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Question:

Verify that x³ + y³ + z³ - 3xyz = ½(x + y + z)[(x - y)² + (y - z)² + (z - x)²]

Solution:

We have,L.H.S.=x³+y³+z³-3xyz=(x+y+z)(x²+y²+z²-xy-yz-zx)[by polynomial identity]=½(x+y+z)(2x²+2y²+2z²-2xy-2yz-2zx)=½(x+y+z)(x²-2xy+y²+y²-2yz+z²+z²-2zx+x²)=½[(x+y+z)(x-y)²+(y-z)²+(z-x)²]Hence proved.