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\title{Explorations 1 - MAT 517/MAE 303}
\date{}
\author{}
\begin{document}


\maketitle

Our goal is to study some methods to find the roots of polynomials.


\section{First software explorations}




\begin{exer} Write a short description about what each of the following demonstrations do.
\end{exer}
 
\url{http://demonstrations.wolfram.com/PolynomialRootsInTheComplexPlane/}

\url{http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/}

\section{The roots of polynomials of degree two}

The following demonstration shows you how to locate the complex roots of a quadratic equation.


\url{http://demonstrations.wolfram.com/LocatingTheComplexRootsOfAQuadraticEquation/}
\begin{exer} Prove that what the demonstration do is (mathematically) correct.
\end{exer}





\section{The roots of polynomials of degree three}

The goal of this section is to study how to find the roots of a polynomials of degree three, that is, polynomials of the form $$Ax^3+Bx^3+cX^2+dX+e,$$ where $A,B,C,D$ and $E$ are real numbers and $A \ne 0$. 


\begin{exer} Using the Wolfram Demostration "Sliding the Roots of Cubics" 
\url{http://demonstrations.wolfram.com/SlidingTheRootsOfCubics/}
\begin{enumerate} 
\item Find a few (approximate) values of $p$ and $q$ for which the polynomial  $x^3+px+q$ has double roots. Can you guess what are all those values?
\item Find a few (approximate) values of $p$ and $q$ for which the polynomial  $x^3+px+q$ has three real roots. Can you guess what are all those values?
\item Come back to this problem after reading Subsection~\ref{ct}, and state for which values of $p$ and $q$ the equation $x^3+px+q$ has double roots and for which values has three real roots. Does this answer coincide with what you found in the previous two questions?
\end{enumerate} 
\end{exer}

 

\begin{exer} Finding the roots of a polynomial of degree three is equivalent   to find the roots of polynomials of the form $$P(x)=x^3+a x^2 + b x + c.$$
Can you explain why?
\end{exer}

 

\begin{exer} Check (using technology) that that by substituting  $x$ by $t-\frac{a}{3}$ in the polynomial $$P(x)=x^3+a x^2 + b x + c$$   a polynomial of degree three, with no quadratic term, is obtained. 
Explain why this implies that in order to find the roots of any polynomial of degree three, it is enough to find the roots of a polynomial of the form $t^3+pt+q$. 
\end{exer}



\subsection{The Cardano-Tartaglia method}\label{ct}
\begin{enumerate}
\item In the equation
\begin{equation}\label{depressed}
x^3+P.x+Q=0,
\end{equation} 
Substitute $x$ by $u+v$, and show that the equation you obtained is satified for all pairs $(u,v)$  such that
\begin{eqnarray}\label{uv}
3\cdot u \cdot v&=-P\\
u^3+v^3&=-Q \nonumber
\end{eqnarray}
\item Solving  the system of equations
\begin{eqnarray}\label{uv1}
 u^3 \cdot v^3&=&-P/27\\
u^3+v^3&=&-Q\nonumber
\end{eqnarray}
for $u$ and $v$ gives nine pairs $(u,v)$ of solutions. If $(u_0,v_0)$ is one of these nine pairs, is  $u_0+v_0$ a solution of Equation \ref{depressed})? Justify your answer.
\item Find all the solutions of Equation~\ref{depressed}. (Hint: The system \ref{uv1} will yield a quadratic equation with discriminant $\Delta$. Find the solutions of Equation~\ref{depressed} for each of the cases, $\Delta>0$, $\Delta=0$ and $\Delta<0$.
\end{enumerate}






A detailed explanation of the method to find the roots fo cubit polynomial can be found in:


\url{http://www.shsu.edu/~kws006/Math_History/4_Calculus_files/
SolvingCubics.pdf}



\section{Solving a cubic equation with technology}

\begin{exer} Solve the equation $x^3+x+1=0$ with 
\begin{enumerate}
\item Geogebra
\item Wolfram-Alpha
\item The TI-84 in three ways: using "trace", using the "solver" in Math, using the "calc" key.
 Explain why the results you obtain in the TI-84 are different are different than those of Wolfram-Alpha.
 \item In Section \ref{ct} an algorithm to find the \emph{exact} roots of a cubic equation was given. We just saw that technology also finds roots of cubics. Discuss the pros and cons of each method (technology vs. algebraic).
\end{enumerate}
\end{exer}

\section{Netwon's method}

\begin{figure}[h]
 \begin{center}
\includegraphics[scale=0.4]{newtonMethod.pdf}%


 \caption{Basins of atraction}
   \label{fh2}
   \end{center}
 \end{figure}
 


To find the roots of polynomials numerically, Newton's method can be applied (see \url{http://archives.math.utk.edu/visual.calculus/3/newton.5/} for a brief description of the method.)


 Do you know the mathematics behind the intricate graphs below? 

\url{http://demonstrations.wolfram.com/ComplexNewtonIterationForACubicPolynomial/}
 
 
 \begin{figure}[htbp]
 \begin{center}

\includegraphics[width=8cm]{nm1.jpg}
 \caption{Basins of atraction}
   \label{fh}
   \end{center}
 \end{figure}
 
 
 
 
 
Here \url{http://demonstrations.wolfram.com/LearningNewtonsMethod/} are some examples

Here \url{http://facstaff.unca.edu/mcmcclur/mathematicaGraphics/Newton/code.html} is the code to generate these pictures in Mathematica.

\begin{exer} Use Geogebra to explore Newton's method to find roots of quadratic equations.

\end{exer}



\begin{figure}[htbp]
 \begin{center}
%\includegraphics[scale=0.25]{newtonMethod.pdf}
%\includegraphics[scale=0.25]{nm.gif}

\includegraphics[scale=0.25,angle=45]{nmMan.jpg}
   \caption{The Mandelbrot set appears in the Basins of atraction of the Newton method for find roots of cubics!}
   \label{fh1}
   \end{center}
 \end{figure}

\section{Extra credit problem}
\begin{exer} (Not with technology) Two trains part simultaneosly, in opposite directions,  from A and B. When they cross the train departing from A has traversed 20km more than the other, and arrives to B 45 minutes afterwards. The other train arrives to A one hour and 20 minutes after crossing. What is the distance between A and B?
\end{exer}


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