I asked this question already, but I cannot find it anymore. If it is a duplicate, I will delete it.
Is there a polynomial
$$p(x)=x^4+ax^3+bx^2+cx+d$$
such that p and all the derivates upto the third all have the maximal possible number of simple integer roots.
In other words, the polynomials can be written as follows :
$p(x)=(x-e)(x-f)(x-g)(x-h)$ with distinct integers $e,f,g,h$
$p'(x)=4(x-i)(x-j)(x-k)$ with distinct integers $i,j,k$
$p''(x)=12(x-l)(x-m)$ with distinct integers $l,m$
and finally
$p'''(x)=24(x-n)$ with integer n.
I conjecture that no such polynomial exists, but I have no idea how to prove it. I call such polynomials perfect polynomials ( Hopefully, there is no other meaning for perfect polynomials, otherwise "ideal polynomials" might work). There are perfect polynomials of degree 3.
This is open: http://www.openproblemgarden.org/op/quartic_rationally_derived_polynomials
From looking at a few of the relevant papers, this appears to be an extremely difficult problem. For example, the full classification of cubics with this property required the theory of elliptic curves.
Note that if we allow two of the roots of $p$ to coincide, examples do exist, such as $X^2(X-308)(X-360)$. According to a paper of Buchholz and Kelly, equivalence classes of such polynomials are parametrized by a certain elliptic curve.