Question

if w (omega) is cube root of unity then (x-y)(xw-y)(xw^2-y)=.........
pls solve this with explanation​

Answer 1

Answer:

x^3 – y^3

Step-by-step explanation:

(x – y) (xw – y) (xw^2 – y)

= (x^2w – xy – xyw + y^2) (xw^2 – y)

= (x^2w – xy (1 + w) + y^2) (xw^2 – y)

= (x^2w – xy (– w^2) + y^2) (xw^2 – y)           (∵ 1+w + w^2 = 0 ⇒ 1 + w = – w^2)

= (x^2w + xyw^2 + y^2) (xw^2 – y)

= x^3w^3 –x^2yw + x^2yw^4 – xy^2w^2 + xy^2w^2 – y^3

= x^3w^3 –x^2yw + x^2yw^3, w – y^3

= x^3 – x^2yw + x^2yw – y^3                             (∵ w^3 = 1)

= x^3 – y^3