A problem on inner product space of polynomials

Question

I have to find a polynomial $q \in P_2(\Bbb R)$ (vector space of all polynomials of degree less than or equal to 2) such that $$p\left(\frac{1}{2}\right)=\int_{0}^{1}p(x)q(x)dx$$ for every $p \in P_2(\Bbb R)$. Now, if this inner product is same for all $p \in P_2(\Bbb R)$ then it has to be zero. Am I wrong ?

Answer 1

Just check the condition for the basis vectors $\{1, t, t^2\}\subset P_2(\mathbb{R})$. If $$ q(t) = xt^2 + yt + z $$ then you will get $$ \int_0^1 t^2q(t) = \frac{1}{4} $$ which amounts to $$ \frac{x}{5} + \frac{y}{4} + \frac{z}{3} = \frac{1}{4} $$ Similarly, $$ \frac{x}{4} + \frac{y}{3} + \frac{z}{2} = \frac{1}{2} $$ and $$ \frac{x}{3} + \frac{y}{2} + z = 1 $$ Can you solve these simultaneously?