Let $M\in \mathbb{N}$ and $c\in(0,1)$. I want to solve in $x$ \begin{align} c(1-x)^M +Mx-1 = 0, \\ 0 \leq x \leq 1. \end{align}
Even if it is a root-finding problem of a polynomial of degree $M$, I suspect (and hope) than it may exist an explicit solution to this problem. I cannot ensure the existance nor uniqueness of the solution of the problem but I strongly think that we have both.
Any idea how to tackle the problem ?