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 ith 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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