Question

Let f(x) be a polynomial with integer coefficients. Suppose that there are distinct integer a1, a2, a3, a4, sich that f(a1) = f(a2) = f(a3) = f(a4) = 3, show that there does not exist any integer b with f(b) = 14

Answer 1

f is of the form

f(x)=q(x)(x-a1)(x-a2)(x-a3)(x-a4)+3

Answer 2

if f(b)=14 then

q(b)(b-a1)(b-a2)(b-a3)(b-a4)+3=14

q(b)(b-a1)(b-a2)(b-a3)(b-a4)=11

but 11 is prime, thus ...

Answer 3

Thank you so much!!!