Three-variable system of simultaneous equations x + y + z = 4 x 2 + y 2 + z 2 = 4 x 3 + y 3 + z 3 = 4Any ideas on how to solve for ( x , y , z ) satisfying the three simultaneous equations, provided there can be both real and complex solutions?

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Three-variable system of simultaneous equations  x  +  y  +  z  =  4      x    2    +      y    2    +      z    2    =  4      x    3    +      y    3    +      z    3    =  4Any ideas on how to solve for   (  x  ,  y  ,  z  ) satisfying the three simultaneous equations, provided there can be both real and complex solutions?

Allyson Gonzalez · Accepted Answer

For a fixed number of variables and a fixed power n the sum of powers      x    n    +      y    n    +      z    n    +  .  .  .  +      w    n  is a symmetric polynomial.It is expressible in terms of elementary symmetric polynomials. The elementary symmetric polynomials for three variables are-       e    1    =  x  +  y  +  z-       e    2    =  x  y  +  x  z  +  y  z-       e    3    =  x  y  zand your polynomials expressed in terms of them are-   x  +  y  +  z  =      e    1  -       x    2    +      y    2    +      z    2    =      e    1    2    −  2      e    2  -       x    3    +      y    3    +      z    3    =      e    1    3    −  3  (      e    1        e    2    −      e    3    )Now we can find the values of       e    1    ,      e    2    ,      e    3   evaluated at the given   x  ,  y  ,  z:      e    1    =  4,       e    2    =  6,       e    3    =  4Now consider the polynomial  (  t  −  x  )  (  t  −  y  )  (  t  −  z  )  =      t    3    −      e    1        t    2    +      e    2    t  −      e    3    =      t    3    −  4      t    2    +  6  t  −  4It has the solutions   t  =  2  ,  1  +  i and   1  −  i.So now we can check if these are correct:-   (  2  )  +  (  1  +  i  )  +  (  1  −  i  )  =  4-   (  2      )    2    +  (  1  +  i      )    2    +  (  1  −  i      )    2    =  4  +  2  i  −  2  i  =  4-   (  2      )    3    +  (  1  +  i      )    3    +  (  1  −  i      )    3    =  8  +  −  2  +  2  i  −  2  −  2  i  =  4

lasquiyas5loaa · Accepted Answer

You can write the elementary functions in terms of these sums of powers, thus getting the coefficients for a polynomial equation whose roots are   x,   y, and   z. For instance, squaring the first equation and subtracting the second, you get the value of   x  y  +  y  z  +  x  z. Cubing the first equation and using what you now know, you get the value of   x  y  z. In general you can use Newton&#039;s identities.

Three-variable system of simultaneous equations x + y + z = 4 x

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