Question 41
Let $a_{1}, \ldots, a_{n}$ be distinct integers. Prove that the polynomial
$$
\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)-1
$$
cannot be written as the product of two nonconstant polynomials with integer coefficients (i.e. it is irreducible over the integers).
$$
\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)-1
$$
cannot be written as the product of two nonconstant polynomials with integer coefficients (i.e. it is irreducible over the integers).