IIT JEE , AIEEE Forum - Mathematics , Algebra - if a + b + c = 0 then a(b-c)^2 + b(c-a)^2 + c(a-b)^2 / abc = ??

harsha_27

a[b-c]² + b[c-a]² +c[a-b]² /abc = a{[b+c]²-4bc} + b{[c+a]²-4ca} + c{[a+b]²-4ab}/abc

                                           =a{a²-4bc} + b{b²-4ca} + c{c²-4ab}/abc

                                           =a³+b³+c³-12abc/abc

                                           =3abc-12abc/abc   [ since if a+b+c=0,a³+b³+c³=3abc]

                                           =-9abc/abc

                                           =- 9

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