Clever Yet Simple Polynomial Subsitution

Algebra Level 4

Let \( f(x) \) be a monic polynomial of degree \( 2009 \) such that \( f(m) = 1 \) for \( m = 1, 2, 3,\cdots,2009 \). Find \( 2010(f(2010)-2009!) \).

Note that monic means that the leading coefficient of the polynomial is 1.

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