Question

If w is a complex cube root of unity than show that (a-b) (a-bw) (a-bw2)=a3-b3

Answer 1

Answer:

Conside LHS:

Given, w is a complex cube root of unity (that is 1).

This implies w^3 = 1.

Also, w^4 = (w^3) (w) = (1) (w) = w

That is, w^4 = w

Also, the sum of the roots, 1 + w + w^2 = 0.

Separating w^2 & Substituting w^4 = w, we get,

(1 + 5w^2 + w^4) = (1+ w^2+4w^2+w)

= (1 + w + w^2 + 4w^2)

= (0 + 4w^2)

= 4w^2

(1 + 5w^2 + w^4) = 4w^2

Similarly,

(1 + 5w + w^2) = 4w &

(5 + w + w^2) = 4

Now, L.H.S. = (4w^2 ) (4w) (4)

= 64 w^3

= (64) (1)

= 64.

Thus Proved....