Showing that a polynomial is zero given that a sum containing its coefficients sum to zero

Question

I've been trying to solve this exam question on an exam in real analysis. Thus, only such methods may be used. The problem is as follows.

Assume that $c_0,c_1,\dots,c_n$ are real numbers so that $$\sum_{k=0}^{n}\frac{c_k}{k+1}=0.$$ Prove that the polynomial $$p(x)=\sum_{k=0}^{n}c_kx^k$$ has a zero in $[0,1]$.

I have played around with shorter samples of the sum to see how it works to give us coefficients with the desired properties, but I don't really have any clue as of how to prove it.

Answer 1

Hint: Define$$q(x)=\sum_{k=0}^n\frac{c_kx^{k+1}}{k+1}.$$What is $q(0)$? And $q(1)$? How are $p(x)$ and $q(x)$ related?