Question

dimensions for the set of all polynomial of degree greater than or equal to 4​

Answer 1

Answer:

Well, considering it is a subset of a vector space, really there's only three things that can go wrong:

1) It isn't closed under addition, 2) It isn't closed under scalar multiplication, or 3) It's the empty set.

As it turns out, it isn't empty, and it is closed under scalar multiplication (a degree n polynomial scaled by a non-zero scalar is still degree n, and when scaled by 0, it is the zero polynomial). Therefore, really all you can do is show that it isn't closed under addition, and by far the simplest way of doing this is with a counterexample.