Question

proove that :- x3+y3+z3-3xyz= 1/2 (x+y+z) [ (x-y)^2 + (y-z)^2 + (z-x)^2]

Answer 1

Hi ,

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

=1/2(x+y+z)[x²+y²-2xy+y²+z²-2yz+z²+x²-2zx]

=1/2(x+y+z)[2x²+2y²+2z²-2xy-2yz-2zx]

=1/2(x+y+z){2[x²+y²+z²-xy-yz-zx]}

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

= x³ + y³ + z³ - 3xyz

= LHS

I hope this helps you.

: )