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Polynomial

Source: Vietnam National Olympiad 2015 Problem 5

January 11, 2015
algebrapolynomialinductionalgebra unsolved

Problem Statement

Let {f(x)}{\left\{ {f(x)} \right\}} be a sequence of polynomial, where f0(x)=2{f_0}(x) = 2, f1(x)=3x{f_1}(x) = 3x, and
fn(x)=3xfn−1(x)+(1−x−2x2)fn−2(x){f_n}(x) = 3x{f_{n - 1}}(x) + (1 - x - 2{x^2}){f_{n - 2}}(x) (n≥2)(n \ge 2)
Determine the value of nn such that fn(x){f_n}(x) is divisible by x3−x2+xx^3-x^2+x.
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