Question 7
Consider two polynomials $P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ and $Q(x) = b_n x^n + b_{n-1} x^{n-1} + \dots + b_1 x + b_0$ with integer coefficients such that $a_n - b_n$ is a prime, $a_{n-1} = b_{n-1}$ and $a_n b_0 - a_{n-1} b_n \neq 0$. Suppose there exists a rational number $r$ such that $P(r) = Q(r) = 0$. Prove that $r$ is an integer.