Question 177
Given two non-constant polynomials $P(x), Q(x)$ with real coefficients. For a real number $a$, we define
$$
P_{a} = \{ z \in \mathbb{C} : P(z) = a \} ; \quad Q_{a} = \{ z \in \mathbb{C} : Q(z) = a \} .
$$
Denote by $K$ the set of real numbers $a$ such that $P_{a} = Q_{a}$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.
$$
P_{a} = \{ z \in \mathbb{C} : P(z) = a \} ; \quad Q_{a} = \{ z \in \mathbb{C} : Q(z) = a \} .
$$
Denote by $K$ the set of real numbers $a$ such that $P_{a} = Q_{a}$. Suppose that the set $K$ contains at least two elements, prove that $P(x) = Q(x)$.