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EspañolEnglish

x + y + z = 0
x ³ + y ³ + z ³ = π

¿ Qué vale xyz  ?

 

 

   

Todas las funciones involucradas son simétricas .
Como intervienen tres variables, { x y z }, podemos definir una ecuación de tercer orden con coeficientes las funciones siméticas unitarias de { x y z }, relaciones de Cardano-Vieta .

ζ 3 a1 ζ 2 + a2 ζ a3 = 0

a1 = x + y + z   
a2 = xy + xz + yz
a3 = xyz     

∴  x 3 a1 x 2 + a2 x a3 = 0
  y 3 a1 y 2 + a2 y a3 = 0
  z 3 a1 z 2 + a2 z a3 = 0

π − 3a3 = 0

π
a3 = ――  
3


EnglishEspañol

x + y + z = 0
x ³ + y ³ + z ³ = π

calculate xyz

 

   

All functions are symmetric functions.
We have three variables, { x y z }, we could setup a cubic equation , Vieta's formula .

ζ 3 a1 ζ 2 + a2 ζ a3 = 0

a1 = x + y + z   
a2 = xy + xz + yz
a3 = xyz     

∴  x 3 a1 x 2 + a2 x a3 = 0
  y 3 a1 y 2 + a2 y a3 = 0
  z 3 a1 z 2 + a2 z a3 = 0

π − 3a3 = 0

π
a3 = ――  
3


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