How do you prove whether a set of polynomials is in the vector space P³

This may not be all that helpful, but you must prove the 10 axioms of a vector space using what you are given.

1. For all x,y contained in V, then x + y is contained in V. (closure under addition)
2. For all x,y,z contained in V, (x + y) + z = x + (y + z). (associativity in addition)
3. There exists a "0" vector such that for all x contained in V, x + "0" = "0" + x = x. (additive identity)
4. For all x contained in V, then there exists a -x in V such that -x + x = x + -x = "0". (additive inverse)
5. For all x,y contained in V, then x + y = y + x. (commutativity in addition)
6. For all x contained in V and a (a scalar), then ax is contained in V. (closure under scalar multiplication)
7. For all x,y in V and a scalars, then a(x + y) = ax + ay. (scalar distribution)
8. If x is in V and a and b are scalars, then (a + b)x = ax + bx. (vector distribution)
9. If x is in V and a and b are scalars, a(bx) = (ab)x. (associativity in scalar multiplication)
10. For all x in V, 1x = x. (identity of scalar multiplication)

Generally speaking, it is easiest (from my experience) to prove in this order: 1, 6, 5, 2, 3, 4, 7, 8, 9, 10 since if something is going to break, it is likely to be closure, and then commutativity will help you with the identity and inverses proofs.

If you can give me a specific example of a set of polynomials, I can work through it for you.

Edit: I will actually use this easy set of equations. The set S = {cx^3 | c is a positive integer}.

1. Let ax^3 and bx^3 be in S.
ax^3 + bx^3 = (a+b)x^3
Since a and b are positive integers, a+b is a positive integer, so (a+b)x^3 is in S.

6. This fails. We can use a counterexample.
Let your scalar be -3 and your polynomial in the set be 2x^3.
-3 * 2x^3 = -6x^3, which is not contained in S. So, the set is not close under scalar multiplication.

Therefore, S is not a vector space. Q.E.D.