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Part of 1992 Nordic

Problems(1)

f(x) = (x -a_1)(x - a_2) ... (x -a_n) - 1,not divisible

Source: Nordic Mathematical Contest 1992 #2

10/5/2017
Let n>1n > 1 be an integer and let a1,a2,...,ana_1, a_2,... , a_n be nn different integers. Show that the polynomial f(x)=(x−a1)(x−a2)⋅...⋅(x−an)−1f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1 is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than nn and such that the highest power of xx has coefficient 11.
Integer Polynomialpolynomial divisionalgebra