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Algebra Level 5

Let \(g(x) \) denote a quintic \(\left(5^\text{th} \text{ degree}\right)\) polynomial such that \(g(n) = 2^n\) for \(n = 1,2,3,4,5\) and \(g(x) \) has a constant term of \(5!\). Find \(g(6) \).

Bonus: Generalize this.

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