Question

if x+y+z=0,then x^3+y^3+z^3=??

Answer 1

If ( x + y + z ) = 0

Then, x³ + y³ + z³ = 3 xyz.

Proof :

Given,

⇒ x + y + z = 0

⇒ x + y = ( -z )   ------- ( 1 )

By squaring both sides ,

⇒ ( x + y )³ = ( -z )³

⇒ x³ + y³ + 3 xy ( x + y ) = -z³

⇒ x³ + y³ + z³ + 3xy ( x + y ) = 0

By substituting the value of ( 1 )

⇒ x³ + y³ + z³ + 3 xy ( -z ) = 0

⇒ x³ + y³ + z³ - 3 xyz = 0

∴ x³ + y³ + z³ = 3 xyz.

Proved.

Answer 2

If, x+y+z=0

then, x^3+y^3+z^3 = 3 xyz

as we know,

x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)

Put x+y+z =0 (Given)

x^3 + y^3 + z^3 - 3xyz = (0)(x^2 + y^2 + z^2 - xy - xz - yz)

x^3 + y^3 + z^3 - 3xyz = 0

x^3 + y^3 + z^3 = 3xyz (Proved)





Hope it helps