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Let $f \in \mathbb Q [X]$ and not constant or of the form $(x-a)^n$. Suppose:

$f_1 := \frac{f}{gcd(f,D^2f)}$ and;

$f_2 := \frac{f_1}{gcd(f_1,Df_1)}$,

where $Df$ stands for the formal derivative.

Is it true that $gcd(f_2,Df_2)=gcd(f_2,D^2f_2)=1$

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    If I'm not mistaken, you get $\gcd(f_2,Df_2) = 1$, but the difficulties come with the second derivative. It is easy to characterize polynomials that are relatively prime to their derivative, but relatively prime to their second derivative is not so immediate.2011-04-12

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No, it is not true. Try $f(x) = x^2 + x^4 $

$f_1(x) = x^2 + x^4 $

$f_2(x) = x + x^3 $

${\rm gcd}(f_2, D^2 f_2) = x$