Use the rational zeros theorem to list all possible rational zeros of the following. $$ g(x)=-6 x^{3}+9+5 x-9 x^{4}-4 x^{2} $$ Be sure that no value in your list appears more than once.

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We start by writing the polynomial in standard form (with powers of $$ x $$ in descending order):

$$
g(x) = -9x^4 - 6x^3 - 4x^2 + 5x + 9
$$

The Rational Zeros Theorem tells us that any rational zero of the polynomial, written in lowest terms in the form $$\frac{p}{q}$$, must have $$ p $$ as a factor of the constant term and $$ q $$ as a factor of the leading coefficient.

1. The constant term is $$ 9 $$. Its factors are: 
   $$
   \pm 1, \; \pm 3, \; \pm 9.
   $$
   
2. The leading coefficient is $$ -9 $$. Its factors (ignoring the sign since we include $$\pm$$) are:
   $$
   \pm 1, \; \pm 3, \; \pm 9.
   $$

Thus, the possible rational zeros are given by

$$
\frac{p}{q} \quad \text{where} \quad p \in \{\pm 1, \pm 3, \pm 9\} \quad\text{and}\quad q \in \{\pm 1, \pm 3, \pm 9\}.
$$

We list all possible fractions, making sure to reduce to lowest terms and include each distinct value only once.

- When $$ p = \pm 1 $$:
  - Dividing by $$ q = \pm 1 $$ gives: $$ \pm 1 $$.
  - Dividing by $$ q = \pm 3 $$ gives: $$ \pm \frac{1}{3} $$.
  - Dividing by $$ q = \pm 9 $$ gives: $$ \pm \frac{1}{9} $$.

- When $$ p = \pm 3 $$:
  - Dividing by $$ q = \pm 1 $$ gives: $$ \pm 3 $$.
  - Dividing by $$ q = \pm 3 $$ gives: $$ \pm 1 $$ (already listed).
  - Dividing by $$ q = \pm 9 $$ gives: $$ \pm \frac{1}{3} $$ (already listed).

- When $$ p = \pm 9 $$:
  - Dividing by $$ q = \pm 1 $$ gives: $$ \pm 9 $$.
  - Dividing by $$ q = \pm 3 $$ gives: $$ \pm 3 $$ (already listed).
  - Dividing by $$ q = \pm 9 $$ gives: $$ \pm 1 $$ (already listed).

Collecting the distinct values, the list of possible rational zeros is:

$$
\pm 1, \quad \pm \frac{1}{3}, \quad \pm \frac{1}{9}, \quad \pm 3, \quad \pm 9.
$$
The possible rational zeros are $$ \pm1, \pm\frac{1}{3}, \pm\frac{1}{9}, \pm3, \pm9 $$.

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.