InterviewSolution
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InterviewSolution
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x ^ { 3 } + y ^ { 3 } + z ^ { 3 } - 3 x y z = \frac { 1 } { 2 } ( x + y + z ) [ ( x - y ) ^ { 2 } + ( y - z ) ^ { 2 } + ( z - x ) ^ { 2 } ] |
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Answer» R.H.S.=1/2(x+y+z)((x-y)^2+(y-z)^2+(z-x)^2) =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) =(x+y+z)(x^2+y^2+z^2-xy-yz-zx) =x^3+xy^2+xz^×-x^2y-xyz-x^2z+x^2y+y^3+yz^×-xy^2-y^2z-xyz+zx^2+y^2z+z^3-xyz-yz^2-z^2x =x^3+y^3+z^3-3xyz =L.H.S.Hence Proved |
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