Let f(x) be a nonconstant polynomial in x with integer coefficients and suppose that for five distinct integers a1, a2, a3, a4, a5, one has f(a1)= f(a2)= f(a3)= f(a4)= f(a5)= 2.

Find all integers b such that f(b)= 9.

(In reply to Solution (spoiler) by e.g.)

That h(x) is a polynomial with integer coefficients is not exactly obvious.  This is a consequence of Gauss's Lemma: A polynomial with integer coefficients that cannot be factored into irreducible factors with integer coefficients also cannot be factored into irreducible factors with rational coefficients.

Posted by Richard on 2006-05-26 23:12:04