I know that the taylor approx. of a polynomial centered at 0, if n gets big enough, is just the polynomial itself.
But why do people always say "centered at 0"... wouldn't we also get the polynomial back if we approximated around any other x? Why is 0 so special?
On
It's partly about how you write it down, with powers of $(x-a)$ instead of powers of $x$, as noted in the comments. But it's more than that.
If the polynomial is of degree $n$, and you center somewhere else, and get a Taylor approximation of degree less than $n$, it will not be the same as if centered at $0$.
For example, if $f(x)=x^2$, and we center at $1$, then the degree $1$ Taylor approximation is $$1+2(x-1)$$ which expands to $2x-1$, not the same as the degree 1 Taylor approximation centered at $0$ (which is the zero polynomial).
You will still get back the polynomial if you take the Taylor formula of order $n$ centered around any point $a$ , as long as $n \ge \deg$.
Example:
$$t^n = \sum_{k=0}^n \frac{n(n-1) \ldots (n-k+1) a^{n-k}}{k!} (t-a)^k$$