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%TCIDATA{<META NAME="Title" CONTENT="Solving Polynomial Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:12:21 1997}
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\begin{document}


\section{Solving Polynomial Equations 
\index{Exact solutions@Exact solutions}\label{Exact solutions}}

There are four options on the \textsf{Solve} submenu: \textsf{Exact}, 
\textsf{Integer}, \textsf{Numeric}, and \textsf{Recursion}. The option 
\textsf{Exact} 
\index{Solve@Solve!exact@exact}\label{Solve, exact (2)} is general in nature
and is used in most situations. It returns symbolic solutions when it can,
and numerical solutions otherwise. If any of the components of the problem
use floating-point (decimal) notation, the response is a numerical solution.
The other three options are used in more specialized situations, as
discussed next.

\strut

\subsection{\ Equations with One Variable}

\begin{quote}
$\blacktriangleright $ To solve an equation with one variable
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the equation.

\item[2.]  From the \textsf{Solve }submenu, choose \textsf{Exact}.
\end{enumerate}

\textsl{Scientific Notebook} returns a solution. Note that in the following
examples, integer or rational coefficients yield algebraic solutions and
real (floating-point) coefficients yield decimal approximations. All
solutions are found for polynomial equations, including complex solutions.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$5x^{2}+3x=1$, Solution is : $\left\{ x=-%
\dfrac{3}{10}+\dfrac{1}{10}\sqrt{29}\right\} ,\,\left\{ x=-\dfrac{3}{10}-%
\dfrac{1}{10}\sqrt{29}\right\} $

$5x^{2}+3x=1.0$, Solution is : $\left\{ x=.239\right\} ,\,\left\{
x=-.839\right\} $

$s^{2}+10s+\dfrac{1681}{64}=0$, Solution is : $\left\{ s=-5+\dfrac{9}{8}%
i\right\} ,\,\left\{ s=-5-\dfrac{9}{8}i\right\} $

$s^{2}+\left( 10.0\right) s+\dfrac{1681}{64}=0$, Solution is : $\left\{
s=-5.0+1.125i\right\} ,\,\left\{ s=-5.0-1.125i\right\} $
\end{quotation}

\textsl{Scientific Notebook} displays multiple roots in some cases.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$\left( x-5\right) ^{3}=0$, Solution is : $\left\{ x=5\right\} ,\,\left\{
x=5\right\} ,\,\left\{ x=5\right\} $
\end{quotation}

You can solve equations with rational expressions.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$\dfrac{14}{a+2}-\dfrac{1}{a-4}=1$, Solution is : $\left\{ a=5\right\}
,\,\left\{ a=10\right\} $
\end{quotation}

You can solve equations involving absolute values.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$\left| 3x-2\right| =5$, Solution is : $\left\{ x=\dfrac{7}{3}\right\}
,\,\left\{ x=-1\right\} $
\end{quotation}

\subsubsection{Checking the Answer}

The\ \textsf{Define\ }command discussed previously provides a convenient way
of testing solutions.

\strut

\begin{example}
Check the solutions to several of the preceding equations.

\begin{itemize}
\item  Define $a=5$. Evaluate the expression $\dfrac{14}{a+2}-\dfrac{1}{a-4}$
to get $\dfrac{14}{a+2}-\dfrac{1}{a-4}=\,1$.

\item  Define $x=.239$. Evaluation gives $5x^{2}+3x-1=\,2.605\times 10^{-3}$%
. (Note that this solution is not exact . The value of the expression for
this $x$ is ``close to zero.'')

\item  Define $x=-\dfrac{3}{10}+\dfrac{1}{10}\sqrt{29}$. \QTR{MenuDialog}{%
Evaluate} followed by \QTR{MenuDialog}{Simplify} gives 
\[
5x^{2}+3x-1=\,5\left( -\frac{3}{10}+\frac{1}{10}\sqrt{29}\right) ^{2}-\frac{%
19}{10}+\frac{3}{10}\sqrt{29}=\,0 
\]
\end{itemize}
\end{example}

\strut

\subsection{\ Equations with Several Variables \label{Solve, exact (3)}}

If there is more than one variable, enter the \textsf{Variable(s) to Solve
for} in the dialog box that appears when you choose \textsf{Exact} from the 
\textsf{Solve }submenu.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$\dfrac{1}{x}+\dfrac{1}{y}=1$ (Enter $x$), Solution is : $\left\{ x=-\dfrac{y%
}{1-y}\right\} $

