examday

/TMUA

Let $P(x)$ be the unique polynomial of minimal degree with the following properties:

• $P(x)$ has a leading coefficient 1,
• 1 is a root of $P(x)-1$,
• 2 is a root of $P(x-2)$,
• 3 is a root of $P(3x)$, and
• 4 is a root of $4P(x)$.

The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\dfrac mn$, where $m$ and $n$ are relatively prime integers. What is $m+n$?