Question

If $\alpha$ and $\beta$ are zeroes of the quadratic polynomial $f\left(x\right)=3x^2-5x-2$, then find the values of $\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }$

Answer 1

$f\left(x\right)=3x^2-5x-2$

$\alpha +\beta =\frac{5}{3},\alpha \beta =\frac{-2}{3}$

Given, $\frac{{\alpha }^2}{\beta }+\frac{{\beta }^2}{\alpha }$

$=\frac{{\alpha }^3+{\beta }^3}{\alpha \beta }$

$=\frac{(\alpha +\beta )({\alpha }^2+{\beta }^2-\alpha \beta )}{\alpha \beta }$

$=\frac{(\alpha +\beta )\int (\alpha +\beta {)}^2-3\alpha \beta }{\alpha \beta }$

$=\frac{\left(\frac{5}{3}\right)(\frac{25}{9}+2.)}{-2/3}$

$=\frac{-5}{2}\left(\frac{43}{9}\right)=\frac{-215}{16}$