# Vector spaces of polynomials

Algebra Level pending

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 $$p(1) = p(2)$$.

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