The problem reads:

"Let W1 denote the set of all polynomials f(x) in P(F) such that in the representation f(x) = (a_n)(x^n) + (a_(n-1))(x^(n-1)) +...+ (a_1)(x) + a_0, we have a_i = 0 whenever i is even. Likewise let W2 denote the set of all polynomials in g(x) in P(F) such that in the representation g(x) = (b_m)(x^m) + (b_(m-1))(x^(m-1)) +...+ (b_1)(x) + b_0, we have b_i = 0 whenever i is odd. Prove that P(F) = the direct sum of W1 and W2."

Here's my thought's, but I do not know how to prove the 2nd axiom.

So, W1 = all odd polynomials and W2 = all even polynomials.

If P(F) = direct sum of W1 and W2,

- The intersection of W1 and W2 is zero.
- W1 + W2 = V

#2. is obviously true since all odd polynomials plus all even polynomials is the set of all polynomials.

However,?does the intersection of W1 and W2 = 0

If not, then how can I form an example showing this?

If so, how can I prove it?

Please help me understand this!