If we have polynomial with integer coefficients, we can bound the largest and smallest root by an expression that is maximum of abs. value of coefficients to some power. There should be similar bounds for the smallest imaginary value a root can have. If your polynomial have real numbers for coefficients having a formula doesn't help - we have the same problem of being not able to tell if number is equal zero. For example in the quadratic formula, we can have discriminant equal minus epsilon. If epsilon was zero there is no imaginary root but if it isn't we have an imaginary root.
