Answer
$\displaystyle \pm 1,\pm\frac{1}{2},\pm\frac{1}{3},\pm\frac{1}{6},\pm 2,\pm\frac{2}{3}$
Work Step by Step
Rational Zeros Theorem
Let $f$ be a polynomial function of degree 1 or higher of the form
$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\quad a_{n}\neq 0\quad a_{0}\neq 0$
where each coefficient is an integer. If $\displaystyle \frac{p}{q},$ in lowest terms, is a rational zero of $f,$ then
$p$ must be a factor of $a_{0}=2,$
and $q$ must be a factor of $a_{n}=6$ .
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candidates for $p:\quad\pm 1,\pm 2$
candidates for $q:\quad\pm 1,\pm 2,\pm 3,\pm 6$
Possible zeros $\displaystyle \frac{p}{q}$ :
$\displaystyle \pm 1,\pm\frac{1}{2},\pm\frac{1}{3},\pm\frac{1}{6},\pm 2,\pm\frac{2}{3}$
