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\header{Math 3010}{HW 8}{YOUR NAME HERE}
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\begin{document}

\begin{questions}

  \question There are three elementary symmetric polynomials in three unknowns:
  \begin{align*}
    \sigma_1 &= x_1 + x_2 + x_3 \\
    \sigma_2 &= x_1x_2 + x_1x_3 + x_2x_3 \\
    \sigma_3 &= x_1x_2x_3
  \end{align*}
  The following calculation shows how to express the symmetric
  polynomial $x_1^2 + x_2^2 + x_3^2$ in terms of the elementary
  symmetric polynomials above.
  \begin{align*}
    \sigma_1^2 &= (x_1 + x_2 + x_3)^2 \\
    &= (x_1 + x_2 + x_3)(x_1 + x_2 + x_3) \\
    &= x_1^2 + x_2^2 + x_3^2 + 2(x_1x_2 + x_1x_3 + x_2x_3) \\
    &= x_1^2 + x_2^2 + x_3^2 + 2\sigma_2
  \end{align*}
  Thus $x_1^2 + x_2^2 + x_3^2 = \sigma_1^2 - 2\sigma_2$.

  Express the symmetric polynomial $x_1^3 + x_2^3 + x_3^3$ in terms of
  the elementary symmetric polynomials $\sigma_1, \sigma_2$ and
  $\sigma_3$. Hint: start with $\sigma_1^3$, and consider various
  products of $\sigma$s.

  \begin{solution}
    Your answer here...
  \end{solution}

\end{questions}
\end{document}
