2023 AMC 10A Problem 21

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 $\frac{m}{n}$ , where $m$ and $n$ are relatively prime integers. What is $m+n$ ?