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Question
If x+y+z=1,xy+yz+zx=-1 and xyz=-1 , x³+y³+z³=?
If x+y+z=1,xy+yz+zx=-1 and xyz=-1 , x³+y³+z³=?
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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 + z3 = 4 -3 = 1
x3 + y3 + z3 = 1
Answer : 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
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