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for prove without expansion
put x = 1,y=2, z=3
L. H. S = (x-y)³ + (y-z)³ + (z-x)³
= (1-2)³ + (2-3)³+(3-1)³
=(-1)³+(-1)³+(2)³
=-1-1+8
=6
R. H. S = 3(x-y) (y-z) (z-x)
= 3(1-2)(2-3)(3-1)
= 3(-1)(-1)(2)
= 3*2
= 6
L. H. S = R. H. S
hence proved...
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