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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