The polynomial $f(n) = 3n^3 - 39n^2 + 360n + 20$ I saw has one real root and it's $$n_1=\frac{13}{3}-\frac{2}{3} \sqrt{191} \sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac{4913}{191 \sqrt{191}}\right)\right)\approx -0.055223771734378147887$$ First, what formula, or how can that be? Second, how is it so, numerically $$f(n) = 3n^3 - 39n^2 + 360n + 20$$ $$f(n)=3(n-n_1)(n^2-13.055223771734378148 n+120.72095869751148663)$$?
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1Why don't you take a look at this https://en.wikipedia.org/wiki/Cubic_equation ? Concretely here https://en.wikipedia.org/wiki/Cubic_equation#Trigonometric_and_hyperbolic_solutions – Tito Eliatron Jan 13 '20 at 21:04
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1cf. this question – J. W. Tanner Jan 13 '20 at 21:05
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@randomvalue Finding the roots of a cubic in the general case is not easy. – almagest Jan 14 '20 at 09:54
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@almagest which is why I asked on a math profession type website? – dalton atwood Jan 15 '20 at 15:04