Identities VIII

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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