Question 169
For each positive integer $n$, let
$$
P_n(x) = x^n + x^{n+1} + \dots + x + 1.
$$
Show that for each positive integer $a$, there exists a positive integer $n$ and two polynomials $R(x)$ and $Q(x)$ with integer coefficients such that
$$
P_n(x) = (1 + a x + x^2 R(x)) Q(x).
$$
$$
P_n(x) = x^n + x^{n+1} + \dots + x + 1.
$$
Show that for each positive integer $a$, there exists a positive integer $n$ and two polynomials $R(x)$ and $Q(x)$ with integer coefficients such that
$$
P_n(x) = (1 + a x + x^2 R(x)) Q(x).
$$