Question

If $-1$, $2$, $\alpha$ are the roots of the equation $2x^3+x^2-7x-6=0$, then find $\alpha$.

Answer 1

Fact: If $\alpha ,\beta ,\gamma$ are the roots of $a{x}^3+b{x}^2+cx+d=0$, then

$\alpha +\beta +\gamma =-\frac{b}{a}$

$\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$

and $\alpha \beta \gamma =-\frac{d}{a}$

So if $-1,2,\alpha$ are the roots of $2x^3+x^2-7x-6=0$, then

$-1+2+\alpha =-\frac{1}{2}$

$\Rightarrow \alpha =-\frac{1}{2}-1=-\frac{3}{2}$