

$10x-10=10\left( x-1\right) $

$yx+3x-5y-15\!=\left( x-5\right) \left( y+3\right) $

$xz-53x-5z+265\!=\left( x-5\right) \left( z-53\right) $

$x^{2}+3x+yx+3y=\left( x+y\right) \left( x+3\right) $

$x^{2}-4yx-5y^{2}=\left( x+y\right) \left( x-5y\right) $

$3x^{2}+2yx-y^{2}=\left( x+y\right) \left( 3x-y\right) $
\end{quotation}

\index{lcm@lcm}\label{lcm (2)}\label{Least common multiple (2)}The 
\hyperref{least common multiple}{}{}{DM2-2.tex#Greatest common divisor}
function is also available for polynomials.

\begin{quote}
$\blacktriangleright $ To find the least common multiple of two or more
polynomials
\end{quote}

\begin{enumerate}
\item[1.]  Type $lcm$ in mathematics mode. (It will turn gray.)

\item  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{parens.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Parens.wmf';file-properties "XNPEU";}} or
choose \textsf{Insert + Brackets} and select parentheses.

\item  In the input box, type the polynomials separated by (red) commas.

\item  Choose \textsf{Evaluate}.
\end{enumerate}

Apply \textsf{Factor} to the result to reveal the relationship among the
polynomials.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate, } \textsf{Factor}
\end{quote}

\begin{quotation}
$\limfunc{lcm}(yx+3x-5y-15,xz-53x-5z+265)$

$\quad =xyz-53xy+3xz-159x-5yz+265y-15z+795$

$\quad =\left( x-5\right) \left( y+3\right) \left( z-53\right) $
\end{quotation}

\strut

\subsection{\ Roots of Polynomials \label{Polynomials, roots}}

If zero is obtained when a number is substituted for the variable in a
polynomial, then that number is a \emph{root} 
\index{Polynomials@Polynomials!roots@roots} of 
\index{Roots@Roots}\label{Roots} the polynomial. For example, $1$ is a root
of $x^{2}-1$.

\strut

\begin{quotation}
\emph{Note\quad }A number $r$ is a root of a polynomial if and only if $x-r$
is a factor of that polynomial.
\end{quotation}

\strut

The factorization $\,$

\[
x^{3}-%
\frac{8}{3}x^{2}-\frac{5}{3}x+2=\frac{1}{3}\left( x-3\right) \left(
3x-2\right) \left( x+1\right) 
\]
displays the three roots $3$,\ $\frac{2}{3}$, $-1$. Notice that these roots
are precisely the values of the $x$-coordinate where the graph of $y=x^{3}-%
\frac{8}{3}x^{2}-\frac{5}{3}x+2$ crosses the $x$-axis.

\FRAME{dtbpFU}{2.9992in}{1.9995in}{0pt}{\Qcb{$y=x^{3}-\frac{8}{3}x^{2}-\frac{%
5}{3}x+2$}}{}{Plot }{\special{language "Scientific Word";type
"MAPLEPLOT";width 2.9992in;height 1.9995in;depth 0pt;display
"PICT";plot-snapshots TRUE;function
\TEXUX{$x^{3}-\frac{8}{3}x^{2}-\frac{5}{3}x+2$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-2";xmax "4";xviewmin
"-2.12";xviewmax "4.122";yviewmin "-13.93";yviewmax
"17.28";rangeset"X";recompute TRUE;phi 45;theta 45;plottype 4;numpoints
49;axesstyle "normal";xis \TEXUX{x};var1name \TEXUX{$x$};}}

The preceding graph was created within \textsl{Scientific Notebook}. Click 
\hyperref{here}{}{}{DM6.tex} for the chapter discussing techniques for such
displays.

The factorization 
\[
x^3-\frac{13}5ix^2-8x^2+\frac{29}5ix+\frac{81}5x+6i-\frac{18}5 
\]
\[
=\frac 15\left( 5x+2i\right) \left( x-5-3i\right) \left( x-3\right) 
\]
shows the three roots $-\frac 25i$, $5+3i$, $3$.

