Show that ( b + c ) 2 3 b c + ( c + a ) 2 3 a c + ( a + b ) 2 3 a b = 1If a 3 + b 3 + c 3 = 3 a b c and a + b + c = 0 show that ( b + c ) 2 3 b c + ( c + a ) 2 3 a c + ( a + b ) 2 3 a b = 1

Question

Show that             (      b      +      c              )        2                    3      b      c        +            (      c      +      a              )        2                    3      a      c        +            (      a      +      b              )        2                    3      a      b        =  1If       a    3    +      b    3    +      c    3    =  3  a  b  c and   a  +  b  +  c  =  0 show that             (      b      +      c              )        2                    3      b      c        +            (      c      +      a              )        2                    3      a      c        +            (      a      +      b              )        2                    3      a      b        =  1

SweallySnicles3 · Accepted Answer

a  +  b  +  c  =  0    ⟺    b  +  c  =  −  a    ⟹    (  b  +  c      )    2    =      a    2      ⟹                      (        b        +        c                  )          2                            3        b        c              =                    a        3                    3        a        b        c            Actually,   b  +  c  =  −  a    ⟹    (  b  +  c      )    3    =  (  −  a      )    3      ⟹    −      a    3    =      b    3    +      c    3    +  3  b  c  (  b  +  c  )  =      b    3    +      c    3    +  3  b  c  (  −  a  )    ⟺    ∑      a    3    =  3  a  b  c

ziphumulegn · Accepted Answer

(      b      +      c              )        2                    3      b      c        +            (      c      +      a              )        2                    3      a      c        +            (      a      +              b        2            )              3      a      b          =            a      (      b      +      c              )        2                    3      a      b      c        +            b      (      c      +      a              )        2                    3      a      b      c        +            c      (      a      +              b        2            )              3      a      b      c          =            a      3              3      a      b      c        +            b      3              3      a      b      c        +            c      3              3      a      b      c          =                    a        3            +              b        3            +              c        3                    3      a      b      c          =            3      a      b      c              3      a      b      c        =  1

Show that ( b + c ) 2 </msup> </mr

Answered question

Answer & Explanation

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