Question 9
Given the polynomial
$$
f(x) = x^{n} + a_{1} x^{n-1} + a_{2} x^{n-2} + \cdots + a_{n-1} x + a_{n}
$$
with integral coefficients $a_{1}, a_{2}, \ldots, a_{n}$, and given also that there exist four distinct integers $a, b, c$ and $d$ such that
$$
f(a) = f(b) = f(c) = f(d) = 5
$$
show that there is no integer $k$ such that $f(k) = 8$.
$$
f(x) = x^{n} + a_{1} x^{n-1} + a_{2} x^{n-2} + \cdots + a_{n-1} x + a_{n}
$$
with integral coefficients $a_{1}, a_{2}, \ldots, a_{n}$, and given also that there exist four distinct integers $a, b, c$ and $d$ such that
$$
f(a) = f(b) = f(c) = f(d) = 5
$$
show that there is no integer $k$ such that $f(k) = 8$.