Question

If 1, w, w2 are the cube roots of unity, then prove the following :
(x-y)(xw-y)(xw^2-y)=(x^3-y^3)

Answer 1

Answer:

Proved

Explanation:

• $w^3=1$

• $w^2+w+1=0$

Expand the LHS $(x-y)(xw-y)(xw^2-y)$.

$(x-y)(xw-y)(xw^2-y)\\=(x-y)(x^2y^3-xyw-xyw^2+y^2)\\=(x-y)(x^2-xy(w^2+w)+y^2)\\=(x-y)(x^2+xy+y^2)\\=x^3-y^3$

∴$(x-y)(xw-y)(xw^2-y)=x^3-y^3$

Answer 2

w^3=1

Therefore, w^2+w+1=0

Now you can get the answer