Linear Algebra - Vector Spaces

Let P be the set of all polynomials. Show that P, with the usual addition and scalar multiplication of functions, forms a vector space.

I'm just no good at proofs. I know we are supposed to go through and prove the Vector Space Axioms and the C1 and C2 closure properties. I just don't think I'm doing it successfully. I'm just not understanding this. For instance, the closure properties don't seem very different from a couple of the axioms.

Axioms:
A1. x + y = y + x for any x and y in V.
A2. (x + y) + z = x + (y + z) for any x,y,z in V.
A3. There exists an element 0 in V such that x + 0 = x for each x in the set V.
A4. For each x in the set V, there exists an element -x in V such that x + (-x) = 0.
A5. alpha(x + y) = alpha*x + alpha*y for each scalar alpha and any x and y in V.
A6. (alpha + beta)x = alpha*x + beta*x for any scalars alpha and beta and any x that belongs to the set V.
A7. (alpha*beta)x = alpha(beta*x) for any scalars alpha and beta and any x that belongs to the set V.
A8. 1*x = x for all x in V.

Closure properties:
C1: If x is in V and alpha is a scalar, then alpha*x is in V.
C2. If x,y is in V, then x + y is in V.

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...of generality.
<br>Closure properties:
<br>C1: alpha*p1=(alpha*Dn)x^n+...+(alpha*D1)x+(alpha*D0) is still a polynomial. So alpha*p1 is in P
<br>C2. p1+p2=Dnx^n+...+(Dm+Bm)x^m+...+(D1+B1)x+(D0+B0) is still a polynomial. So p1+p2 is in ...