Question

The zeros of $P\left(x\right)=3x^2-x-4$ are $\alpha$ and $\beta$.Then find the value of ${\alpha }^2\beta +\alpha {\beta }^2$.

Answer 1

Sum of roots of $a{x}^2+bx+c=0$ are given as $\frac{-b}{a}$ and product of roots as $\frac{c}{a}$

$\Rightarrow \alpha +\beta =\frac{1}{3}$, $\alpha \beta =-\frac{4}{3}$

${\alpha }^2\beta +\alpha {\beta }^2=\alpha \beta (\alpha +\beta )=\frac{1}{3}(-\frac{4}{3})$

$=-\frac{4}{9}$.