Question

If a + b+c=0 prove that a 4 +b 4 + c 4 =2(a 2 b 2 + b 2 c 2 +c 2 a 2 )

Answer 1

Given a+b+c = 0, a4+b4+c4 = 2a2b2+2b2c2+2c2a2

a+b+c = 0
a+b = -c // find a+b  value  //
(a+b )2 = (-c)2    // squaring on both side  //
a2+b2+2ab = c2​    // use of identity (a+b )2=a2+b2+2ab //
a2+b2- c2​ = 2ab   // arrange the terms to obtain the equation  //  

(a2+b2- c2​ )2= (2ab ) 2   //  square on both the side   and use of identity (a+b+c)2//
(a4+b4+ c4​+2a2b2-2b2c2-2c2a2)  =4a2b2   // simplify   //
a4+b4+ c4​+2a2b2-4a2b2-2b2c2-2c2a2 = 0
a4+b4+ c4​-2a2b2-2b2c2-2c2a2 = 0
a4+b4+ c4​ = 2a2b2+2b2c2+2c2a2 
RHS  = LHS