Question 57
The polynomial $P(x, y)$ of two variables with integer coefficients satisfies the following two conditions: 1) for every integer $a$ there exists exactly one integer $y$ such that $P(a, y) = 0$; and 2) for every integer $b$ there exists exactly one integer $x$ such that $P(x, b) = 0$.
a) Prove that if the degree of $P(x, y)$ is two then this polynomial is divisible by $x - y + C$ or $x + y + C$ where $C$ is integer.
b) Does there exist such polynomial $P(x, y)$ which is not divisible by any polynomial of the form $x - y + C$ and $x + y + C$ where $C$ is integer? (Mikhail Karpuk)
a) Prove that if the degree of $P(x, y)$ is two then this polynomial is divisible by $x - y + C$ or $x + y + C$ where $C$ is integer.
b) Does there exist such polynomial $P(x, y)$ which is not divisible by any polynomial of the form $x - y + C$ and $x + y + C$ where $C$ is integer? (Mikhail Karpuk)