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Question
Verify that `x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`
Solution
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.
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