If α,β,γ are the roots of x3−x2−1=0, then the value of 1+α1−α+1+β1−β+1+γ1−γ=
- 5
x3−x2−1=0α+β+γ=1,αβ+βγ+γα=0,αβγ=1.
Now, 1+α1−α=2−(1−α)1−α=21−α−1∴∑1+α1−α=∑21−α−∑1=2∑11−α−3 (i)Now,1x−α+1x−β+1x−γ=(x−β)(x−γ)+(x−α)(x−β)+(x−α)(x−γ)(x−α)(x−β)(x−γ)1x−α+1x−β+1x−γ=3x2−2xx3−x2−1Let x=1=11−α+11−β+11−γ=3−21−1−1=−1∴(i)=2(−1)−3=−5