Question

If a+b+c=0
Then a4+b4+c4/a2b2+c2(a2+b2) =?????

Answer 1

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Answer 2

Answer:

Squaring on both sides of the relation,

= (a² + b² + c²)² = [-2(bc + ca + ab)²]

= 4 {b²c² + c²a² + a²b² + 2} {bc. ca + ca. ab + ab. bc}

= 4 (b²c² + c²a² + a²b²) + 8abc (a + b + c)

= 4 (b²c² + c²a² + a²b²),

∵ a + b + c = 0

∴ 2 (b²c² + c²a² + a²b²) = $\frac{1}{2}$ (a² + b² + c²)²

And also (a² + b² + c²)² = $(a^{4} + b^{4} +c^{4} )$ + 2 (b²c² + c²a² + a²b²),

⇒ 4 (b²c² + c²a² + a²b²) = $(a^{4} + b^{4} +c^{4} )$ + 2 (b²c² + c²a² + a²b²)

Hence,  $(a^{4} + b^{4} +c^{4} )$ = 2 (b²c² + c²a² + a²b²)