Answer
a. $ \displaystyle \pm 1,\pm 3, \pm\frac{1}{2}, \pm\frac{3}{2}.$
b. $-\displaystyle \frac{1}{2}, 1, 3$
Work Step by Step
If $P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}$
has integer coefficients,
then all the rational zeros of $P$ have the form $x=\displaystyle \pm\frac{p}{q}$
where $p$ is a divisor of the constant term $a_{0}$ and
$q$ is a divisor of the leading coefficient $a_{n}$.
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a.
$a_{0}=-3 \qquad $p: $\pm 1,\pm 3$
$a_{n}=2,\qquad $q: $\pm 1, \pm 2,$
Possible $\displaystyle \frac{p}{q}:\quad \pm 1,\pm 3, \pm\frac{1}{2}, \pm\frac{3}{2}.$
b.
From the graph, actual zeros (x-intercepts):
$-\displaystyle \frac{1}{2}, 1, 3$
