Generating polynomials that are co-prime to their first and second derivatives

Question

Let $f∈Q[X]$ 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 $gcd(f_2,Df_2)=gcd(f_2,D^2f_2)=1$ ?

Answer 1

No.

Let $$f(x)=(x-3x^3)(x+1/3).$$ Then $f''(-1/3)=0$ and $$f_1(x)=x-3x^3.$$ Since $f_1$ and $f_1'$ have no common roots, $f_2=f_1$. But $gcd(f_2,D^2f_2)=x$.

Answer 2

No. Consider polynomials which are also odd functions.

Edit: The above may be useful, but is not correct. If $f_2$ were odd, the answer might be no, but it is unclear that the relations above can ever produce an odd function. End Edit.

Edit2: Thanks to Michael Renardy for producing an odd $f_2$. End Edit2.

Gerhard "Ask Me About System Design" Paseman, 2011.04.12