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Let a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial f(x) =...

Let a1, a2, ..., an be distinct n (≥ 2) integers. Consider the polynomial
f(x) = (x−a1)(x−a2)···(x−an)−1 in Q[x]

(1) Prove that if then f(x) = g(x)h(x)
for some g(x), h(x) ∈ Z[x],
g(ai) + h(ai) = 0 for all i = 1, 2, ..., n

(2) Prove that f(x) is irreducible over Q
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