If x = a - b, y = b - c, z = c - a, then the numerical value of the algebraic expression x³ + y³ + z³ - 3xyz will be?

Solution (By Examveda Team)

$$\eqalign{ & x = a - b \cr & y = b - c \cr & z = c - a \cr & \therefore x + y + z \cr & = a - b + b - c + c - a \cr & = 0 \cr & \therefore {x^3} + {y^3} + {z^3} - 3xyz \cr & = \left( {x + y + z} \right)\left( {{\text{ }}{x^2} + {y^2} + {z^2} - xy - zx - yz} \right) \cr & = \left( 0 \right)\left( {{\text{ }}{x^2} + {y^2} + {z^2} - xy - zx - yz} \right) \cr & = 0 \cr} $$