## Construct a polynomial having a given set of roots

Use elementary symmetric functions to construct a polynomial having as its roots the numbers 1, -1, 2, and 3.

Let $\sigma_i$ denote the $i$th elementary symmetric polynomial in the variables $x_1,x_2,x_3,x_4$. (For instance, $\sigma_1 = x_1+x_2+x_3+x_4$.) Recall that $\prod_{i=1}^4 (z-x_i) = z^4-\sigma_1z^3+\sigma_2z^2-\sigma_3z+\sigma_4$.

Say $x_1 = 1$, $x_2 = -1$, $x_3 = 2$, and $x_4 = 3$. A quick calculation shows that $p(x) = x^4 - 5x^3 + 5x^2 + 5x - 6$.

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