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Problem 837

AIME late

Algebra Difficulty 5.5 Prove it

Show that for any natural number $n>1$ the polynomial $P_{n}=x^{4 n+3}+x^{4 n+1}+x^{4 n-2}+x^{8}$ is divisible by $x^{2}+1$

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

We apply the previous proposition to the number i. The polynomial $x^{2}+1$ is annihilated by i. $x^{2}+1$ has no real roots, so it has no rational roots either. Thus $x^{2}+1$ is irreducible. Since $\mu_{i} \mid x^{2}+1, \mu_{i}=x^{2}+1$ . Since $i^{4}=$ we have $P_{n}(i)=$ $\mathfrak{i}^{3}+\mathfrak{i}^{1}+\mathfrak{i}^{2}+\mathfrak{i}^{4}=-\mathfrak{i}+\mathfrak{i}-1+1=0$ . Since $P_{n}$ is annihilated by $i$ , it is divisible by Ö $m \mathfrak{u}_{i}=x^{2}+1$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.