2013 AMC 12B Problems/Problem 24

Problem

Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$ . What is the minimum perimeter among all the $8$ -sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$ ?

$\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6} \qquad \textbf{(D)}\ 4\sqrt{2} + 4\sqrt{3} \qquad \textbf{(E)}\ 4\sqrt{3} + 6$

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2013 AMC 12B Problems/Problem 24

Problem