I'm sorry if the question is too vague but maybe someone can help me. I am working with a polynomial $f(x)$ with distinct roots $x_1,\ldots,x_d$. I am able to solve the problem I am currently working on if it turns out that $f''(x_i)=0$ for a certain $i\in{1,\ldots,d}$, i.e. $f(x)$ shares a root with its second derivative.

Then I'd like to know what I can say about $f(x)$ knowing that $f'(x_i)$, $f''(x_i)\neq 0$ $\forall i=1,\ldots,d$.

Any suggestion or deduction would be helpful since I fear I am overlooking something or perphaps this condition is too weak to be useful.

EDIT: I already have some algebraic relations among the roots (any cross-ratio lies in a finite set) and I'd like to obtain informations about the coefficients of $f$; of course if I could find an algebraic relation for every single root it would be over.