Use the rational zeros theorem to list all possible rational zeros of the following. $$ f(x)=2 x^{3}+x^{2}-x+5 $$ Be sure that no value in your list appears more than once.

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1. The Rational Zeros Theorem states that any rational zero of the polynomial

$$
   f(x)=2x^3+x^2-x+5
   $$

can be written in the form

$$
   \frac{p}{q},
   $$

where $$ p $$ is a factor of the constant term (5) and $$ q $$ is a factor of the leading coefficient (2).

2. The factors of the constant term, 5, are

$$
   \pm 1,\, \pm 5.
   $$

3. The factors of the leading coefficient, 2, are

$$
   \pm 1,\, \pm 2.
   $$

4. Thus, the possible rational zeros are all combinations of

$$
   \frac{p}{q} = \frac{\pm 1, \pm 5}{\pm 1, \pm 2}.
   $$

This list gives:

$$
   \pm 1,\quad \pm 5,\quad \pm \frac{1}{2},\quad \pm \frac{5}{2}.
   $$

5. No value appears more than once in the list.
The possible rational zeros are $$ \pm1, \pm5, \pm\frac{1}{2}, \pm\frac{5}{2} $$.

Use the rational zeros theorem to list all possible rational zeros of the following.

Be sure that no value in your list appears more than once.

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Use the rational zeros theorem to list all possible rational zeros of the following. Be sure that no value in your list appears more than once.