TITLE: Neat problems using roots of polynomials
QUESTION [8 upvotes]: On a recent test in a course I'm TAing, students were asked to prove that sin(x) is not a rational function by using the fact that polynomials only have finitely many zeroes. During my tutorial, I'd like to show them some other interesting problem whose (reasonably short!) solution uses the fact that polynomials only have finitely many zeroes. What's your favourite?

REPLY [3 votes]: A nice application that deserves to be better known: if a polynomial $\:f(x)\:$ over $\:\mathbb Z/n\:$ has more roots than its degree, then we can quickly compute a nontrivial factor of $\:n\:$ via a simple $\rm\:gcd\:$ calculation.
The quadratic case of this is at the heart of many integer factorization algorithms, which attempt to factor $\:n\:$ by searching for a nontrivial square root in $\rm\: \mathbb Z/n\:,\:$ e.g. a square root of $1$ that is not $\:\pm 1$.