Use the rational zeros theorem to list all possible rational zeros of the following, $$ h(x)=x^{3}-9 x-8 x^{4}+2 x^{2}+4 $$ Be sure that no value in your list appears more thah onç.

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The given polynomial is
$$
h(x) = x^{3} - 9x - 8x^{4} + 2x^{2} + 4.
$$
Step 1. Write the polynomial in standard form (arranged in descending order):
$$
h(x) = -8x^{4} + x^{3} + 2x^{2} - 9x + 4.
$$

Step 2. Identify the leading coefficient and the constant term.  
The leading coefficient is $$-8$$ and the constant term is $$4$$.

Step 3. List the factors of the constant term and the factors of the leading coefficient.  
- Factors of $$4$$ are: $$\pm 1$$, $$\pm 2$$, $$\pm 4$$.  
- Factors of $$-8$$ are: $$\pm 1$$, $$\pm 2$$, $$\pm 4$$, $$\pm 8$$.

Step 4. Use the Rational Zeros Theorem.  
Any possible rational zero can be written in the form
$$
\frac{p}{q},
$$
where $$p$$ is a factor of $$4$$ and $$q$$ is a factor of $$-8$$.

Step 5. Form all possible values $$\frac{p}{q}$$ (and simplify them so that no duplicates occur).  
- For $$p = \pm 1$$:
  - $$\pm \frac{1}{1} = \pm 1$$
  - $$\pm \frac{1}{2}$$
  - $$\pm \frac{1}{4}$$
  - $$\pm \frac{1}{8}$$

- For $$p = \pm 2$$:
  - $$\pm \frac{2}{1} = \pm 2$$
  - $$\pm \frac{2}{2} = \pm 1$$ (duplicate)
  - $$\pm \frac{2}{4} = \pm \frac{1}{2}$$ (duplicate)
  - $$\pm \frac{2}{8} = \pm \frac{1}{4}$$ (duplicate)

- For $$p = \pm 4$$:
  - $$\pm \frac{4}{1} = \pm 4$$
  - $$\pm \frac{4}{2} = \pm 2$$ (duplicate)
  - $$\pm \frac{4}{4} = \pm 1$$ (duplicate)
  - $$\pm \frac{4}{8} = \pm \frac{1}{2}$$ (duplicate)

Step 6. List the distinct possible rational zeros.  
The possible rational zeros are:
$$
\pm \frac{1}{8},\ \pm \frac{1}{4},\ \pm \frac{1}{2},\ \pm 1,\ \pm 2,\ \pm 4.
$$
The possible rational zeros are $$ \pm \frac{1}{8}, \pm \frac{1}{4}, \pm \frac{1}{2}, \pm 1, \pm 2, \pm 4 $$.

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

Be sure that no value in your list appears more thah onç.

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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 thah onç.