Simple Differential

Sam is a maths geek and likes to play with polynomials. He chooses a polynomial of the form $a_1x^n+a_2x^{n-2}+...+a_{n+1}$. For any value of $x$ he selects a number $m$ such that $m$ divides the polynomial completely. When it does not, Sam decided to differentiate it. If it still didn’t divide, he differentiated the polynomial again. He went on with the process $k$ times until $m$ successfully di