Gigantic Polynomial Game Harder Version

Algebra Level 3

\[ \large f(x) = \Box x^{2016} + \Box x^{2015} + \ldots + \Box x^2 + \Box x + \Box \]

The above shows a polynomial of degree 2016, but with all of its coefficients left blank. Two players, Euler and Fermat, take turns to fill in the gaps with any real number. Euler makes the first turn.

Note that players don't have to fill the gaps in order, e.g Euler fills the \(100^\text{th}\) first, then Fermat fills the \(3^\text{rd}\).

When the gaps are all filled, if the equation \(f(x) = 0\) has at least 1 real solution other than 0, Euler wins; otherwise, Fermat wins.

Assume that Euler and Fermat both play optimally. What's the probability that Fermat wins?

Pick the closest answer.


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