Answer
$a.\displaystyle \quad\pm 1,\pm 3,\pm\frac{1}{2},\pm\frac{3}{2}$
$b.\displaystyle \quad-\frac{1}{2},1,$ and $3$
Work Step by Step
Rational Zeros Theorem$:$
$ ... $every rational zero of $P(x)$ is of the form $\displaystyle \frac{p}{q}$
where $p$ and $q$ are integers and
$p$ is a factor of the constant coefficient $a_{0}$
$q$ is a factor of the leading coefficient $a_{n}$
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$a.$
$P(x)=2x^{4}-9x^{3}+9x^{2}+x-3$
candidates for p: $\pm 1,\pm 3$
candidates for q: $\pm 1,\pm 2$
Possible rational zeros $\displaystyle \frac{p}{q}$:$\displaystyle \quad \pm 1,\pm 3,\pm\frac{1}{2},\pm\frac{3}{2}.$
$b.$
From the graph, the actual zeros are $-\displaystyle \frac{1}{2},1,$ and $3$
