2023 AMC 10A Problems/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 $m+n$ ?

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2023 AMC 10A Problems/Problem 21