Use the Rational Zero Theorem to list all possible rational zoros for the given function, $$ f(x)=4 x^{4}-x^{3}+4 x^{2}-2 x-10 $$ Which of the following is the complete list of possible zeros of the given function? A. $$ \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4} $$ B. $$ \pm 1, \pm 2, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4} $$ C. $$ \pm 1, \pm 2, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2} $$ D. The function has no rational zeros.

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To apply the Rational Root Theorem, we need to identify the possible rational zeros of the polynomial function $$ f(x) = 4x^4 - x^3 + 4x^2 - 2x - 10 $$.

1. **Identify the leading coefficient and the constant term:**
   - The leading coefficient (the coefficient of the highest degree term, $$ 4x^4 $$) is $$ 4 $$.
   - The constant term (the term without $$ x $$, which is $$ -10 $$) is $$ -10 $$.

2. **List the factors of the leading coefficient and the constant term:**
   - Factors of $$ 4 $$: $$ \pm 1, \pm 2, \pm 4 $$
   - Factors of $$ -10 $$: $$ \pm 1, \pm 2, \pm 5, \pm 10 $$

3. **Form the possible rational zeros:**
   According to the Rational Root Theorem, the possible rational zeros are given by the ratio of the factors of the constant term to the factors of the leading coefficient. Thus, we can form the following possible rational zeros:

$$
\text{Possible rational zeros} = \frac{\text{Factors of } -10}{\text{Factors of } 4}
$$

This gives us:

- From $$ \pm 1 $$: $$ \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4} $$
- From $$ \pm 2 $$: $$ \pm 2, \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4} $$
- From $$ \pm 5 $$: $$ \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4} $$
- From $$ \pm 10 $$: $$ \pm 10, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4} $$

Combining all these, we get the complete list of possible rational zeros:

$$
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4}
$$

4. **Select the correct option:**
Looking at the options provided:

A. $$ \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4} $$  
B. $$ \pm 1, \pm 2, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4} $$  
C. $$ \pm 1, \pm 2, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2} $$  
D. The function has no rational zeros.

The correct answer is **A**.
The possible rational zeros are $$ \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{4}, \pm \frac{5}{4} $$. The correct answer is A.

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