Consider the equation. x3−9x2+26x−24=0 The roots are such that α=2β So, α+β+γ=9 αβ+βγ+γα=−26 And, αβγ=24 Therefore, 3β+γ=9 And, β2γ=12 Now, β2(9−3β)=12 β3−3β2+4=0 (β+1)(β2−4β+4)=0 (β+1)(β−2)2=0 Therefore, β=−1,2 When β=−1 then α=−2 Then, α3+β3=(−2)3+(−1)3 =−8−1 =−9 When β=2 then α=4. Then, α3+β3=(4)3+(2)3 =64+8 =72