Let P(z) = z^8 + (4*sqrt(3) + 6)*z^4 - (4*sqrt(3) + 7). What is the minimum perimeter among all the 8-sided polygons in the complex plane whose vertices are precisely the zeros of P(z)?
(A) 4*sqrt(3) + 4 (B) 8*sqrt(2) (C) 3*sqrt(2) + 3*sqrt(6)
(D) 4*sqrt(2) + 4*sqrt(3) (E) 4*sqrt(3) + 6
Solution:This is a nice problem because it requires that you work with its data creatively.
If we connect the dots A,B,C,D,E,F,G,H,A in this order, we will get the concave 4-fold symmetrical 8-gon. Due to symmetry, all sides of this 8-gon are congruent. Let's prove that this 8-gon has the smallest perimeter among all other possible 8-gons that connect these eight dots. Each side of this 8-gon has length = sqrt(2). Other possible sides can have lengths of DF = sqrt(2) or EG = sqrt(3) + 1 > sqrt(2).
Q.E.D.
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