Extremely Divisible Polynomial

Consider all degree 4 monic polynomials f(x) f(x) with complex coefficients which satisfy the condition that for all positive integers nn, f(x) f(x) divides f(xn) f(x^n) .

Over all such polynomials, what is the smallest possible positive integer value of f(3) f(-3) ?

(As a followup, can you describe all polynomials such that f(x)f(xn) f(x) \mid f(x^n) for all positive integers nn?)

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