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\homework{6: Polynomials}{Due Thursday, Week 3, at the start of class.}

Solve \textbf{one} of the following \textbf{three} problems.  As always, prove your claims!

The theme for these problems: polynomials!  Sometimes, a problem will boil down to being able to manipulate algebraic expressions until they look like something useful.  These three problems involve this task!
\begin{enumerate}
\item[0.] Solve any un-signed-up-for problems from HW\#5!
\item Take any integer $n$.  Show that we can write $n$ as the sum of $5$ perfect cubes.  

(For example: $1 = 1^3 - 1^3 + 1^3 - 1^3 + 1^3$; $2 = 0^3 + 2^3 + (-2)^3 + 1^3 + 1^3$.  There can be many different ways to write any $n$ as such a sum.)
\item Suppose that $f(x)$ is a polynomial with integer coefficients; moreover, suppose that $f(a) = f(b) = f(c) = -1$, for three distinct integers $a,b,c$.  Prove that there is no integer $d$ such that $f(d) = 0$.
\item (An old USAMO problem.)  Consider the equation
\begin{align*}
x^4 - 18x^3 + kx^2 + 200x - 1984 = 0.
\end{align*}
Because this is a degree-4 polynomial, it has four roots $a,b,c,d$ (where some of these roots may be repeated; i.e. $x^2 - 2x + 1 = 0$ has the root $x=1$ repeated twice.)  Suppose that you know that $ab = -32$.  Find $k$.
\end{enumerate}





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