The Convex Hull Of A Set Of Roots

Algebra Level 5

Let \(F\) denote the set of all monic polynomials \(f(x) \) with complex coefficients, such that the distance between any two distinct complex roots of \( [f(x)]^2-f(x)\) is at least \(1\).

For each polynomial \(f\) in \(F\), let \(S_f\) be the convex hull of the roots of \([f(x)]^2-f(x).\) To 2 decimal places, what is the product of the largest and the smallest possible (positive) area of \(S_f\)?

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