Question

If $\alpha ,\beta$ are roots of $x^2-x+1=0$ then find ${\alpha }^3+{\beta }^3$.

Answer 1

Sum of the roots $\alpha +\beta =-(-1)=1$

product of the roots $\alpha \beta =1$

${\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$

$=(1{)}^3-3\times 1(1)$

$=1-3=-2$

$\therefore {\alpha }^3+{\beta }^3=-2$