jmc

Since $P(0)=0,$ $e=0.$ Let the other $x$-intercepts be $p,$ $q,$ $r,$ and $s,$ so $$P(x)=x(x-p)(x-q)(x-r)(x-s).$$Note that $d=pqrs.$ Since the $x$-intercepts are all distinct, $p,$ $q,$ $r,$ and $s$ are all nonzero, so $d$ must be nonzero. Thus, the answer is $\boxed{\text{(D)}}.$

Any of the other coefficients can be zero. For example, consider $$x(x+2)(x+1)(x-1)(x-2)=x^5-5x^3+4x$$or $$x(x+2)(x-1)(x-2)(x-4)=x^5-5x^4+20x^2-16x.$$