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

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Answer 2

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

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

taking 2 as common

=1/2×2(x+y+z)(x^2+y^2+z^2-xy-yz-zx)

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

=(x^3+xy^2+xz^2-x^2y-xyz-x^2z)+

(x^2y+y^3+z^2y-y^2x-y^2z-xyz)+

(x^2z+y^2z+z^3-xyz-z^2y-z^2x)

=x^3+y^3+z^3-3xyz

Hence, LHS=RHS