Unexpected use of polynomials in combinatorics Can someone please post some (relatively easy, say high school level) combinatorial problems which can be solved with polynomials but NOT generating functions.
Related to this post in ME.SE.

Here is an example I was talking about:
We have $2n$ different numbers $a_1,...a_n, b_1,...b_n$. A table $n\times n$ is divided on $n^2$ unit cells and in cell $(i,j)$ we write a number $a_i+b_j$. Suppose that all products of numbers written in cells in each column are the same. Prove that then all products of numbers written in cells in each row are the same.
Idea for a solution: Observe a polynomial $$P(x) = (x+a_1)...(x+a_n)-(x-b_1)...(x-b_n)$$
