VMO 2015 Problem 5

Let ${\left\{ {f(x)} \right\}}$ be a sequence of polynomial, where ${f_0}(x) = 2$ , ${f_1}(x) = 3x$ , and ${f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x)$ $(n \ge 2)$ Determine the value of $n$ such that ${f_n}(x)$ is divisible by $x^3-x^2+x$ .