Polynomials with factors of the form $(x^q-a)$ with roots modulo every integer

Given an odd prime $q$, a natural number $l$ and non-zero $q$-free integers $a_1,a_2,\dots ,a_l$, none of which are equal to $1$ or $-1$, we give necessary and sufficient conditions for the polynomial ${\prod }_{j=1}^l(x^q-a_j)$ to have roots modulo every positive integer. Consequently: (i) if $l\le q$ and none of $a_1,a_2,\dots ,a_l$ is a perfect $q^{th}$ power, then the polynomial ${\prod }_{j=1}^l(x^q-a_j)$ fails to have roots modulo some positive integer; $(ii)$ For every $l\in \mathbb{N}$, and every $(c_j{)}_{j=1}^l\in ({\mathbb{F}}_q\setminus \{0\}{)}^l$, the polynomial ${\prod }_{j=1}^l(x^q-a_j)$ has roots modulo every positive integer if and only if ${\prod }_{j=1}^l(x^q-{\text{rad}}_q(a_j^{c_j})))$ has roots modulo every positive integer. Here ${\text{rad}}_q(a_j)$ denotes the $q$-free part of the integer $a_j$.