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2 $begingroup$ PT $(x-a_{1})(x-a_{2})..(x-a_{n})+1$ can not be factored into two smaller polynomial $P(x)$ and $Q(x)$ with integer coefficients, where $a_{i}$ 's are all different integers. This problem can be solved by considering the root of equation $P(x)Q(x)-1=0$ This problem comes from Terry Tao's book Solving mathematical problem (page 47), in which he gives a hint as if P(x) and Q(x) are such factors then what can you say about $P(x)-Q(x)$ How does one solve this problem this hint? Edit: This appears not to be true as pointed out by Darji and Eric. For interested readers, The actual problem can be found here, page 47 Excercise 3.7 polynomials contest-math irreducible-polynomials