I assume $p(x)=a_nx^n+...+a_0$ where $a_i\in F$ and $x$ is variable. To verify it as vector space, I think we need to check axioms.
i.e. $\forall x, y,z\in V,\ (x+y)+z=x+(y+z)$ and $\forall x, y\in V,\ c\in F \ ,\ c(x+y)=cx+cy$. But since it is a polynomial this time, how are we supposed to check it? How do we disintegrate the equation?
Before we verify that $P_n(F)$ is a vector space, it is essential to know what $P_n(F)$ is: For a given $n$, $P_n(F)$ is defined as the set of all polynomials with degree of at most $n$, i.e.:
$P_n(F):=\{a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\mid a_0,a_1,\ldots,a_n\in F\}$
Please note that in this context $x$ is not a variable; $x$ is only a formal symbol!
Now I will give you some properties of this vector space so that you can get some intuition:
The elements of $P_n(F)$ are linear combinations of $\{1,x,x^2,\ldots,x^n\}$; therefore $\{1,x,x^2,\ldots,x^n\}$ is a basis of $P_n(F)$. So the vector space $P_n(F)$ has dimension $n+1$.
Knowing a basis, you can use coordinates: For example, you could write
$[a_0,a_1,\ldots,a_n]$
instead of
$a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$.
Given two polynomials
$a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$
and
$b_nx^n+b_{n-1}x^{n-1}+\ldots+b_1x+b_0$,
their sum is
$(a_n+b_n)x^n+(a_{n-1}+b_{n-1})x^{n-1}+\ldots+(a_1+b_1)x+(a_0+b_0)$.
Or, by using coordinates, you can say that the sum of
$[a_0,a_1,\ldots,a_n]$
and
$[b_0,b_1,\ldots,b_n]$
is
$[a_0+b_0,a_1+b_1,\ldots,a_n+b_n]$
I hope that you now see why polynomials have the structure of a vector space.