AMC 10 2023 Part A Problem 21

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

  • P(x) has 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(3 x), and
  • 4 is a root of 4 P(x).

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

Answer Choices
A. 41
B. 43
C. 45
D. 47
E. 49