Roots of a Random Polynomial

Calculus Level 4

\[a_{2016}x^{2016} + a_{2015}x^{2015} + \cdots + a_1x + a_0= 0\]

The coefficients of the equation above are independently sampled from a uniform distribution over \([0,1],\) and the 2016 roots are plotted in the complex plane. Which of the following graphs is most likely to depict these roots?

Bonus: Explain this phenomenon, extend it to polynomials of arbitrary degree, and/or discuss what happens if the distribution of coefficients is changed.

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