Question

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

Answer 1

take x = 1, y = 2 ,and z = 3
now,

LHS = x³ + y³ + z³ -3xyz

= ( 1 )³ + ( 2 )³ + (3 )³ -3× 1 × 2 × 3

= 1 + 8 + 27 - 18

=18

RHS = 1/2(x + y + z){ (x -y)² + (y - z)² + (z - x)²}

= 1/2( 1 + 2 + 3){ (1 - 2)² + (2 -3)² + (3 -1)²}

= 1/2( 6) { 1 + 1 + 4 }

= 3 × 6 = 18

LHS = RHS
hence, verified //