Question 11
Let $n \ge 2$ be a positive integer. For any integer $a$, let $Q_a(x)$ denote the polynomial $x^n + ax$. Let $p$ be a prime number and $S_a$ be the set
$$
S_a = \{b \mid 0 \le b \le p-1, \exists c \in \mathbb{Z}, Q_a(c) \equiv b \pmod{p}\}.
$$
Show that the expression $\frac{1}{p-1} \sum_{a=1}^{p-1} |S_a|$ is an integer.
$$
S_a = \{b \mid 0 \le b \le p-1, \exists c \in \mathbb{Z}, Q_a(c) \equiv b \pmod{p}\}.
$$
Show that the expression $\frac{1}{p-1} \sum_{a=1}^{p-1} |S_a|$ is an integer.