Question

If x+y+z = 1 , xy +yz+zx=1 and xyz=-1 find value of x×x×x + y×y×y + z×z×z

Answer 1

Given
x+y+z=1,xy+yz+zx=−1 and xyz=−1

x
3
+y
3
+z
3
−3xyz=(x+y+z)(x
2
+y
2
+z
2
−xy−yz−zx)

x
3
+y
3
+z
3
−3(−1)=1(x
2
+y
2
+z
2
)−(−1)

x
3
+y
3
+z
3
+3=x
2
+y
2
+z
2
+1

x
3
+y
3
+z
3
+2=x
2
+y
2
+z
2


(x+y+z)
2
=x
2
+y
2
+z
2
+2xy+2yz+2xz

1
2
=x
2
+y
2
+z
2
+2(−1)

x
2
+y
2
+z
2
=1+2

x
2
+y
2
+z
2
=3

x
3
+y
3
+z
3
+2=3

x
3
+y
3
+z
3
= 1

Hope it helps u thanks a lot