How Can I Fully Analyze a Sixth Order Algebraic Equation?

[traianus]

I have the following equation (a,b,c real coefficients):

x^6 + a*x^4 + b*x^2 + c = 0

How do I do a complete discussion? For example I would like to know if there are six coincident roots, 2 triple roots, distint roots, a pair of complex solution and 4 real coincident solutions and so on.

 

[river_rat]

Well for starters that is just a monic cubic in disguise :)

 

[traianus]

I have noticed it. It can be transformed in the form of a cubic equation (it is sufficient to replace x^2 with y). But I would like if somebody does the complete study in detail.

 

[mathwonk]

well that means do a complete study of cubics and then take square roots of all the roots. i am sure you can do this. or read my algebra book on my website.