Question

Verify that `x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

 

Answer 1

It is known that,

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

`therefore x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

                                     `= 1/2(x+y+z)[2x^2+2y^2+2z^2-2xy-2yz-2zx]`

                                     `= 1/2(x+y+z)[(x^2+y^2-2xy)+(y^2+z^2-2yz)+(x^2+z^2-2zx)]`

                                     `= 1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`