TITLE: How to prove polynomials with degree $n$ does not form a vector space?
QUESTION [6 upvotes]: This is one of my linear algebra problems:

Prove that polynomials of degree $n$ does not (The professor made these words bold intentionally) form a vector space.

From what I read, the set of polynomials of degree $n$ should be a vector space, because:

There is an "One" and a "Zero" in this set;
We can find inverse for addition and multiplication from this set;
It follows all the axioms of addition.
It follows all the axioms of scalar multiplication.

Then can someone give me some hints to prove it does not form a vector space?

REPLY [10 votes]: Polynomials of degree $n$ does not form a vector space because they don't form a set closed under addition. 
For instance:
$$X^n-X^n=0$$
which is not of degree $n$.
So, don't get confused with the set of polynomials of degree less or equal then $n$, which form a vector space of dimension $n+1$. We often work with this space.