It should be written as pn (x) = a1x n+a2x (n-1)+...+an+1.

Obviously, when the derivative of Pn(x) is of order n, P 'n (n) (x) = a1n (n-1) (n-2) ...1.

When calculating the n+ 1 derivative, because p' n (n) (x) = a1n (n-1) (n-2) ...1is a constant, the derivative is zero.

Both places marked with red lines can be explained in this way.