Vector spaces of polynomials

Algebra Level 2

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

A: All polynomials of the form p(x)=ax2+bx+cp(x)=ax^2 + bx + c with a,b,cRa,b,c\in\mathbb{R} and a+b+c=1a+b+c=1.

B: All polynomials of the form p(x)=ax2+bx+cp(x) = ax^2 + bx + c with a,b,cRa,b,c\in\mathbb{R} and a+b+c=0a+b+c=0.

C: All polynomials of the form p(x)=ax2+bx+cp(x)=ax^2 + bx + c with a,b,cRa,b,c\in\mathbb{R} and p(1)=p(2)p(1) = p(2).

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