Quartic with complex coefficients

[ab959]

I am trying to solve a fourth order polynomial which is in the following form

$$x^4+Ax^3+(B_1+B_2p)x^2-(C+Ap)x+D+Ep=0$$

Where $$A$$, $$B_1$$, $$B_2$$, $$C$$, $$D$$, $$E$$, are real parameters and p is a complex parameter.

I have investigated many ways of solving this equation however there does not seem to be much information regarding complex coefficients. My solution is very messy for real coefficients but it still exists and I derived it using Ferrari's method. I am not sure if I can use this method in the case where p is complex.

Any suggestions will be much appreciated.

 

[HallsofIvy]

Yes, Ferrari's method works for both real and comples coefficients.

 

[ab959]

OK great thanks I realized this soon after I posted. What I am more interested in knowing is if you classify the roots in terms of the parameters. I.e. knowing when there will be 4 real roots or complex roots etc. My intuition tells me however that if Im(p) not equal to zero then all solutions will be complex. This is possible for real coefficients however I don't know if this can be done for complex coefficients.