If 1, , ^2 are the cube roots of unity then the value of (x+y)^2+ (x +y ^2 )^2+ (x ^2+y )^2 is

Step-by-step solution

Given that 1, w , w ^2 are the cube roots of unity. 1+ w + w ^2=0 and w ^3=1 . Now, (x+y)^2 (x w+y w^2 )+ (x w^2+y w )^2 aligned & =x^2+y^2+2 x y+x^2 w^2+y^2 w^4+2 x y w^3 & +x^2 w^4+y^2 w^2+2 x y w^3 & =x^2+y^2+2 x y+x^2 w^2+y^2 w+2 x y+x^2 w+y^2 w^2+2 x y & =x^2+y^2+6 x y+x^2 (w^2+w )+y^2 (w+w^2 ) & =x^2+y^2+6 x y-x^2-y^2=6 x y aligned