prove that (x+y)³ +(y+z)³ + (z+x)³ -3(x+y) (y+z) (z+x)=2(x³+y³+z²-3x y z)
Rishabh01:
Friend, this question is from Sample Question Paper-2 from R.D. Sharma. I have tried it today only but i am not getting the solution . If you got to know then please tell me also'
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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
=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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