jmc

Question

Let f(x) be a polynomial such that f(x^2 + 1) = x^4 + 4x^2.Find f(x^2 - 1).

Accepted Answer

Let y = x^2 + 1. Then x^2 = y - 1, and x^4 = y^2 - 2y + 1, so f(y) = (y^2 - 2y + 1) + 4(y - 1) = y^2 + 2y - 3.Hence, f(x^2 - 1) = (x^2 - 1)^2 + 2(x^2 - 1) - 3 = x^4 - 4.