Linear algebra, polynomial problem Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(
Let $S$ be the following subset of the vector space $P_3$ of all real polynomials $p$ of degree at most 3:
$$S=\{p\in P_3\mid p(1)=0, p^\prime (1)=0\}$$
where $p^\prime$ is the derivative of $p$.
a) Determine whether $S$ is a subspace of $P_3$
b) determine whether the polynomial $q(x)= x-2x^2 +x^3$ is an element of S
Attempt:
I know that for the first part I need to proof that it's none empty, closed under addition and multiplication right?
will this give me full mark for the part a if I answer like this:
$(af+bg)(1)=af(1)+bg(1)=0+0=0$ and 
$(af+bg)′(1)=af′(1)+bg′(1)=0+0=0$ 
so therefore it's a subspace of $P_3$?
b) i got no idea...
Thank you very much!

Alright, for part a), you have to look at the definition of a subspace. That is, it must contain the additive identity (the zero-polynomial), which is trivial. It must be closed under multiplication by scalar, which it is, since its degree will not change, regardless of what real number $C$ you multiply a polynomial $P$ with. Lastly, we need it to be closed under addition, which it is, by similar argument as for multiplication by scalar. 
For part b), you simply need to check. $q(1) = 1 - 2 + 1 = 0$, $q'(x) = 1 - 4x + 3x^2, q'(1) = 1 - 4 + 3 = 0$. So yes, it is in $S$.
