x+y+z=1, xy+yz+zx=2 and xyz=-2 find the value of x³+y³+z³
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Step-by-step explanation:
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
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