per wiki
A polynomial equation, also called algebraic equation, is an equation of the form
$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0}$
and
a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms
so, as long as the n is a finite number, such as 2 billion, 3 trillion,
$P = x^n$
is a polynomial equation, do I understand this right?
A polynomial equation is of the form
$$a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0} \color{red}{ \boldsymbol{=0}}$$
while what you wrote is just the general form of a polynomial.
And indeed, if you check the definition and plug in $n=100000000000$ with $a_n=1$ and $a_k=0$ for all $k \neq n$ then we see that $x^{100000000000}$ is a polynomial. (But by itself not a polynomial equation.)