can someone help explain how to determine if something is a vector space or not

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can someone help explain how to determine if something is a vector space or not

Mathematics
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http://en.wikipedia.org/wiki/Vector_space#Definition
you have to check it for the axioms and see if they are true or not
how do you use the axioms to determine for instance if the set of all fifth degree polynomials with the standard operations is a vestor space

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Other answers:

Is that space closed under addition
i guess there is a subtlety here. if it means the polynomial must have degree 5, then there is no zero vector. if it means degree 5 or less, then yes
right...as stated it is not a vector space...no zero vector and no closure of addition
that is the very thing that Im confused about how do you determine that there is no zero vector and no closure of addition
the zero vector has to be just the number 0 0 is not a 5th degree polynomial
\[(x^5+3x)+(-x^5+x^2)=3x-x^2\] which is not a 5th degree polly
is there a vector (fifth degree polynomial in this case) say \[p(x)\] with \[p(x)+v(x)=v(x)\] for all fifth degree polynomials v the answer is no, because the only polynomial that would work would be the zero polynomial, which does not have degree 5
further more, \[x^5-2x+(-x^5)\] does not have degree 5, so it is not closed under addition
what zarkon said
ok that makes sense so what is the differance in saying it has fifth degree and fifth degree or less
is one a vector and one is not and why
5th degree or less is a vector space
so what is the step by step method to determing the answer to these types of problems. I mean when talking about continuous functions and things like that wouldnt get a little tricky
use the difinition I provided in my first post
i know it has to adhere to the list of axioms but can you explain it a little.

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