I'd just like to make an annoying mathematician comment on the beginning (paragraph 3): you might want to point out that the first definition you give of Bézier(n,t) is, for any n and t, nothing else than the constant function 1 (that, by the way, is an absolutely fine polynomial, unlike the statement above that if f(x) has no x then it's not a polynomial).
You can see this by noting that it's nothing else than $((1-t)\cdot t)^n = 1^n = 1$. If you insist on being fancy, you can say that the i-th term in the sum is the probability of a discrete random variable X with binomial law being equal to i, so the sum over all i must be 1 (here the binomial law has parameters (n,t)).
By the way, this last thing has the nice interpretation that each point of the Bézier curve is just the expected value of $p(X)$, where $p(0), \dots, p(n)$ are the points you are interpolating - by changing t you are just changing the parameter of the random variable.