A sequence of points is a polynomial if and only if after a certain amount of finite differences it becomes constant.

Thus, for example,

1,2,3,4,5,6,...

The finite difference is,

1,1,1,1,1,...

It is constant, thus there exists a polynomial that defines it.

Furthermore, the number of finite differences that we made were 1. In this case the polynomial is a polynomial of degree 1.