jmc

Question

Let x^8 - 98x^4 + 1 = p(x) q(x),where p(x) and q(x) are monic, non-constant polynomials with integer coefficients. Find p(1) + q(1).

Accepted Answer

We can write align* x^8 - 98x^4 + 1 &= (x^8 + 2x^4 + 1) - 100x^4 &= (x^4 + 1)^2 - (10x^2)^2 &= (x^4 + 10x^2 + 1)(x^4 - 10x^2 + 1). align*Setting x = 1 in each factor, the final answer is (1 + 10 + 1) + (1 - 10 + 1) = 4.