Answer:
[tex]\dfrac{-215}{4}[/tex]
Step-by-step explanation:
f(x) = 3x² - 5x - 2
a = 3; b = -5; c = -2
[tex]\sf \alpha + \beta = \dfrac{-b}{a}[/tex]
[tex]\sf = \dfrac{-(-5)}{3}\\\\=\dfrac{5}{3}[/tex]
[tex]\sf \alpha \beta = \dfrac{c}{a}[/tex]
[tex]\sf = \dfrac{-2}{3}[/tex]
[tex]\sf \dfrac{\alpha ^2}{\beta }+ \dfrac{ \beta ^2}{\alpha }= \dfrac{\alpha ^3+\beta ^3}{\alpha \beta }[/tex]
[tex]\sf = \dfrac{(\alpha +\beta )[(\alpha+\beta)^2-3 \alpha \beta] }{\alpha \beta }\\\\\\= \dfrac{ \frac{5}{3}*[(\frac{5}{3})^2-3* \frac{-2}{3}]}{ \frac{-2}{3}}\\\\\\= \dfrac{ \frac{5}{3}*[ \frac{25}{9}+2]}{ \frac{-2}{3}}\\\\\\= \dfrac{ \frac{5}{3}* \frac{25+18}{2}}{ \frac{-2}{3}}\\\\\\= \dfrac{ \frac{5}{3}* \frac{43}{2}}{ \frac{-2}{3}}\\\\\\= \dfrac{5}{3}* \dfrac{43}{2}* \dfrac{-3}{2}\\\\\\= \dfrac{-215}{4}[/tex]
