Prove that for \(n> 0\) the polynomial \(x^{2n + 1} - (2n + 1)x^{n + 1} + (2n + 1)x^n - 1\) is divisible by \((x - 1)^3\).
The water level in a pool is given by a quadratic function \(h(t) = at^2 + bt + c\), where \(t\) is measured in hours.
At the moment when the pool is completely drained, say at time \(t_0\), we have \(h(t_0) = 0\) and \(h'(t_0) = 0\).
It is also known that after the first hour, the water level has dropped to exactly half of its original value: \(h(1) = \tfrac{1}{2} h(0)\).
How many hours does it take for the pool to drain completely?