A Rational-Derived Cubic

The above cubic curve (red) has the remarkable property that its first derivative (green), its second derivative (blue) as well as the cubic itself all have integer roots. While such cubics have been completely characterised the general problem for arbitrary degree polynomials is still an open problem. See When Newton met Diophantus for more details. In fact, simply finding a quartic with four distinct integer roots for which all three derivatives also have all integer roots would be a major step forward.