Show $S$ is a subspace of $P_2$

Question

Let $P_2$ be the set of real polynomials of degree $\ge 2$ in the variable $x$. That is $P_2 = \{f(x) = a_0+a_1x+a_2x^2 | a_0, a_1, a_2 \in\mathbb{R}\}$. We have shown that $P_2$ is a vector space under polynomial addition and scalar multiplication. Consider the subset $S = \{f \in P_2 | f(1) = 0\}$ (that is for example that $x - 1$ is an element of $P_n$). (1) Show the $S$ is a subspace of $P_2$ and (2) find a set that spans $S$.

I am just learning subspaces and spans so I am not quite sure about how to go about starting the solution to this problem.

Answer 1

For (1) you need to show it is closed under summation and scalar multiplication, meaning that if you take $f,g \in P_2$ that satisfy it and an $\alpha \in \mathbb{R}$ then $f+g$ and $\alpha f$ satisfy it.
For (2) you need to find the solution to the linear system $f(1)=0$ for arbitrary $f$ and see how you can write the answer as a combination of two distinct vectors which you rewrite as polynomials, if you learnt about dimension/basis you need to find two distinct solutions to it and that will do it. Also $P_2$ is generally defined as polynomials with degree less than $2$ so you probably made a typo.