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A quick check of some particular situations shows that the following makes sense. I'm not sure if it is true though. So, any insight welcomed.

Let $a_1,...,a_m$ and $b_1,...,b_n$ be positive integers such that for any integer $q$ the number of $a_{i}$'s divisible by $q$ is greater than the number of $b_{i}$'s divisible by $q$. Then, $\left.\prod_i (f(x)^{a_{i}}-x) \mathrel{}\middle|\mathrel{} \prod_j (f(x)^{b_{j}} - x)\right..$

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    sorry for the post; I tried to close it but apparently I can't.2012-05-02

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How about $f(x) = x\,$, $a_1 = 6\;$ and $b_1 = 3\;$?

($x+1$ divides $x^3-x$, but not $x^6-x$, except in characteristic 2)