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`
2. If `x+y+z=12,xy+yz+zx=47` and `x^3+y^3+z^3=216`, then find `xyz`
Solution:
Here `x+y+z=12,xy+yz+zx=47` and `x^3+y^3+z^3=216`
We know that
`x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)`
`x^2+y^2+z^2=50`
Here `x+y+z=12` and `xy+yz+zx=47`
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=12^2-2*47`
`:.x^2+y^2+z^2=144-94`
`:.x^2+y^2+z^2=50`
`:.216-3xyz=(12)(50-47)`
`:.216-3xyz=12*3`
`:.216-3xyz=36`
`:.3xyz=216-36`
`:.3xyz=180`
`:.xyz=60`
3. If `x+y+z=10,xyz=15` and `x^3+y^3+z^3=75`, then find `x^2+y^2+z^2-xy-yz-zx`
Solution:
Here `x+y+z=10,xyz=15` and `x^3+y^3+z^3=75`
`x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)`
`:.75-3*15=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.75-45=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.30=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.(x^2+y^2+z^2-xy-yz-zx)=(30)/(10)`
`:.x^2+y^2+z^2-xy-yz-zx=3`
4. If `x+y+z=10,xyz=15` and `x^3+y^3+z^3=75`, then find `xy+yz+zx`
Solution:
Here `x+y+z=10,xyz=15` and `x^3+y^3+z^3=75`
`x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)`
`:.75-3*15=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.75-45=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.30=10*(x^2+y^2+z^2-xy-yz-zx)`
`:.(x^2+y^2+z^2-xy-yz-zx)=(30)/(10)`
`:.x^2+y^2+z^2-xy-yz-zx=3`
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)`
Substitute this in above equation
`:.(x+y+z)^2-2(xy+yz+zx)-xy-yz-zx=3`
`:.(10)^2-3(xy+yz+zx)=3`
`:.100-3(xy+yz+zx)=3`
`:.3(xy+yz+zx)=100-3`
`:.3(xy+yz+zx)=97`
`:.(xy+yz+zx)=97/3`
`:.xy+yz+zx=32.33`
This material is intended as a summary. Use your textbook for detail explanation.
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