# A problem of factoring a polynomial with a hint

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

• What if $n=2$, $a_1 = 1$ and $a_2 = -1$? – darij grinberg Feb 3 at 4:01

Notice that $$P(a_i)Q(a_i)=1$$ for all $$i$$. But since $$P$$ and $$Q$$ have integer coefficients and the $$a_i$$ are integers, this means $$P(a_i)=Q(a_i)=1$$ or $$P(a_i)=Q(a_i)=-1$$ for each $$i$$. In particular, $$a_i$$ is a root of $$P-Q$$ for each $$i$$, so $$P-Q$$ is either $$0$$ or has degree at least $$n$$.
But, since $$\deg(PQ)=n$$, the only way $$P-Q$$ can have degree at least $$n$$ is if one of $$P$$ and $$Q$$ has degree $$n$$ and the other is a constant. Presumably this possibility is meant to be excluded by the requirement that $$P$$ and $$Q$$ are "smaller" polynomials.
The only remaining possibility is that $$P-Q=0$$, so $$P=Q$$. This actually is possible--for instance, as darij grinberg commented, you could have $$n=2$$, $$a_1=1$$, and $$a_2=-1$$ and so the polynomial is $$(x-1)(x+1)+1=x^2$$ and so we can have $$P(x)=Q(x)=x$$. Another example (with $$n=4$$) is $$(x-1)(x-2)(x-3)(x-4)+1=(x^2-5x+5)^2.$$ So, the problem statement is not quite correct without some additional assumption to rule out this case.
• Great example. So it appears if $(x-a_{1})(x-a_{2})..(x-a_{n})+1$ reducible then it is also a square of a polynomial? – piyush_sao Feb 3 at 7:33
• More precisely, it is the square of an irreducible polynomial (since otherwise it would have a nontrivial factorization with $P\neq Q$). – Eric Wofsey Feb 3 at 7:52