Question

Find the number of polynomials f(x) that satisfy all of the
following conditions:
f(x) is a monic polynomial,
f(x) has degree 1000,
f(x) has integer coefficients,
f(x) divides
f(2x^3+x)

Answer 1

brilliant qn as usual. Since the live period is over, here is a hint:
We only ned to consider irreducibles of degree≤1000 and consider how many ways there are to multiply such polynomials together to obtain a polynomial of degree 1000. Next show that |å| ≤ 1 where å is a root of f(x).
Easy to finish off.