Polynomial Degree Problem

Hello, this problem is again from Gelfand and Shen's Algebra textbook.

**Problem 171:** The highest coefficient of $\displaystyle P(x)$ is 1, and we know that $\displaystyle P(1)=0, P(2)=0, P(3)=0,\ldots,P(9)=0,P(10)=0.$ What is the minimal possible degree of $\displaystyle P(x)$? Find $\displaystyle P(11)$ for this case.

Any hints would be appreaciated.

Re: Polynomial Degree Problem

I solved it, I don't know why I didn't see it, this was very easy.

Since $\displaystyle P(1)=0, P(2)=0, P(3)=0,\ldots,P(9)=0,P(10)=0.$, this means that $\displaystyle P(x)$ has at least 10 roots so it is at least degree 10 polynomial.

$\displaystyle P(x)=Q(x-1)(x-2) \ldots (x-9)(x-10)$, $\displaystyle Q$ has to be 1 since maximum coefficient is 1, so

$\displaystyle P(x)=(11-1)(11-2) \ldots (11-9)(11-10)=10!=3628800.$