# 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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