Hi (Hi)

Challenging (more or less) question. I put it in this subforum as to avoid giving hints on the way to solve it.

My friend told me a method, but I'm curious to know if there are others.

So we have $\displaystyle P$ a polynomial such that its degree $\displaystyle \text{deg}(P)\leq n$, $\displaystyle n \in \mathbb{N}^*$

If $\displaystyle \int_0^1 P(t)t^k ~dt=0 \quad \forall k \in \mathbb{N} ~:~ 0\leq k\leq n$,

prove that $\displaystyle P(x)=0$.