If x + y + z = 0, prove that x 2 2 x 2 + y z + y 2 2 y 2 + z x + z 2 2 z 2 + x y = 1A problem in my homework had asked me:When x + y + z = 0, evaluate x 2 2 x 2 + y z + y 2 2 y 2 + z x + z 2 2 z 2 + x y Without too much difficulty, one can see that the value should be 1 using ( x , y , z ) = ( 1 , 0 , − 1 )I decided to use x = − y − z, which turned out not to be as difficult as initially thought. However, would someone care to enlighten me to some other methods of doing this?

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If   x  +  y  +  z  =  0, prove that             x      2              2              x        2            +      y      z        +            y      2              2              y        2            +      z      x        +            z      2              2              z        2            +      x      y        =  1A problem in my homework had asked me:When   x  +  y  +  z  =  0, evaluate            x      2              2              x        2            +      y      z        +            y      2              2              y        2            +      z      x        +            z      2              2              z        2            +      x      y      Without too much difficulty, one can see that the value should be   1 using   (  x  ,  y  ,  z  )  =  (  1  ,  0  ,  −  1  )I decided to use   x  =  −  y  −  z, which turned out not to be as difficult as initially thought. However, would someone care to enlighten me to some other methods of doing this?

billyfcash5n · Accepted Answer

HINT:  2      x    2    +  y  z  =  2  (  y  +  z      )    2    +  y  z  =  (  2  y  +  z  )  (  y  +  2  z  )Now   2  y  +  z  =  x  +  y  +  z  +  y  −  x  =  y  −  x

ban1ka1u · Accepted Answer

The first term is            x      2              2              x        2            −      x      y      −              y        2              =            x      2              (      2      x      +      y      )      (      x      −      y      )        =            x      2              (      x      −      z      )      (      x      −      y      )      Do the same transformation on each of the fractions and add them up.So the whole expression is            x      2              (      x      −      z      )      (      x      −      y      )        +            y      2              (      y      −      z      )      (      y      −      x      )        +            z      2              (      z      −      y      )      (      z      −      x      )        =  .  .  .  =  1

If x + y + z = 0 , prove that x 2 </msup>

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