If a + b + c = 0, then what is the value of a4 + b4 + c4 – 2a2b2 –

1.	If a + b + c = 0, then what is the value of a4 + b4 + c4 – 2a2b2 – 2b2c2 – 2c2a2 ?
Answer» Given, a + b + c = 0. Now, a⁴ + b⁴ + c⁴ – 2a²b² – 2b²c² – 2c²a² = (a² + b² + c²)² – 4a²b² – 4b²c² – 4c²a² = [(a + b + c)² – 2ab – 2bc – 2ca]² – 4a²b² – 4b²c² – 4c²a² = [0² – 2ab – 2bc – 2ca]² – 4a²b² – 4b²c² – 4c²a² = 4a²b² + 4b²c² + 4c²a² + 8ab²c + 8abc² + 8a²bc – 4a²b² – 4b²c² – 4c²a² = 8ab²c + 8abc² + 8a²bc = 8abc (b + c + a) = 8abc. 0 = 0.

If a + b + c = 0, then what is the value of a4 + b4 + c4 – 2a2b2 – 2b2c2 – 2c2a2 ?

Answer»

Given, a + b + c = 0.

Now, a⁴ + b⁴ + c⁴ – 2a²b² – 2b²c² – 2c²a² = (a² + b² + c²)² – 4a²b² – 4b²c² – 4c²a²

= [(a + b + c)² – 2ab – 2bc – 2ca]² – 4a²b² – 4b²c² – 4c²a²

= [0² – 2ab – 2bc – 2ca]² – 4a²b² – 4b²c² – 4c²a²

= 4a²b² + 4b²c² + 4c²a² + 8ab²c + 8abc² + 8a²bc – 4a²b² – 4b²c² – 4c²a²

= 8ab²c + 8abc² + 8a²bc = 8abc (b + c + a) = 8abc. 0 = 0.