Algebra Level 4

Let $$f(x)$$ denote a $$10^\text{th}$$ degree polynomial such that $$f(x) = 10$$ has roots $$x=1,2,3,\ldots,10$$ and $$f(x)$$ has a constant term of $$10!$$. Find $$f(11)$$.

Bonus: Generalize this.

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