$\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{x}=1$ (Enter $z$), Solution is : $%
\left\{ z=-\dfrac{yx}{x+y-yx}\right\} $

$\dfrac{1}{r_{1}}+\dfrac{1}{r_{2}}=\dfrac{1}{R}$ (Enter $R$), Solution is : $%
\left\{ R=\dfrac{r_{1}r_{2}}{r_{1}+r_{2}}\right\} $
\end{quotation}

\strut

\subsection{\ Systems of Equations \label{Systems of equations}}

You can create a system of equations either by entering equations in a
one-column matrix or by entering equations in a multirow display.

\begin{quote}
$\blacktriangleright $ To create a system of equations using a matrix 
\index{Systems of equations@Systems of equations}
\end{quote}

\begin{enumerate}
\item[1.]  From the \textsf{Insert} menu, choose \textsf{Matrix}, or click 
\FRAME{itbpF}{19.5625pt}{18.8125pt}{2pt}{}{}{matrix.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;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 'Matrix.wmf';file-properties
"XNPEU";}}.

\item[2.]  Set the number of rows equal to the number of equations.

\item[3.]  Set the number of columns to $1$.

\item[4.]  Choose \textsf{OK}.

\item[5.]  Type the equations in the matrix, one equation to a row.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To create a system of equations using a display
\end{quote}

\begin{enumerate}
\item[1.]  Click \FRAME{itbpF}{0.32in}{0.3312in}{0.0701in}{}{}{display.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.32in;height 0.3312in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Display.wmf';file-properties "XNPEU";}} or
choose \textsf{Insert + Display.}

\item[2.]  Type the equations in the display, one equation to a row, adding
rows as needed with the \textsc{enter} key.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To solve a system of equations
\end{quote}

\begin{enumerate}
\item[1.]  Create a system of equations.

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

\item[3.]  From the \textsf{Solve} submenu, choose\textsf{\ Exact}.

\item[4.]  If a dialog box appears asking \textsf{Variable(s) to Solve for},
type the variable name(s) in the box, separated by commas.
\end{enumerate}

Following are two examples for systems of two equations.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
2x-y=5 \\ 
x+3y=4
\end{array}
$, Solution is : $\left\{ x=\frac{19}{7},y=\frac{3}{7}\right\} $

$
\begin{array}{c}
x^{2}-y^{2}=5 \\ 
x+y=1
\end{array}
$, Solution is : $\left\{ x=3,y=-2\right\} $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
\begin{eqnarray*}
x^{2}-3y &=&7 \\
6x+4y &=&9
\end{eqnarray*}

Solution is : $\left( 
\begin{array}{cc}
y=\frac{45-3\sqrt{301}}{8} & x=\frac{-9+\sqrt{301}}{4} \\ 
y=\frac{45+3\sqrt{301}}{8} & x=\frac{-9-\sqrt{301}}{4}
\end{array}
\right) $
\end{quotation}

When the number of unknowns is larger than the number of equations, you are
asked to specify\textsf{\ }variables in a dialog box.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
2x-y=1 \\ 
x+3z=4
\end{array}
$, \textsf{Variable(s) to Solve for}: $x,y$

\qquad \qquad \qquad \quad Solution is : $\{x=-3z+4$, $y=-6z+7\}$

\ 

$
\begin{array}{c}
2x-y=1 \\ 
x+3z=4 \\ 
w+x=-3
\end{array}
$, \textsf{Variable(s) to Solve for}: $x,y,z$

\qquad \qquad \qquad \quad Solution is : $\{x=-w-3,$ $y=-7-2w,$ $z=\frac{1}{3%
}w+\frac{7}{3}\}$
\end{quotation}

The next two examples illustrate the difference between symbolic and
numerical solutions. The symbolic solution to the following system is given
in terms of the roots of a cubic.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
2x-y^{2}=1 \\ 
\;x+3z=4 \\ 
\,\;y+z^{2}=2
\end{array}
$, Solution is : $\left\{ y=1,z=1,x=1\right\} ,$

$\qquad \left\{ x=-3\rho +4,y=-\rho ^{2}+2,z=\rho \right\} $ where $\rho $
is a root of $\left( Z^{3}+Z^{2}-3Z+3\right) $
\end{quotation}

For numerical approximations to the roots, add a decimal to the $3,$ and
from the \textsf{Polynomials} submenu, choose \textsf{Roots}$.$

