The Convex Hull Of A Set Of Roots

Algebra Level 5

Let FF denote the set of all monic polynomials f(x)f(x) with complex coefficients, such that the distance between any two distinct complex roots of [f(x)]2f(x) [f(x)]^2-f(x) is at least 11.

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

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