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Identity (x+y)(x-y)

Prove that. x...

Question

Prove that.
(x+y)³+(y+z)³+(z+x)³ —3(x+y)(y+z)(z+x)=2(x³+y³+z³—3xyz)

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Solution

LHS=[X³+Y³+3XY(X+Y)]+[Y³+Z³+3YZ(Y+Z)]+[Z³+X³+3ZX(Z+X)]-3XYZ-3X²Y-3XZ²-3X²Z-3Y²Z-3Y²X-3YZ²-3XYZ
=X³+Y³+3X²Y+3XY²+Y³+Z³+3Y²Z+3YZ²+Z³+X³+3Z²X+3X²Z-3XYZ-3X²Y-3XZ²-3X²Z-3Y²Z-3Y²X-3YZ²-3XYZ
=2X³+2Y³+2Z³-6XYZ
=2(X³+Y³+Z³-3XYZ)
LHS=RHS
HENCE PROVED

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