Let α and α2 be the roots ofx2+x+1=0 then the equation whose roots are α31 and α62, is
x2−x+1=0
x2+x−1=0
x2+x+1=0
x60+x30+1=0
Since, α,α2 be the roots of the equation x2+x+1=0∴ α+α2=−1 and α3=1 ...(i) Now, α31+α62=α311+α31⇒ α31+α62=α30⋅α1+α30⋅α⇒ α31+α62=α310⋅α1+α310⋅α⇒ α31+α62=α(1+α) [from Eq. (ii)] ⇒ α31+α62=−1 [ from Eq. (i)] Again, α31⋅α62=α93⇒ α31⋅α62=α331=1∴ Required equation is,x2−α31+α62x+α31⋅α62=0⇒x2+x+1=0