% Math 370 hw5 Spring 2014
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\title{History of Mathematics \\
Homework 6
}
\author{Ethan Bolker}

\begin{document}

\maketitle

These exercises provide a chance to explore (in modern terms) some of
the ways the Italians and then Lagrange worked on cubic equations. 

\begin{enumerate}

\item Find all three roots of this cubic polynomial (page 231 in our text):
%
\begin{equation}\label{eq:cubic}
c(x) = x^3 - 15x - 4 .
\end{equation}

You don't need the formula for solving cubics. Since 
$x=4$ is one root, $(x-4)$ is a factor of $c(x)$. The other
factor is a quadratic polynomial so you can use the quadratic formula
to find its roots.

Sketch the graph of $c(x)$, showing the roots, the local maxima and
minima and the inflection point. (This is an easy calculus 1
exercise.) You can use pencil and paper and scan the result,
or, if you're ambitious, do it in \LaTeX{} with tikz/pgfplots.


\item Show that 
%
\begin{equation}\label{eq:weird}
\sqrt[3]{ 2 + \sqrt{-121}} +
\sqrt[3]{ 2 - \sqrt{-121}} = 4
\end{equation}

Hint: Calculate $(2+i)^3$.

\item Every complex number has three cube roots -- when you know one
  you get the other two by multiplying it by the other two cube roots
  of $1$. So, for example, the three cube roots of $2 + 11i$ are
\begin{equation*}
2+i, \frac{-1 + \sqrt{3}}{2}(2+i) \text{ and }
\frac{-1 - \sqrt{3}}{2}(2+i).
\end{equation*}

Using all three cube roots in each term 
on the left in Equation~\ref{eq:weird}
  would seem to give nine answers. Show that there are really
  only three and that they are  the three roots of
  Equation~\ref{eq:cubic} you found in the first problem.% 

\footnote{I haven't checked this. I'm pretty sure it's right.}

\item Do Exercise 5.4 (page 219) in our text.

\end{enumerate}

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Here is the \LaTeX{} source for this document. You can cut it from the
pdf and use it to start your answers. I used the \verb!\jobname! macro
for the source file name, so you can call your file by any name you like.

\verbatiminput{\jobname}

\end{document}