You can find all real and complex roots directly by applying \textsf{Roots}
from the \textsf{Polynomials} menu.

\begin{quote}
$\blacktriangleright $ To find the roots of a polynomial
\end{quote}

\begin{enumerate}
\item[1.]  Type the polynomial.

\item[2.]  Leave the insertion point in the expression.

\item[3.]  From the \textsf{Polynomials} submenu, choose \textsf{Roots}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{3}-\frac{13}{5}ix^{2}-8x^{2}+\frac{29}{5}ix+\frac{81}{5}x+6i-\frac{18}{5}
$, roots: $
\begin{array}{c}
-\frac{2}{5}i \\ 
5+3i \\ 
3
\end{array}
$
\end{quotation}

\label{FTA}It follows from the Fundamental Theorem of Algebra that the
number of roots (including complex roots and multiplicities) is the same as
the degree of the polynomial.

For polynomials with rational (real or complex) coefficients, \textsl{%
Scientific Notebook} uses the usual formulas for finding roots symbolically
for polynomials of degree $4$ or less, and it finds the roots numerically
for polynomials of higher degree. This behavior is due to the mathematical
phenomenon that there is no general formula in terms of radical expressions
for the roots of polynomials of degree $5$ and higher.

\subsubsection{Second-Degree Polynomials\label{Quadratic polynomials}}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$ax^{2}+bx+c$, roots: $
\begin{array}{c}
\frac{1}{2a}\left( -b+\sqrt{\left( b^{2}-4ac\right) }\right) \vspace{6pt} \\ 
\frac{1}{2a}\left( -b-\sqrt{\left( b^{2}-4ac\right) }\right) \vspace{6pt}
\end{array}
$

$5x^{2}+2x-3$, roots: $
\begin{array}{c}
\frac{3}{5} \\ 
-1
\end{array}
$

$5x^{2}+x+3$, roots: $
\begin{array}{c}
-\frac{1}{10}+\frac{1}{10}i\sqrt{59}\vspace{6pt} \\ 
-\frac{1}{10}-\frac{1}{10}i\sqrt{59}\vspace{6pt}
\end{array}
$
\end{quotation}

\subsubsection{Third- and Fourth-Degree Polynomials\label{Cubic polynomials}}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{3}+3x^{2}+3x+1$, roots: $
\begin{array}{c}
-1 \\ 
-1 \\ 
-1
\end{array}
$\quad

$x^{4}+4x^{3}-12x^{2}-32x+64$, roots: $
\begin{array}{r}
-4 \\ 
-4 \\ 
2 \\ 
2
\end{array}
$

$x^{3}+cx-1$, roots: $
\begin{array}{l}
-\frac{1}{6}\frac{-\alpha ^{2}+12c}{\alpha } \\ 
\frac{1}{12}\frac{-\alpha ^{2}+12c}{\sqrt[3]{\alpha }}+\frac{1}{2}i\sqrt{3}%
\left( \frac{1}{6}\frac{\alpha ^{2}+12c}{\sqrt[3]{\alpha }}\right) \vspace{%
6pt} \\ 
\frac{1}{12}\frac{-\alpha ^{2}+12c}{\sqrt[3]{\alpha }}-\frac{1}{2}i\sqrt{3}%
\left( \frac{1}{6}\frac{\alpha ^{2}+12c}{\sqrt[3]{\alpha }}\right) \vspace{%
6pt}
\end{array}
$,

$\qquad \qquad \qquad \qquad \alpha =\sqrt[3]{\left( 108+12\sqrt{\left(
12c^{3}+81\right) }\right) }$
\end{quotation}

Factor the polynomials to observe the repeated roots.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$x^{3}+3x^{2}+3x+1=\,\left( x+1\right) ^{3}\vspace{6pt}$

$x^{4}+4x^{3}-12x^{2}-32x+64=\left( x+4\right) ^{2}\left( x-2\right) ^{2}%
\vspace{6pt}$

