For a complex polynomial $p(x)\in \Bbb{C}[x]$ of degree $n$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $1\le k\le n-1$, show that $p(\lambda)=0$ means $\lambda\in [0,1]$.
I haven't come by any theoretical direction so far. Since it is a question given in an Introduction to Operator Theory and Hilbert Spaces, I think that approaching Complex Analysis only will be very tedious. The problem is that I can't really tell what of the the course themes it is that I should use. I was thinking Min-Max theorem, Spectral Theory and things in that area of the course syllabus. I still can't see how it can be done; the question seems irrelevant. I would appreciate it if you could give any hint or guide me a little.
I think the intended question was something like:
We may notice that $p(x)$ has $n+1$ coefficients and we have $n-1$ linear constraints on them, so the space of solutions has dimension $2$. We may also notice that $$ \int_{0}^{1} x^k\,P_n(2x-1)\,dx = 0 $$ for any $k\in\{0,1,2,\ldots,n-1\}$, with $P_n(x)$ being a Legendre polynomial.
Another solution is given by $$ p(x) = \frac{P_{n+1}(2x-1)-P_{n-1}(2x-1)}{x}, $$ so the integral constraints translate into: $$ p(x) = A\cdot P_n(2x-1)+B\cdot \frac{P_{n+1}(2x-1)-P_{n-1}(2x-1)}{x} $$ and the claim follows from Turan's inequality for Legendre polynomials, or by bounding the eigenvalues of a Sturm-Liouville problem associated with the Legendre differential equation.