Question

If x+y+z=1,xy+yz+zx=-1 and xyz=-1 , x³+y³+z³=?

Answer 1

Given x + y + z = 1 and xy + yz + zx = -1 and xyz = -1

Consider, x + y + z = 1

Squaring on both sides, we get

(x + y + z)² = 1²

x² + y² + z² +2(xy + yz +zx) = 1

⇒ x² + y² + z² = 1 − 2(xy + yz +zx)

= 1− 2(-1) = 1+2 = 3

∴ x² + y² + z² = 3

We know that x³ + y³ + z³ − 3xyz = (x + y + z)( x² + y² + z² − xy − yz − zx)

= 1*(3+1) = 1(4) = 4

x³ + y³ + z³ − 3xyz = 4

x³ + y³ + z³ − 3*(-1) = 4

x³ + y³ + z³ + 3 = 4

x³ + y³ + z³ = 4 -3 = 1

x³ + y³ + z³ = 1

Answer : 1