Identity VII is
- a ^{ 3 } + b ^{ 3 } + c ^{ 3 } − 3abc = (a + b + c) (a ^{ 2 } + b ^{ 2 } + c ^{ 2 } − ab − bc − ac)
Lets take an example
a ^{ 3 } + b ^{ 3 } + c ^{ 3 } − 3abc = (a + b + c) (a ^{ 2 } + b ^{ 2 } + c ^{ 2 } − ab − bc − ac)
2
^{
3
}
+ 3
^{
3
}
+ 4
^{
3
}
– 3(2)(3)(4)
= (2 + 3 + 4) (2
^{
2
}
+ 3
^{
2
}
+ 4
^{
2
}
– (2)(3) – (3)(4) – (4)(2))
8 + 27 + 64 – 72 = (9) (4 + 9 + 16 – 6 – 12 – 8)
99 – 72 = (9) (29 – 26)
27 = (9) (3)
27 = 27
If a + b + c = 0
Then,
a ^{ 3 } + b ^{ 3 } + c ^{ 3 } − 3abc = (a + b + c) (a ^{ 2 } + b ^{ 2 } + c ^{ 2 } − ab − bc − ac)
a ^{ 3 } + b3 + c ^{ 3 } − 3abc = (0) (a ^{ 2 } + b ^{ 2 } + c ^{ 2 } − ab − bc − ac)
a ^{ 3 } + b ^{ 3 } + c ^{ 3 } − 3abc = 0
a ^{ 3 } + b ^{ 3 } + c ^{ 3 } = 3abc
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