Answer the following questions about the equation below. (a) List all rational roots that are possible according to the Rational Zero Theorem. Choose the correct answer below. A. $$ \pm 1 $$ B. $$ \pm 1, \pm 2, \pm 4, \pm 8 $$ D. $$ -1,-2,-4,-8 $$ (b) Use synthetic division to test several possible rational roots in order to identify one actual root. One rational root of the given equation is (Simplify your answer.) (

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To solve the problem, we will follow these steps:

### Part (a): List all rational roots according to the Rational Zero Theorem.

The Rational Root Theorem states that any rational solution, expressed as a fraction $$ \frac{p}{q} $$, has $$ p $$ as a factor of the constant term and $$ q $$ as a factor of the leading coefficient.

1. **Identify the constant term and leading coefficient**: 
   - Let's assume the polynomial is of the form $$ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 $$.
   - The constant term is $$ a_0 $$ and the leading coefficient is $$ a_n $$.

2. **List the factors**:
   - Find the factors of the constant term $$ a_0 $$.
   - Find the factors of the leading coefficient $$ a_n $$.

3. **Form the possible rational roots**:
   - The possible rational roots are given by $$ \frac{p}{q} $$, where $$ p $$ are the factors of $$ a_0 $$ and $$ q $$ are the factors of $$ a_n $$.

Since the specific polynomial is not provided, I will assume a general case. Let's say the polynomial is $$ x^3 - 8 $$ (as an example).

- The constant term $$ a_0 = -8 $$ has factors: $$ \pm 1, \pm 2, \pm 4, \pm 8 $$.
- The leading coefficient $$ a_n = 1 $$ has factors: $$ \pm 1 $$.

Thus, the possible rational roots are $$ \pm 1, \pm 2, \pm 4, \pm 8 $$.

### Answer for Part (a):
B. $$ \pm 1, \pm 2, \pm 4, \pm 8 $$

### Part (b): Use synthetic division to test several possible rational roots.

Let's test the possible rational roots using synthetic division. We will start with $$ 1, -1, 2, -2, 4, -4, 8, -8 $$.

Assuming the polynomial is $$ x^3 - 8 $$:

1. **Testing $$ x = 2 $$**:
   - Set up synthetic division with $$ 2 $$:
   ```
   2 | 1  0  0  -8
     |    2  4   8
     ----------------
       1  2  4   0
   ```
   - The remainder is $$ 0 $$, which means $$ x = 2 $$ is a root.

### Answer for Part (b):
One rational root of the given equation is $$ 2 $$.

If you have a specific polynomial in mind, please provide it, and I can adjust the calculations accordingly.
Possible rational roots are $$ \pm 1, \pm 2, \pm 4, \pm 8 $$. Testing these, $$ x = 2 $$ is a root.

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