Is $x^k$ in a polynomial subspace $\mathcal{P}_n$ where $n>k$? So I'm pretty confused about polynomial subspaces. Say I have a polynomial $ p(x) = x^3 + 2x^2 + 3x + 4 $, whose highest degree is 3, would this be in $\mathcal{P}_4$? I have been reading other answers on this site saying that since the largest degree in $ x^3 + 2x^2 + 3x + 4 $ is 3, it is not in $\mathcal{P}_4$, however, my teacher has told me that any polynomial of a degree smaller than the subspace in question is in fact in that subspace because the coefficient in front of the largest degree could be 0, for example
$ p(x) = x^3 $ is in $\mathcal{P}_5$ because $ p(x) $ could be $ x^3 + 0x^4 + 0x^5 $.
From answers on here, for example, proving that $\mathcal{P}_n$ is not a subspace:
Let $ p(x) = x^3 + x^2 $ and $ q(x) = -x^3 + x^2$, then $ p(x) + q(x) = 2x^2$ which is not of degree 3, therefore it is not $\mathcal{P}_n$ as it is not closed under addition. But couldn't you express $ p(x) + q(x) $ as $ 2x^2 + 0x^3 $?
Could I get some clarification on this? Thanks!
 A: $\mathcal P_n$ is the space of polynomials with degree $\leq n$. Otherwise it would not be a vector space. The reason is that you can take a polynomial $f=X^n+a_{n-1}X^{n-1}+\dots+a_0$ and subtract the polynomial $X^n$ to get
$$a_{n-1}X^{n-1}+\dots+a_0=f-X^n,$$
which has to be in $\mathcal P_n$ for it to be a vector space. You can construct polynomials of lower degree in a similar way to find that a vector space containing all polynomials of degree $n$ must also contain all polynomials of lower degree than $n$. Also keep in mind that while you can express a polynomial as $2X^2$ or as $0X^3+2X^2$, taking that to mean that the degree depends on how we write it down really defeats the purpose of defining a degree. The degree of a polynomial is the highest exponent of $X$ with non-zero coefficient.
Of course, you could define $\mathcal P_n$ differently if you're not interested in it being a vector space, but I've only seen this notation for vector spaces of polynomials.
A: $\mathcal{P}_{n}$ is vector space of dimension $(n+1)$ over any field $\mathbb{F}$ with the basis $\{1,x,x^2,\cdot\cdot\cdot,x^n\}$.
That means, any polynomial $a_{n}x^{n}+a_{n-1}x^{n-1}+\cdot\cdot\cdot+a_0$ over the field $\mathbb{F}$ are the elements of the vector space $\mathcal{P}_{n}$.
So, any polynomial of degree $\le n $ over $\mathbb{F}$  belong into $\mathcal{P}_{n}$.
