Finding the basis of a subset of polynomials 
Let $W$ be a subspace of the polynomials with maximum degree of $3$ and $p(1) = p(2) = 0$. Find the basis and the dimension of the subspace. The field is the real numbers.

My attempt:
My first thought is that it is the usual basis of $\mathbb{P}_3 = \{1, x, x^2, x^3\}$.  However, on second thought, is it a trick question? We will never have $p(1) = p(2)$, will we? So would the basis be the empty set and the dimension be $0$?

You need a basis that incorporates $p(1)=p(2)=0$ while still having polynomials of degree three. So you won't have a constant term, and you need some polynomials like $(x-1)(x-2)$ in your basis set. Then because you need degree three polynomials, maybe include $x(x-1)(x-2)$. So our set could be 
$$S=\{(x-1)(x-2),x(x-1)(x-2)\}$$
can anybody confirm this might be right? I'm not that confident, I'm just throwing out ideas for you.
edit: another answer is basically the same as this, so I'll assume that this is right and say, there you go!
