2023 AMC 10A Problem 21

There is a unique polynomial P(x) of least degree with leading coefficient 1 satisfying all of the following:

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).

All the roots of P(x) except one are integers. If the one non-integer root can be written as \frac{m}{n}, where m and n are relatively prime positive integers, what is m + n?

\textbf{(A)} 41 \qquad \textbf{(B)} 43 \qquad \textbf{(C)} 45 \qquad \textbf{(D)} 47 \qquad \textbf{(E)} 49

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.