The correct options are
A (2a−b−c)(2b−a−c)(2c−a−b)
C 27abc ,if a+b+c=0
Let a+bω+cω2=x and a+bω2+cω=y
∴(a+bω+cω2)3+(a+bω2+cω)3=x3+y3=(x+y)(x+ωy)(x+ω2y)
Now,
x+y=(a+bω+cω2)+(a+bω2+cω)
=2a+b(ω+ω2)+c(ω+ω2)
=(2a−b−c)⋯(1)
x+ωy=(a+bω+cω2)+ω(a+bω2+cω)
=(1+ω)a+(1+ω)b+2ω2c
=ω2(2c−a−b)⋯(2)
x+ω2y=(a+bω+cω2)+ω2(a+bω2+cω)
=ω(2b−a−c)⋯(3)
Hence from (1),(2) and (3)
x3+y3=(2a−b−c)(2c−a−b)(2b−a−c)⋯(4)
If
a+b+c=0⇒b+c=−a,c+a=−b,a+b=−c
Putting these values in the equation (4), we get
x3+y3=(3a)(3b)(3c)=27 abc