2022 SSMO Tiebreaker Round Problems/Problem 2

Problem

Let $P(x) = x^3 - 7x^2 + 9x - 2$. If $P(x) = (x - a)^3 + b(x - a) + c$ where $a, b, c$ are real numbers, then the value of $a - b - c$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

Solution