Finding the roots of a 6-degree polynomial

[ZeroKool007]

Hello,
I am having trouble finding the roots of a 6-degree polynomial equation in expression form. I hope anyone can help reduce it to a form where the expression for each root can be determine.



Qx amd mx are defined values, Fx is the variable. Fx^6 and Fx^2 are the degree polynomial...Fx^5, Fx^4, Fx^3, Fx^1 are multiply by 0.

Example: mx=6.3, Qx=0.4 ... this would give the results of
A = 0.4890380
B = 0.788825-0.86129i
C = 0.788825+0.86129i

Thanks,
ZeroKool

 

[tkhunny]

Hello,
I am having trouble finding the roots of a 6-degree polynomial equation in expression form. I hope anyone can help reduce it to a form where the expression for each root can be determine.

View attachment 10745

Qx amd mx are defined values, Fx is the variable. Fx^6 and Fx^2 are the degree polynomial...Fx^5, Fx^4, Fx^3, Fx^1 are multiply by 0.

Example: mx=6.3, Qx=0.4 ... this would give the results of
A = 0.4890380
B = 0.788825-0.86129i
C = 0.788825+0.86129i

Thanks,
ZeroKool

Why do you think it can be done, generally?

 

[Dr.Peterson]

Hello,
I am having trouble finding the roots of a 6-degree polynomial equation in expression form. I hope anyone can help reduce it to a form where the expression for each root can be determine.

View attachment 10745

Qx amd mx are defined values, Fx is the variable. Fx^6 and Fx^2 are the degree polynomial...Fx^5, Fx^4, Fx^3, Fx^1 are multiply by 0.

Example: mx=6.3, Qx=0.4 ... this would give the results of
A = 0.4890380
B = 0.788825-0.86129i
C = 0.788825+0.86129i

Thanks,
ZeroKool

This is a very poorly written equation, so that at first I thought it was not even a polynomial. It appears that you intended all the x's to be subscripts or something; they should all be removed and F treated as the variable. Replacing the complicated coefficients with M and N, the equation is just

F^6 - MF^2 - N = 0

and in fact is a third-degree polynomial in F^2. Letting u = F^2, it becomes

u^3 - Mu - N = 0,

which is called a depressed cubic.

There is a very complicated formula for one solution in terms of M and N, and the others can be found by dividing by the corresponding factor to reduce it to a quadratic. You can find this method by searching for "depressed cubic equation". Here are a few of the pages I found, presenting these ideas in different ways. (You can skip the parts about depressing the equation.)

https://brilliant.org/wiki/cardano-method/
http://www.sosmath.com/algebra/factor/fac11/fac11.html
http://www.ms.uky.edu/~corso/teaching/math330/Cardano.pdf

I'll let you carry all that out.