Answer
$a.\displaystyle \quad\pm 1,\pm\frac{1}{2},\pm\frac{1}{4}$
$b.\displaystyle \quad\frac{1}{4}$ and $1$
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 $a_{n}$
---
$a.$
$P(x)=4x^{4}-x^{3}-4x+1$
candidates for p:$\quad \pm 1$
candidates for q:$\quad \pm 1,\pm 2,\pm 4$
Possible rational zeros $\displaystyle \frac{p}{q}$:$\displaystyle \quad \pm 1,\pm\frac{1}{2},\pm\frac{1}{4}.$
$b.$
From the graph, the actual zeros are $\displaystyle \frac{1}{4}$ and 1.
