Common roots of polynomial and its derivative Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th  derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots then, Is it true that $f$ has to be a zero polynomial? Here $m<k$ is a positive integer. This statement is true for $m=1$ but is it true for larger $m$ also?

Assume $a,b,c \in \mathbb{R}$ solve
$$2(a^3+b^3+c^3)-3(a^2b+ab^2+b^2c+bc^2+a^2c+ac^2)+12abc=0,$$
e.g. $(a,b,c)=(-1,1,3)$. Then 
$$
\begin{eqnarray}
f(x)&:=&(x-a)(x-b)(x-c)(3x^2-2(a+b+c)x+3(ab+bc+ca)-2(a^2+b^2+c^2))\\
&=&3x^5-5(a+b+c)x^4+10(ab+bc+ca)x^3\\
&&+(2(a^3+b^3+c^3)-3(a^2b+ab^2+\dots)-18abc)x^2+\dots
\end{eqnarray}
$$
satisfies
$$f^{(2)}(x)=60(x-a)(x-b)(x-c)$$
and is therefore a counterexample for $m=2$ and $k=3$.
