Question

this is factor theorem........ my doubt is how do we get the posiibilities that are given ? pl see attached

Answer 1

You may get the possible rational roots by using rational root theorem :
The rational roots of any polynomial with integer coefficients are of form : $\frac{p}{q}$
where
$p$ = factor of constant term
$q$ = factor of leading coefficient

In our case
$p=5$
$q=6$

factors of $5$ : $\pm{1},~\pm{5}$
factors of $6$ : $\pm{1},~\pm{2},~\pm{3},~\pm{6}$

therefore the possible rational roots are :
$\pm\frac{1}{2},~\pm\frac{1}{3},~\pm\frac{1}{6},~\pm\frac{5}{2},~\pm\frac{5}{3},~\pm\frac{5}{6},~\pm{1}$