How to solve this algebraic problem about remainder of polynomials? Question :

My approach :
Now as I had to obtain a remainder of $\frac{(x-1)^{2017}}{x^2 - x +1}$
So, I could write this as $\frac{(x-1)^{2017}}{(x - 1)^2 + x}$
now I substitute $t = (x-1)$, so $\frac{(x-1)^{2017}}{(x - 1)^2 + x}$ could be written as $\frac{t^{2017}}{t^{2} + t + 1}$
From here we can easily obtain the remainder by plain division itself, so $P^{1}(x) = -(t+1)t^{2015} = -x(x-1)^{2015}$
Since we need $P^{2017}(2016)$, using the above obtained $P^{1}(x)$, I first obtained $P^{1}(2016) = -2016(2015)^{2015}$.
Similarly $P^{2}(x) = -P^{1}(x)(P^{1}(x) - 1)^{2015}$ by question definition, but there's no further way to simplify it, and also $P^{n}(x)$ gets bigger and bigger as $n$ increases.
So I genuinely feel this approach is wrong way, so could you help me out upon this question ?

Hint:
You mean remainder right, not quotient.
$t^3\equiv1\pmod{t^2+t+1}$
As $3|2016, t^{2016}\equiv1$
$\implies t^{2017}\equiv t$
