19. [11] Let $P$ be a polynomial with $P(1)=P(2)=\cdots=P(2007)=0$ and $P(0)=2009$ !. $P(x)$ has leading coefficient 1 and degree 2008. Find the largest root of $P(x)$.
Answer: $4034072 P(0)$ is the constant term of $P(x)$, which is the product of all the roots of the polynomial, because its degree is even. So the product of all 2008 roots is 2009 ! and the product of the first 2007 is 2007 !, which means the last root is $\frac{2009!}{2007!}=2009 \cdot 2008=4034072$.