Question 5
Suppose quadratic polynomials $P$, $Q$, $R$ with integer coefficients satisfy the following properties:
* $P(1) = P(2) = Q(3) = 0$.
* $P(x)^2 + Q(x)^2 = R(x)^2$ holds for any real number $x$.
* There exists no integer greater than or equal to $2$, which divides all of the numbers appearing as a coefficient of any of $P$, $Q$, $R$.
* The coefficients of the quadratic terms of $P$ and $Q$ are not $0$, and the coefficient of the quadratic term of $R$ is positive.
Find all possible forms of $R(x)$ satisfying the conditions above.
* $P(1) = P(2) = Q(3) = 0$.
* $P(x)^2 + Q(x)^2 = R(x)^2$ holds for any real number $x$.
* There exists no integer greater than or equal to $2$, which divides all of the numbers appearing as a coefficient of any of $P$, $Q$, $R$.
* The coefficients of the quadratic terms of $P$ and $Q$ are not $0$, and the coefficient of the quadratic term of $R$ is positive.
Find all possible forms of $R(x)$ satisfying the conditions above.