# Polynomial Degree Problem

• November 13th 2012, 10:38 AM
Polynomial Degree Problem
Hello, this problem is again from Gelfand and Shen's Algebra textbook.

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

Any hints would be appreaciated.
• November 13th 2012, 01:07 PM
Since $P(1)=0, P(2)=0, P(3)=0,\ldots,P(9)=0,P(10)=0.$, this means that $P(x)$ has at least 10 roots so it is at least degree 10 polynomial.
$P(x)=Q(x-1)(x-2) \ldots (x-9)(x-10)$, $Q$ has to be 1 since maximum coefficient is 1, so
$P(x)=(11-1)(11-2) \ldots (11-9)(11-10)=10!=3628800.$