if w is cube root of unity , then prove that
( x - y ) ( x w - y ) (x w 2 - y ) = x3 - y3
(x – y) (xw – y) (xw2 – y)
= (x2w – xy – xyw + y2) (xw2 – y)
= (x2w – xy (1 + w) + y2) (xw2 – y)
= (x2w – xy (– w2) + y2) (xw2 – y) (∵ 1+w + w2 = 0 ⇒ 1 + w = – w2)
= (x2w + xyw2 + y2) (xw2 – y)
= x3w3 –x2yw + x2yw4 – xy2w2 + xy2w2 – y3
= x3w3 –x2yw + x2yw3, w – y3
= x3 – x2yw + x2yw – y3 (∵ w3 = 1)
= x3 – y3