2010/041) If (x-k)^2 is a factor of x^3 + 3px + q

prove that 4p^3 + q^2 =0
proof:
(x-k)^2 is a factor of x^3+3px + q so x-k is factor of this as well as derivative of this

so (x-k) is a factor so k^3 + 3pk + q = 0

and (x-k) is a factor of d/dx (x^3+3px + q) or 3x^2 + 3p or x^2+p =0

so k^2 + p = 0 ... 1 or p = -k^2

sum of zeros zero so 3rd factor = -2k

product = -2k^3 = q ...2

from 1 p = - k^2 or p^3 = -k^6 or 4p = -4k^6
and q^2 = 4k^6by adding we get 4p^3+q^2 = 0