Prove that

$\displaystyle{ \left(\int_0^1 x^k P(x) dx = 0,\ (1\le k\le n = \deg(P))\right)}$

$\displaystyle{ \implies \left(\int_0^1 (P(x))^2dx = (n+1)\bigg(\int_0^1 P(x)dx \bigg )^2 \right ) }$

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