jmc

Question

Let x^8 + 3x^4 - 4 = p_1(x) p_2(x) p_k(x),where each non-constant polynomial p_i(x) is monic with integer coefficients, and cannot be factored further over the integers. Compute p_1(1) + p_2(1) + + p_k(1).

Accepted Answer

First, we can factor x^8 + 3x^4 - 4 as (x^4 - 1)(x^4 + 4). Then x^4 - 1 = (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x - 1)(x + 1),and by Sophie Germain, x^4 + 4 = x^4 + 4x^2 + 4 - 4x^2 = (x^2 + 2)^2 - (2x)^2 = (x^2 + 2x + 2)(x^2 - 2x + 2).Thus, the full factorization is x^8 + 3x^4 - 4 = (x^2 + 1)(x - 1)(x + 1)(x^2 + 2x + 2)(x^2 - 2x + 2).Evaluating each factor at x = 1, we get 2 + 0 + 2 + 5 + 1 = 10.