Question

If $\alpha ,\beta$ are the roots of $x^2-px+q=0$, find the value of
(1) ${\alpha }^2+{\beta }^2$
(2) ${\alpha }^3+{\beta }^3$

Answer 1

Given that $\alpha ,\beta$ are roots of equation $x^2-px+q=0$

We know that $\alpha +\beta =p$ and $\alpha \beta =q$

$\Rightarrow {\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta =p^2-2q$

$\Rightarrow {\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )=p^3-3qp=p(p^2-3q)$