I'm interested in solving the following equation in $[0,1]$ $$px^n - x + (1-p)=0$$
where $p \in [0,1]$ and $n \in \mathbb N -\{1,2 \} $ both constant.
To start with, we can easily see that $x=1$ is a solution and I also know there is another solution in $[0,1]$ for every $n \in \mathbb N -\{1,2 \}$ and for all $p> p_c (n)$ . I've tried using the Horner method with $(x-1)$ we get :
$$(x-1)(px^{n-1} + px^{n-2}+..+ px^2 + px+p-1)=0 $$
So we get $(px^{n-1} + px^{n-2}+..+ px^2 + px+p-1)=0 $ . Then we can do : $$ x^{n-1} + x^{n-2}+..+ x=\frac{1-p}{p}$$
or $$\frac{x^n-1}{x-1}=\frac{1-p}{p}+1 $$
But this doesn't seem to give anything useful. Any ideas on how we can solve this?