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Question

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


Solution

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 + z= 4 -3 = 1 

x3 + y3 + z= 1

Answer : 1 

 







 

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