Question

If 1, ω and ω² are the cube roots of unity, prove that

(x + y +z)(x+ωy+ω²z)(x+ω²y+ωz) = x³ +y³ +z³–3xyz.​

Answer 1

LHS-

(x+y+z)(x+wy+w²z)(x+w²y+wz)

(x+y+z)(x²+w²xy+wxz+wxy+w³y²+w²yz+w²zx+w⁴yz+w³z²)

w³=1, 1+w+w²=0=>w+w²=-1

(x+y+z){x²+y²+z²+xy(w²+w)+yz(w²+w³.w)+zx(w+w²)}

(x+y+z)(x²+y²+z²-xy-yz-zx)

x³+y³+z³-3xyz

(using identity a³+b³+c³-3abc=(a+b+c)(a²+b²+c²-ab-bc-ca)

hope its help u