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

\begin{quotation}
$Z^{3}+Z^{2}-3Z+3.0$, roots: $
\begin{array}{l}
-2.5987 \\ 
.79934-.71798i \\ 
.79934+.71798i
\end{array}
$
\end{quotation}

You find the complete numerical solution by substituting these three values
for $\rho $ in the preceding solution.

\begin{quotation}
$
\begin{array}{l}
\left\{ y=1,\;z=1,\;x=1\right\} \\ 
\left\{ x=11.796,\;y=-4.7532,\;z=-2.5987\right\} \\ 
\left\{ x=1.602+2.1539i,\;y=1.8766+1.1478i,\;z=.79934-.71798i\right\} \\ 
\left\{ x=1.602-2.1539i,\;y=1.8766-1.1478i,\;z=.79934+.71798i\right\}
\end{array}
$
\end{quotation}

For this same set of equations, if you want only real roots, add a decimal
to an entry in any one of the equations, and choose \textsf{Exact} from the 
\textsf{Solve} submenu to obtain the following result.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
2x-y^{2}=1.0 \\ 
x+3z=4 \\ 
y+z^{2}=2
\end{array}
$, Solution is : $
\begin{array}{l}
\left\{ x=1.0,\;y=1.0,\;z=1.0\right\} \\ 
\left\{ x=11.796,\;y=-4.7531,\,\;z=-2.5987\right\}
\end{array}
$
\end{quotation}

\strut

\subsection{\ Numerical Solutions \label{Numeric, solve}}

You can find numerical solutions in two ways. You can apply \textsf{Exact}
from the \textsf{Solve} submenu after entering at least one coefficient in
floating-point form---that is, with a decimal. You can apply \textsf{Numeric}
on the \textsf{Solve} 
\index{Solve@Solve!numeric@numeric}\label{Solve, numeric}submenu. \textsf{%
Solve + Numeric} 
\index{Numeric@Numeric}gives all real solutions to a polynomial equation,
but gives only one solution to a system of equations. You use it primarily
when solving (systems of) transcendental equations or when you want to
specify a search interval for the solution.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$x^{2}+7x-5.2=0$, Solution is : $\left\{ x=-7.\,6773\right\} ,\allowbreak
\left\{ x=.\,67732\right\} \medskip $

$x^{3}-3.\,8x-15.\,6=0$, Solution is : $\left\{ x=3.0\right\} ,\allowbreak
\left\{ x=-1.\,5-1.\,7176i\right\} ,\allowbreak \left\{
x=-1.\,5+1.\,7176i\right\} $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Numeric}
\end{quote}

\begin{quotation}
$x^{2}+7x-5.2=0$, Solution is : $\left\{ x=-7.\,6773\right\} ,\allowbreak
\left\{ x=.\,67732\right\} \medskip $

$x^{3}-3.\,8x-15.\,6=0$, Solution is : $\left\{ x=3.0\right\} \medskip $

$\left[ 
\begin{array}{c}
x^{2}+y^{2}=5 \\ 
x^{2}-y^{2}=1
\end{array}
\right] $, Solution is : $\left\{ x=1.\,7321,y=-1.\,4142\right\} $
\end{quotation}

\begin{quote}
$\blacktriangleright $ To find a numerical solution within a specified range
of the variable
\end{quote}

\begin{enumerate}
\item[1.]  Add a row to the bottom of the matrix, or press \textsc{enter} to
generate a new input box in a display.

\item[2.]  Write the interval of your choice, and use the membership symbol $%
\in $ to indicate that the variable lies in that interval.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Numeric}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
x^{2}+y^{2}=5 \\ 
x^{2}-y^{2}=1 \\ 
x\in \left( -2,0\right)
\end{array}
$, Solution is : $\left\{ x=-1.7321,y=-1.4142\right\} $

\strut

$
\begin{array}{c}
x^{2}+y^{2}=5 \\ 
x^{2}-y^{2}=1 \\ 
x\in \left( -2,0\right) \\ 
y\in \left( 0,2\right)
\end{array}
$, Solution is : $\left\{ y=1.4142,x=-1.7321\right\} $
\end{quotation}

\begin{quote}
$\blacktriangleright $ To find all numerical solutions to a system of
polynomial equations
\end{quote}

\begin{enumerate}
\item[1.]  Change at least one of the coefficients to floating-point form.

