If x=p+q, y=p +q ^2 and z=p ^2+q , where is a complex cube root of unity, then x y z equals to

Step-by-step solution

Given, x=p+q, y+p +q ^2 and z=p ^2+q aligned x y z= & (p+q) (p +q ^2 ) (p ^2+q ) = & (p+q) (p^2 ^3+p q ^2+p q ^4+q^2 ^3 ) = & (p+q) (p^2+p q ^2+p q +q^2 ) & ( ^3=1 ) = & (p+q) (p^2+p q ( + ^2 )+q^2 ) = & (p+q) (p^2-p q+q^2 )=p^3+q^3 & ( 1+ + ^2=-1 ) aligned