Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.0 AIME, harder Find the answer

The roots of $z^{6}+z^{4}+z^{2}+1=0$ are the vertices of a convex polygon in the complex plane. Find the sum of the squares of the side lengths of the polygon.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Factoring the polynomial as $(z^{4}+1)(z^{2}+1)=0$ , we find that the 6 roots are $e^{ \pm i \pi / 4}$ , $e^{ \pm i \pi / 2}, e^{ \pm i 3 \pi / 4}$ . The calculation then follows from the Law of Cosines or the distance formula.

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