Let . Show that is irreducible in unless .

Note that if is reducible in , then it must have either a linear or a quadratic factor.

Suppose has a linear factor in ; say , with these coefficients in . Comparing coefficients, we have the following system of equations: , , , , . Since and are integers, the last equation yields the two cases and . Note also that is a root of , so that . We can see that in the first case, , and in the second, . Indeed, this yields the factorizations and . For no other does have a linear factor.

Now suppose has a quadratic factor in ; say . Again comparing coefficients, we have , , , , and . Substituting the first two equations into the third, we see that is a rational root of .

If , then is a rational root of ; by the rational root test, . Neither of these is a root of , a contradiction.

If , then is a rational root of ; by the rational root theorem, . Then and , and we have . Indeed, this corresponds to the factorization . For no other does have a quadratic factor.