Write a polynomial in terms of elementary symmetric polynomials

Let p(x) = \prod_{i,j = 1}^2 (x - a_ib_j). Verify that the coefficients of p(x) can be written as polynomials in the elementary symmetric polynomials in \alpha_i and \beta_j.


Let \sigma_1 = a_1+a_2, \sigma_2 = a_1a_2, \tau_1 = b_1+b_2, and \tau_2 = b_1b_2. Evidently, p(x) = x^4 - \sigma_1\tau_1x^3 + (\sigma_1^2\tau_2 + \sigma_2\tau_1^2 - 2\sigma_2\tau_2)x^2 - \tau_1\tau_2\sigma_1\sigma_2 x + \sigma_2^2 \tau_2^2.

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