14. List the potential rational zeros for the polynomial: $$ f(x)=10 x^{4}-x^{2}+25 $$

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1. The Rational Root Theorem tells us that any potential rational zero of the polynomial

$$
   f(x)=10x^4-x^2+25
   $$

is of the form

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

where $$ p $$ divides the constant term (25) and $$ q $$ divides the leading coefficient (10).

2. The factors of the constant term 25 are

$$
   \pm 1,\ \pm 5,\ \pm 25.
   $$

3. The factors of the leading coefficient 10 are

$$
   \pm 1,\ \pm 2,\ \pm 5,\ \pm 10.
   $$

4. Thus, the potential rational zeros are all numbers of the form

$$
   \pm \frac{p}{q} \quad = \quad \pm \frac{\text{factor of }25}{\text{factor of }10}.
   $$

5. Listing the possibilities:

- For $$ p = 1 $$:
     
     $$
     \pm \frac{1}{1} = \pm 1,\quad \pm \frac{1}{2},\quad \pm \frac{1}{5},\quad \pm \frac{1}{10}.
     $$
   
   - For $$ p = 5 $$:
     
     $$
     \pm \frac{5}{1} = \pm 5,\quad \pm \frac{5}{2},\quad \pm \frac{5}{5} = \pm 1,\quad \pm \frac{5}{10} = \pm \frac{1}{2}.
     $$
   
   - For $$ p = 25 $$:
     
     $$
     \pm \frac{25}{1} = \pm 25,\quad \pm \frac{25}{2},\quad \pm \frac{25}{5} = \pm 5,\quad \pm \frac{25}{10} = \pm \frac{5}{2}.
     $$

6. Removing duplicates, the complete list of potential rational zeros is:

$$
   \pm 1,\quad \pm \frac{1}{2},\quad \pm \frac{1}{5},\quad \pm \frac{1}{10},\quad \pm 5,\quad \pm \frac{5}{2},\quad \pm 25,\quad \pm \frac{25}{2}.
   $$
The potential rational zeros of the polynomial $$ f(x) = 10x^4 - x^2 + 25 $$ are: $$ \pm 1, \quad \pm \frac{1}{2}, \quad \pm \frac{1}{5}, \quad \pm \frac{1}{10}, \quad \pm 5, \quad \pm \frac{5}{2}, \quad \pm 25, \quad \pm \frac{25}{2} $$

14. List the potential rational zeros for the polynomial: \( f(x)=10 x^{4}-x^{2}+25 \)

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