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

R.H.S.

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

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

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

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

= `2 xx 1/2 xx (x + y + z)(x^2 + y^2 + z^2 − xy − yz − zx)`

= (x + y + z)(x² + y² + z² − xy − yz − zx)

= x³ + y³ + z³ − 3xyz

= L.H.S.

Hence, it is verified.