Supose we have a polynomial function (what follows is the definition of the book i'm reading) $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $f(x)=\sum_{i=0}^n a_ix^i$ $(0 \leq i \leq n \in \mathbb{N}$ $\wedge a_i \in \mathbb{C})$. My (high school) book defines "numerical evaluation of a number $a$ in $f$ is the image of $a$ by the function $f$" $($i.e $f(a))$ and then proceeds to give some examples without talking about $f(0)$. My question is, we define $f(0)=a_0 \cdot 0^0 =a_0$, even if $0^0$ is undefined? If so, then $0^0=1$? But if this last one question is true (at least for polynomials) then why in some other subjects $0^0$ is undefined?
The question could be reformulated to: could we avoid making $0^0=1$ (because it is, at least supposedly, undefined), but still mantain $f(0)=a_0$?
I would love to know an answer to this, even if it is very difficult or very easy to understand.
When defining a polynomial the convention is that the "$x^0$" in the constant term is $1$ before you do any evaluation. You need that convention so that formal polynomial multiplication works as expected.
So the $a_0x^0$ term is just $a_0$.