1.

If a+b+c=2, ab+bc+ca=-1 and abc=-1 and abc=-2 , find the value of `a^(3)+b^(3)+c^(3)`.

Answer» We know that,
`a^(3)+b^(3)+c^(3)-3abc=(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)`
`=(a+b+c)(a^(2)+b^(2)+c^(2)+2ab+2bc+2ca-3ab-3bc-3ca)`
`=(a+b+c)[(a+b+c)^(2)-3(ab+bc+ca)]`
`=2[(2)^(2)-3(-1)]=2(4+3)=14`
`therefore a^(3)+b^(3)+c^(3)-3(-2)=14`
`implies a^(3)+b^(3)+c^(3)=8`


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