How can I determine whether polynomials under a predetermined condition satisfy the subspace of *all* of the space P of all polynomials? (My approach so far in the comments).

Question

anonymous · Answer

I want to say that if the polynomial function f(x) follows a patter of coefficients and variables in increasing powers, that I should be able to create a g(x) and an h(x) and add them together to say whether they're closed on addition. Is that the right direction?

anonymous · Answer

Here's what I'm thinking:

kirbykirby · Answer

Yes you're going in the right direction.

kirbykirby · Answer

The condition though is saying you're looking at all polynomials of degree 3 (that's why the coefficient of $a_3$ can't be 0.

kirbykirby · Answer

Now you need to also look if it's closed under scalar multiplication

kirbykirby · Answer

The last thing to check is if the 0 vector  (i.e the zero polynomial) is in the subspace.

anonymous · Answer

That helps, thanks. :)