algebra precalculus - Domain of a Polynomial function

A polynomial function is given as,

$P(x)= a_nx^n+a_{n-1}x^{n-1}+.....+a_1x+a_0$

Notice the last but one term $a_1x$. This term is a simplified form of $a_{n-(n-1)}x^{n-(n-1)}$.

Now let us take the last term of the Polynomial. The term $a_0$ is a simplified form of $a_{n-n}x^{n-n}$. Notice that $x^{n-n} = x^0 = 1$ only when $x\neq0$. This is because $0^0$ is indeterminate. It is evident that $x=0$ is clearly not in the domain of $P(x)$. But by definition, the polynomial function given above is defined for all values of $x$, $x\in(-\infty,\infty)$.

Was I right to frame the last term of the polynomial the way I did above? If no, I would like to know why.

Answer

On pondering this good question further, I think that part of the problem is that we have no name for the functions $x\mapsto x^n$. A clean way of getting around the difficulty might be the following:

Define functions $P_n$ for nonnegative integers $n$ inductively as follows: for all $x$, $P_0(x)=1$, and for $n\ge0$, define $P_{n+1}(x)=xP_n(x)$. You see that this makes $P_0$ the constant function $1$, and for $n>0$, $P_n(x)=x^n$.

Then your function can be written $\sum_{i=0}^na_iP_i\>$.