If `alpha,beta and gamma` are roots of equation `x^(3)-3x^(2)+1=0`, then the value of `((alpha)/(1+alpha))^(3)+((beta)/(1+beta))^(3)+((gamma)/(1+gamma))^(3)` is __________.

Answer

-1

Explanation

Let `alpha_(1)=(alpha)/(1+alpha),beta_(1)=(beta)/(1+beta),andgamma_(1)=(gamma)/(1+gamma)`,
Also, let `y=alpha_(1),beta_(1) and gamma_(1)` be roots of an equation is y.
Then `y=(x)/(1+x)`
`therefore x =(y)/(1-y)`
Now,`x^(3)-3x^(2)+1=0`
`rArr((y)/(1-y))^(3)-3((y)/(1-y))^(2)+1=0`
`rArr3y^(3)-3y+1=0`
`rArralpha_(1)+beta_(1)+gamma_(1)=0 and alpha_(1)beta_(1)gamma_(1)=-(1)/(3)`
`therefore alpha_(1)^(3)+beta_(1)^(3)+gamma_(1)^(3)=3alpha_(1)beta_(1)gamma_(1)=3(-(1)/(3))=-1`