Octic Equations
===============
No, I'm not going to show you how to solve one. You can see some
solutions on YouTube. Eight is greater than four, so according to
the Abel-Ruffini theorem, not all of the octic equations can be
solved using radicals. If you see an octic equation on YouTube,
that's must be because the presentor has composed it.
Most of the times, the are solved by transforming them into quartic equations
that are easy to solve.
I am now solving one for the regular heptadecagon I want to draw.
The value I'm interested in is cos(360/17). The equation I solve for
that purpose is cos(8t)=cos(9t), and the value I'm looking for is
cos(t). Dropping cos(t) yields an octic equation. The value given in
nested radicals appears in the Wikipedia[1], but I want to show the
factorization. WolframAlpha[2] does not show the solution in nested radicals,
yet, and showing solutions step by step requires an upgrade that costs
money. Finding a solution somewhere else is difficult. All I find is the
equation to solve. So, I'm trying now to solve it pretending I don't
lnow the solution. With a little bit of trial-and-error, it seems that
the equation solves itself.


[1] https://en.wikipedia.org/wiki/Heptadecagon
[2] https://www.wolframalpha.com/

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