$x^{3}-\frac{3}{2}\sqrt[3]{2}x-1=\frac{1}{4}\left( x-\left( \sqrt[3]{2}%
\right) ^{2}\right) \left( 2x+\left( \sqrt[3]{2}\right) ^{2}\right) ^{2}%
\vspace{6pt}$
\end{quotation}

The roots of third- and fourth-degree polynomials can be complicated, with
multiple embedded radicals in the expressions. To put those roots in simpler
form, you may want numerical approximations. You get numerical results if
you enter at least one coefficient in decimal notation. You can also get a
numerical form directly from the symbolic one by applying \textsf{Evaluate
Numerically} to the matrix of roots. The following example shows both a
symbolic solution and a numerical solution (with \textsf{Digits Used in
Display} set to $6$).

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{3}+3x+1$, roots:

$
\begin{array}{l}
-\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }+\frac{1}{\sqrt[3]{%
\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }}\vspace{6pt} \\ 
\frac{1}{2}\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }-\frac{1}{%
2\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }}+\frac{1}{2}i\sqrt{%
3}\left( -\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }-\frac{1}{%
\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }}\right) \vspace{6pt}
\\ 
\frac{1}{2}\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }-\frac{1}{%
2\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }}-\frac{1}{2}i\sqrt{%
3}\left( -\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }-\frac{1}{%
\sqrt[3]{\left( \frac{1}{2}+\frac{1}{2}\sqrt{5}\right) }}\right)
\end{array}
$

\ \ \vspace{6pt}

$x^{3}+3x+1.0$, roots: $
\begin{array}{l}
-.322185 \\ 
\;\,.161093-1.75438i \\ 
\;\,.161093+1.75438i
\end{array}
$
\end{quotation}

Substituting the exact roots for $x$ in the polynomial $x^{3}+3x+1$ gives
zero, as it should. Applying \textsf{Evaluate} has little effect, but both 
\textsf{Simplify} and \textsf{Factor} give the following result. 
\index{Simplify@Simplify}

\begin{quote}
$\blacktriangleright $ \textsf{Simplify} or \textsf{Factor}
\end{quote}

\begin{quotation}
$\left( -\sqrt[3]{\left( 
\tfrac{1}{2}+\tfrac{1}{2}\sqrt{5}\right) }+\frac{1}{\sqrt[3]{\left( \frac{1}{%
2}+\frac{1}{2}\sqrt{5}\right) }}\right) ^{3}+3\left( -\sqrt[3]{\left( \tfrac{%
1}{2}+\tfrac{1}{2}\sqrt{5}\right) }+\frac{1}{\sqrt[3]{\left( \frac{1}{2}+%
\frac{1}{2}\sqrt{5}\right) }}\right) +\,1=\,0$
\end{quotation}

Using the numerical approximations to the roots, you get a very small, but
nonzero, value. You can get closer approximations to the roots by increasing
the number of displayed digits before finding the roots.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( -.322185\right) ^{3}+3\left( -.322185\right)
+1.0=\,1.174312318\times 10^{-6}\vspace{6pt}$

$\left( -.32218535462608559291\right) ^{3}+3\left(
-.32218535462608559291\right) +1$

$\qquad \qquad \qquad =\,4.870126439\times 10^{-21}$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{4}-2x-3$, roots: $
\begin{array}{l}
-1\vspace{6pt} \\ 
\sqrt[3]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }-\frac{2}{9\sqrt[3%
]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }}+\frac{1}{3}\vspace{6pt}
\\ 
-\frac{1}{2}\sqrt[3]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }+%
\frac{1}{9\sqrt[3]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }}+\frac{%
1}{3} \\ 
\qquad \qquad +\frac{1}{2}i\sqrt{3}\left( \sqrt[3]{\left( \frac{37}{27}+%
\frac{1}{3}\sqrt{17}\right) }+\frac{2}{9\sqrt[3]{\left( \frac{37}{27}+\frac{1%
}{3}\sqrt{17}\right) }}\right) \vspace{6pt} \\ 
-\frac{1}{2}\sqrt[3]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }+%
\frac{1}{9\sqrt[3]{\left( \frac{37}{27}+\frac{1}{3}\sqrt{17}\right) }}+\frac{%
1}{3} \\ 
\qquad \qquad -\frac{1}{2}i\sqrt{3}\left( \sqrt[3]{\left( \frac{37}{27}+%
\frac{1}{3}\sqrt{17}\right) }+\frac{2}{9\sqrt[3]{\left( \frac{37}{27}+\frac{1%
}{3}\sqrt{17}\right) }}\right)
\end{array}
$
\end{quotation}

