Question

prove that (a^2 -b^2)^3+(b^2-c^2)^3 +(c^2-a^2)^3=3(a+b)(b+c)(c+a)(a-b)(b-c)(c-a)

Answer 1

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Let , ( a^2 - b^2 ) = X  ---------------------- equation 1

        ( b^2 - c^2 ) = Y  ---------------------- equation 2 
      
        ( c^2 - a^2 ) = Z ----------------------- equation 3

Adding all these three equations,

a^2 - b^2 + b^2 - c^2  + c^2 - a^2 = X + Y + Z

0 = X + Y + Z.

We know that when X+ Y + Z = 0, then X³ + Y³ + Z³ = 3XYZ

Now,

X³ + Y³ + Z³ = 3XYZ

By the the values of X, Y and Z in the above equation,

( a^2 - b^2)³ + ( b^2 - c^2 )³ + ( c^2 - a^2 )³ =3 ( a^2 - b^2 ) ( b^2 - c^2 ) ( c^2 - a^2 ) 

      = 3 ( a + b ) ( a - b ) ( b + c ) ( b -c ) ( c + a ) ( c - a ) 

     =3 ( a + b ) ( b + c ) ( c + a ) ( a -b ) ( b - c ) ( c - a ) 

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