Usually, we define a polynomial as
$a_n x^n + \cdots + a_1 x + a_0$
where $x$ is called indeterminate.
Would it be better to define it as
$a_n x^n + \cdots + a_1 x + a_0 x^0$
where $x^0$ means the identity element in the structure which $x$ belongs to.
For the study of the polynomial itself, I think these two definition make no difference. But when you treat a polynomial (expression) as a polynomial function, and if we use the second definition, we can just substitute all the $x$ with some variable (eg. some square matrix), rather than define it in a "adhoc" way, i.e. if we are using the first definition, we have to explicitly define the function to add an identity element to the constant term $a_0$ to make $a_1 x$ and $a_0$ addable.
Is the second definition equivalents to the first one?
If yes, then is it true that the authors of those texts actually means def 2 when the define the polynomial using def 1, they just omit the identity element?
If not, why? Would it be nicer to eliminate the non-addable $a_1x + a_0$ with $a_1x + a_0 x^0$, (even though we really don't need to add these two terms together when we are studying the knowledge of polynomial itself)?