You can evaluate the roots numerically or enter a coefficient in
floating-point notation before finding roots to get the following.

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$x^{4}-2x-3.0$, roots: $\, 
\begin{array}{l}
-1.0 \\ 
\;\,1.574743074 \\ 
-0.2873715369+1.349996398i \\ 
-0.2873715369-1.349996398i
\end{array}
$
\end{quotation}

\subsubsection{Polynomials of Degree 5 and Higher}

Numerical approximations are always returned for roots of polynomials of
degree $5$ and higher. You can change the number of digits used in the
display of these roots by making changes in the \textsf{Settings} box.

\begin{quote}
$\blacktriangleright $ \textsf{Polynomials + Roots}
\end{quote}

\begin{quotation}
$5x^{5}+5x^{4}-10x^{3}-10x^{2}+5x+5$, roots: $
\begin{array}{r}
1.0 \\ 
1.0 \\ 
-1.0 \\ 
-1.0 \\ 
-1.0
\end{array}
$

\ 

$\,x^{8}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+x+1$, roots: $
\begin{array}{r}
-.93969262-.34202014i \\ 
-.93969262+.34202014i \\ 
-.5-.8660254i \\ 
-.5+.8660254i \\ 
.17364818-.98480775i \\ 
.17364818+.98480775i \\ 
.76604444-.64278761i \\ 
.76604444+.64278761i
\end{array}
$
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{enumerate}
\begin{itemize}
\item  \hyperref{Numerical Approximations}{}{}{DM2-3.tex#Numerical
approximations}

\item  \hyperref{The Display of Numerical Results}{}{}{DM2-3.tex#Maple
settings}

\item  \hyperref{Integration by Partial Fractions}{}{}{DM7-5.tex#Partial
fractions}

\item  \hyperref{Polynomials Modulo m}{}{}{DM12-3.tex#Polynomials modulo m}

\item  \hyperref{Congruence Modulo Polynomials}{}{}{DM12-4.tex#Polynomials
modulo polynomials}

\item  \hyperref{Greatest Common Divisor of Polynomials}{}{}{DM12-4.tex#gcd,
polynomials}

\item  \hyperref{Multiplicity of Roots of Polynomials}{}{}{%
DM12-4.tex#Multiple roots}

\item  \hyperref{Exercises and Solutions}{}{}{DM3-6.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 38.875pt;height 24.5pt;depth
6pt;original-width 36.125pt;original-height 22.5625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Addition}{}{}{DMIndex.tex#Addition}, \hyperref{Completing a square%
}{}{}{DMIndex.tex#Completing a square}, \hyperref{Division}{}{}{%
DMIndex.tex#Division}, \hyperref{Expand}{}{}{DMIndex.tex#Expand}, \hyperref{%
Factor}{}{}{DMIndex.tex#Factor}, \hyperref{Greatest common divisor}{}{}{%
DMIndex.tex#Greatest common divisor}, \hyperref{Least common multiple}{}{}{%
DMIndex.tex#Least common multiple}, \hyperref{Long division}{}{}{%
DMIndex.tex#Long division}, \hyperref{Partial fractions}{}{}{%
DMIndex.tex#Partial fractions}, \hyperref{Polynomials}{}{}{%
DMIndex.tex#Polynomials}, \hyperref{Rational expressions}{}{}{%
DMIndex.tex#Rational expressions}, \hyperref{Roots of polynomials}{}{}{%
DMIndex.tex#Roots} }
\end{enumerate}

\end{document}
