If x+y+z=1,xy+yz+zx=-1 and xyz=-1 then find x^3+y^3+z^3 calculator

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1. If `x + 1/x = 2` then find `x^2 + 1/x^2`

2. If `x - 1/x = 6` then find
(1) `x^2 + 1/x^2` (2) `x^4 + 1/x^4` (3) `x^3 + 1/x^3` (4) `x + 1/x`
(5) `x^2 - 1/x^2` (6) `x^4 - 1/x^4` (7) `x^ 3 - 1/x^3`

3. If `x^2 + 1/x^2 = 23` then find (1) `x + 1/x` (2) `x^2 + 1/x^2`

4. If `x + y = 5` and `xy = 6` then find
(1) `x^2 + y^2` (2) `x^3 + y^3` (3) `x^4 + y^4` (4) `x - y`
(5) `x^2 - y^2` (6) `x^3 - y^3` (7) `x^4 - y^4`

5. If `x + y = 3` and `x - y = 15` then find `x^2 + y^2` and `xy`

6. If `x + y + z = 9` and `x^2 + y^2 + z^2 = 29` then find `xy + yz + zx`

7. If `x^2 + y^2 + z^2 = 29` and `xy + yz + zx = -14` then find `x + y + z`

8. If `x+y+z=1,xy+yz+zx=-1` and `xyz=-1`, then find `x^3+y^3+z^3`

9. If `x+y+z=12,xy+yz+zx=47` and `x^3+y^3+z^3=216`, then find `xyz`

Example

1. If `x+y+z=1,xy+yz+zx=-1` and `xyz=-1`, then find `x^3+y^3+z^3`

Solution:
Here `x+y+z=1,xy+yz+zx=-1` and `xyz=-1`

We know that
`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=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)+3xyz`

`:.x^3+y^3+z^3=(1)(x^2+y^2+z^2-(-1))+3*(-1)`

`x^2+y^2+z^2=3`

Here `x+y+z=1` and `xy+yz+zx=-1`

We know that
`(x+y+z)^2=(x^2+y^2+z^2)+2(xy+yz+zx)`

`:.(x^2+y^2+z^2)=(x+y+z)^2-2(xy+yz+zx)`

`:.x^2+y^2+z^2=1^2-2*(-1)`

`:.x^2+y^2+z^2=1+2`

`:.x^2+y^2+z^2=3`

`:.x^3+y^3+z^3=1*(3+1)-3`

`:.x^3+y^3+z^3=1*4-3`

`:.x^3+y^3+z^3=4-3`

`:.x^3+y^3+z^3=1`