f(x) = x^n = P(x)*x*(x-1)^2 + ax^2 + bx + c
として
f'(x) = n*x^(n-1) = P(x)*(3x^2 - 4x + 1) + P'(x)*x*(x-1)^2 + 2ax + b

f(0) = 0 = c
f(1) = 0 = a + b + c
f'(1) = n = 2a + b

a = n
b = -n
c = 0