how can one comment on the nature of roots of an equation of nth degree

for eg. (x-3)⁹+(x²-3²)⁹+............(x⁹-3⁹)⁹ =0 has got 8 imaginary and 1 real root..the only sol that i can think of is by diff. it it once and noting the fact that 1st differential has got no solutions so the eqn has 8 imaginary and one real sol.

 

another thing that needs to be discussed is ----let there be an eqn of 4th degree. and its second differential (quadritic eqn) has no real roots then how many real and imginary roots does the orignal eqn(of 4th degree) has.

 

CAN THIS RESULT BE GENERALISED?????????