Question 5
Define polynomials by
$$f_0(x)=2,\quad f_1(x)=3x,\quad f_n(x)=3xf_{n-1}(x)+(1-x-2x^2)f_{n-2}(x)\quad(n\ge2).$$
Find all positive integers $n$ for which $f_n(x)$ is divisible by $x^3-x^2+x$.
$$f_0(x)=2,\quad f_1(x)=3x,\quad f_n(x)=3xf_{n-1}(x)+(1-x-2x^2)f_{n-2}(x)\quad(n\ge2).$$
Find all positive integers $n$ for which $f_n(x)$ is divisible by $x^3-x^2+x$.