Vector spaces of polynomials

Algebra Level 2

Which of the following sets of polynomials are vector spaces over $\mathbb{R}$ with the usual polynomial addition and scalar multiplication?

A: All polynomials of the form $p(x)=ax^2 + bx + c$ with $a,b,c\in\mathbb{R}$ and $a+b+c=1$ .

B: All polynomials of the form $p(x) = ax^2 + bx + c$ with $a,b,c\in\mathbb{R}$ and $a+b+c=0$ .

C: All polynomials of the form $p(x)=ax^2 + bx + c$ with $a,b,c\in\mathbb{R}$ and $p(1) = p(2)$ .