On the factorization of $x^{d_r}+x^{d_{r-1}}+\cdots +x^{d_2}+x^{d_1}+p$ and a conjecture of C. Nicol

For an arbitrary prime $p$ and arbitrary positive integers $r$ and $d_1,\dots ,d_r$ with $d_r>\cdots >d_1$, we show that $x^{d_r}+x^{d_{r-1}}+\cdots +x^{d_2}+x^{d_1}+p$ removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials ${\mathrm{\Phi }}_n(x)+{\mathrm{\Phi }}_m(x)$ is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where $n$ and $m$ are integers exceeding $1$. We also establish similar results for ${\mathrm{\Phi }}_n(x){\mathrm{\Phi }}_m(x)+1$.