\item[2.]  From the \textsf{Solve} submenu, choose \textsf{Exact.}
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
x^{2}+y^{2}=5.0 \\ 
x^{2}-y^{2}=1.0
\end{array}
$, Solution is : $
\begin{array}{l}
\left\{ y=-1.4142,x=1.7321\right\} 
\vspace{4pt} \\ 
\left\{ y=-1.4142,x=-1.7321\right\} \vspace{4pt} \\ 
\left\{ y=1.4142,x=1.7321\right\} \vspace{4pt} \\ 
\left\{ y=1.4142,x=-1.7321\right\} \vspace{4pt}
\end{array}
$
\end{quotation}

These four solutions are illustrated in the following graph as the four
points of intersection of two curves. See the section \hyperref{Implicit
Plots}{}{}{DM6-4.tex#Implicit equations} for guidelines on making such
graphs.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Implicit}
\end{quote}

\begin{quotation}
$x^{2}+y^{2}=5,x^{2}-y^{2}=1$
\end{quotation}

\FRAME{dtbpF}{3.0441in}{2.0384in}{0pt}{}{}{example.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
3.0441in;height 2.0384in;depth 0pt;original-width 216.8125pt;original-height
144.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Example.wmf';file-properties "XNPEU";}}

\strut \ 

\subsection{\ Inequalities \label{Inequalities}}

With \textsl{Scientific Notebook} you can solve inequalities 
\index{Inequalities@Inequalities}directly.

\begin{quote}
$\blacktriangleright $ To solve an inequality
\end{quote}

\begin{enumerate}
\item[1.]  Leave the insertion point in the inequality.

\item[2.]  From the \textsf{Solve} submenu, choose \textsf{Exact}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Exact}
\end{quote}

\begin{quotation}
$16-7y\geq 10y-4$, Solution is : $y\leq 
\frac{20}{17}\medskip $

$x^{3}+1>x^{2}+x$, Solution is : $\left\{ -1<x,x\neq 1\right\} \medskip $

$\left| 2x+3\right| \leq 1$, Solution is : $\left\{ -2\leq x,x\leq
-1\right\} \medskip $

$\dfrac{7-2x}{x-2}\geq 0$, Solution is : $\{2\leq x,x\leq \frac{7}{2}%
\}\medskip $

$x^{2}+2x-3>0$, Solution is : $\left\{ x<-3\right\} ,\,\left\{ 1<x\right\}
\medskip $
\end{quotation}

\strut

Note the different format of this last response. When a list of inequalities
is enclosed in a single pair of braces and separated by commas, such as $%
\left\{ -1<x,\;x\neq 1\right\} $, all the inequalities must hold for $x$ to
be a solution. However, when the inequalities are enclosed in separate
braces, such as $\left\{ x<-3\right\} ,\,\left\{ 1<x\right\} $, any point
that satisfies any of the inequalities is a solution.

The solution to this last inequality can also be read from the graph of the
polynomial $y=x^2+2x-3$. In the next plot, you see that the graph passes
through the $x$-axis at $x=-3$ and $x=1$, and the solution includes every
point to the left of $-3$ or to the right of $1$.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Rectangular}
\end{quote}

\begin{quotation}
$x^{2}+2x-3$
\end{quotation}

\FRAME{dtbpF}{2.7717in}{1.8472in}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.7717in;height 1.8472in;depth
0pt;display "PICT";function \TEXUX{$x^{2}+2x-3$};linecolor "black";linestyle
1;linethickness 1;pointstyle "point";xmin "-4.46045";xmax "4.49691";xviewmin
"-4.64";xviewmax "4.68";yviewmin "-4.598";yviewmax "26.83";rangeset"X";phi
45;theta 45;plottype 4;numpoints 49;axesstyle "normal";xis
\TEXUX{x};var1name \TEXUX{$x$}}}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Factoring Polynomials}{}{}{DM3-1.tex#Polynomials, factor}

\item  \hyperref{Roots of Polynomials}{}{}{DM3-1.tex#Polynomials, roots}

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

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

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

\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{Exact solutions}{}{}{DMIndex.tex#Exact solutions}, \hyperref{%
Inequalities}{}{}{DMIndex.tex#Inequalities}, \hyperref{Numeric solutions}{}{%
}{DMIndex.tex#Numerical solutions}, \hyperref{Solve}{}{}{DMIndex.tex#Solve}, 
\hyperref{Systems of equations}{}{}{DMIndex.tex#Systems of equations}}

\end{document}
