How to Factor a Sextic Equation into Two Cubics?

[elfboy]

Anyone know of a good way to factor

x^6+x^4(13^.5/2-13/2)+x^2(13-2*13^.5)+3*13^.5/2-13/2 into 2 cubics?

without having to manipulate the messy cubic equation?

 

[epenguin]

These things are intended as exercises in recognising patterns of which you have already dealt with simpler examples, woven into something more elaborate.

The rules are you are supposed to show an effort. Try and deal with different bits of it. For instance can you not do something with the part that has as factor a straight 13? The rest of it looks more difficult. I would ask you check whether you have really transcribed it exactly right. These things (I cannot imagine it is anything but an artificially constructed problem) are meant to work out to something that rhymes with sense.

 

[elfboy]

The equation looks arbitrary but its roots are plus minus sin(pi/13), sin(3pi/13), and sin(4pi/13)

 

[elfboy]

After a month of working on this problem on and off I finally factored it in the way desired

(x^3+bx^2+c*x+(-3*13^.5/2+13/2)^.5)*(x^3-bx^2+c*x-(-3*13^.5/2+13/2)^.5)

13^.5/2-13/2=-b^2+2c
13-2*13^.5=c^2-b(26-6*13^.5)^.5solve for b and c and plug into the above expression to obtain the two cubics:

(x^3+(13/2+3*13^.5/2)^.5*x^2+13^.5*x+(-3*13^.5/2+13/2)^.5)*
(x^3-(13/2+3*13^.5/2)^.5*x^2+13^.5*x-(-3*13^.5/2+13/2)^.5)=0

i didn't think this was possible to get a nice expression but I've done it

 

[CellCoree]

There is no simple or universal method for factoring a sextic equation into two cubics without manipulating the equation. However, there are certain techniques and strategies that can be used to simplify the process. One approach could be to try and identify common factors or patterns within the equation, such as grouping terms or using substitution. Another approach could be to use numerical methods such as the Rational Root Theorem or the Quadratic Formula to find potential roots of the equation, which can then be used to factor it into simpler terms. Ultimately, factoring a sextic equation can be a complex and time-consuming process, but with careful analysis and the use of appropriate techniques, it can be achieved without too much difficulty.