Solutions to x 3 + y 3 + z 3 = x 2 + y 2 + z 2 = x + y + z = 0I need to prove that x y z = 1. dealing trew this problem I get that x + y = 2 3 and that results that z = − 2 3 . after all i got x y z = − 10 27 When I was before dealing with this, I over and over get that x y z = 0.

Question

Solutions to      x    3    +      y    3    +      z    3    =      x    2    +      y    2    +      z    2    =  x  +  y  +  z  =  0I need to prove that   x  y  z  =  1. dealing trew this problem I get that   x  +  y  =      2    3   and that results that   z  =  −      2    3  . after all i got   x  y  z  =  −      10    27  When I was before dealing with this, I over and over get that   x  y  z  =  0.

Jamiya Costa · Accepted Answer

Note that for any reasonable (including complex) x,y,z we have  x  y  +  y  z  +  z  x  =                                                  (          x          +          y          +          z                      )            2                    −                      x            2                    −                      y            2                    −                      z            2                                                                        2              =  0So we have that   x  ,  y  ,  z are roots of the a cubic with unknown constant term   a, and since the terms in   x and       x    2   have zero coefficient we have      x    3    +  a  =  0  ;      y    3    +  a  =  0  ;      z    3    +  a  =  0Adding these together we get   3  a  =  0, and since   x  y  z  =  −  a we have   x  y  z  =  0, and separately, we conclude   x  =  y  =  z  =  0.

Logan Wyatt · Accepted Answer

x    2    +      y    2    +      z    2    =  0 has only the trivial zero solution   (  0  ,  0  ,  0  ) as the left hand side is always positive otherwise.

Solutions to x 3 </msup> + y 3 </msup> + z 3

Answered question

Answer & Explanation

New Questions in Algebra I