If the graph of the cubic polynomial has rotational symmetry then a maximum point must be rotated to become a minimum and vice versa so the center of rotation will be the midpoint of the line joining the maximum and minimum points. If there are no maximum and minimum points then consider the point of inflexion.

The graph of a function has rotational symmetry about the origin if and only if $f(-x) = -f(x)$. You can do this question without calculus if you can find the transformation of coordinates that removes the quadratic term from the polynomial equation.