Question

If $\alpha ,\beta$ and $1$ are roots of $x^3-2x^2-5x+6=0$, then find $\alpha$ and $\beta$

Answer 1

If $1$ is one root of $x^3-2x^2-5x+6=0$, then $(x-1)$ will be a factor

of polynomial $x^3-2x^2-5x+6$

Other factor can be obtained by using long division or synthetic division

$1|1-2-56$

$|\downarrow 1-1-6$

${}^{--}|{}^{-------------------}$

$|1-1-60$

So other factor is $(x^2-x-6)$

Hence $\alpha ,\beta$ are the roots of $x^2-x-6=0$

$\Rightarrow (x+2)(x-3)=0$

$\Rightarrow x=-2,3$