Question

x^3+y^3+z^3-3xyz=1/2(x+y+z)​

Answer 1

We know that,

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

⇒ x3 + y3 + z3 - 3xyz = 1/2×(x + y + z) 2(x2 + y2 + z2 - xy - yz - xz)

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

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

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

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