Complex Algebra/Examples/z^4 - 3z^2 + 1 = 0

Example of Complex Algebra
The roots of the equation:
 * $z^4 - 3z^2 + 1 = 0$

are:
 * $2 \cos 36 \degrees, 2 \cos 72 \degrees, 2 \cos 216 \degrees, 2 \cos 252 \degrees$

Proof
Then we have:

and similarly:

Hence:

which doesn't seem to help much.

What about binomial theorem on $\paren {z - 1}^{10}$ or $\paren {z + 1}^{10}$? See how that goes:

Or we could use the Quintuple Angle Formula for Cosine:


 * $-1 = \cos \pi = 16 \cos^5 \dfrac \pi 5 - 20 \cos^3 \dfrac \pi 5 + 5 \cos \dfrac \pi 5$

Let $z = \cos \dfrac \pi 5$, which leads to:


 * $16 z^5 - 20 z^3 + 5 z + 1 = 0$

Next thing: try to factor by $z^4 - 3z^2 + 1$ -- but the day job calls.