Let p(x) be a polynomial with integer coefficients for
which p(0) = p(1) = 1. Let a0 be any nonzero integer and define
an+1 = p(an) for n ³ 0. Prove that, for distinct
nonnegative integers i and j, the greatest common divisor of
ai and aj is equal to 1.
(a) Let n be a positive integer and let f(x) be a
with real coefficients and real roots that differ by at least n.
Prove that the polynomial
g(x) = f(x) + f(x + 1) + f(x + 2) + ¼+ f(x + n)
also has real roots.
(b) Suppose that the hypothesis in (a) is replaced by
positing that f(x) is any polynomial for which the difference
between any pair of roots is at least n. Does the result