Question

If $\alpha ,\beta ,\gamma$ are the roots of $x^3+a{x}^2+bxx+c=0$, then find $\sum {\alpha }^2\beta +\sum \alpha {\beta }^2$.

Answer 1

Given that,

$\alpha ,\beta ,\gamma$ are the roots of $x^3+a{x}^2+bx+c=0$

$⟹\alpha +\beta +\gamma =-a$

$\alpha \beta +\beta \gamma +\gamma \alpha =b$

and $\alpha \beta \gamma =-c$

$\therefore \sum {\alpha }^2\beta +\sum \alpha {\beta }^2$

$={\alpha }^2\beta +{\beta }^2\gamma +{\gamma }^2\alpha +\alpha {\beta }^2+\beta {\gamma }^2+\gamma {\alpha }^2$

$=\alpha \beta (\alpha +\beta )+\beta \gamma (\beta +\gamma )+\gamma \alpha (\gamma +\alpha )$

$=\alpha \beta (-a-\gamma )+\beta \gamma (-a-\alpha )+\gamma \alpha (-a-\beta )$

$=-a\alpha \beta -\alpha \beta \gamma -a\beta \gamma -\alpha \beta \gamma -a\gamma \alpha -\alpha \beta \gamma$

$=-a(\alpha \beta +\beta \gamma +\gamma \alpha )-3\alpha \beta \gamma$

$=-ab-3(-c)=-ab+3c$

$=3c-ab$ (Ans)