Let F(x) = x^81 + x^49 + x^25 + x^9 + x, G(x) = x^3 - x

Let F(x) = G(x) * Q(x) + R(x) where R(x) is the remainder when F(x) is divided by G(x), therefore R(x) at most has a degree of 2. Let R(x) = ax^2 + bx + c.

for any value t, if G(t) = 0, then F(t) = R(t), plug in t = 0, 1, -1, and solve for a, b, c.

Alternetively, when G(x) = 0, which means x^3 = x, F(x) becomes R(x), so substitute x^3 with x in F(x) until the degree is less than 3 which yields R(x).