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

We start by using the Rational Zero Theorem. For a polynomial \[ f(x)=10x^4 - x^2 + 25, \] the potential rational zeros are of the form \[ \frac{p}{q}, \] where: - \( p \) is a factor of the constant term (25). - \( q \) is a factor of the leading coefficient (10). **Step 1. Find factors of the constant term.** The constant term is \( 25 \). Its factors are: \[ \pm 1, \quad \pm 5, \quad \pm 25. \] **Step 2. Find factors of the leading coefficient.** The leading coefficient is \( 10 \). Its factors are: \[ \pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10. \] **Step 3. List the possible rational zeros.** Using the Rational Zero Theorem, the potential rational zeros are \[ \frac{p}{q} = \frac{\text{factor of } 25}{\text{factor of } 10}. \] This gives us: \[ \pm \frac{1}{1}, \quad \pm \frac{1}{2}, \quad \pm \frac{1}{5}, \quad \pm \frac{1}{10}, \] \[ \pm \frac{5}{1}, \quad \pm \frac{5}{2}, \quad \pm \frac{5}{5}, \quad \pm \frac{5}{10}, \] \[ \pm \frac{25}{1}, \quad \pm \frac{25}{2}, \quad \pm \frac{25}{5}, \quad \pm \frac{25}{10}. \] **Step 4. Simplify and remove duplicate values.** Let’s simplify each fraction: - \(\pm \frac{1}{1} = \pm 1\) - \(\pm \frac{1}{2} = \pm \frac{1}{2}\) - \(\pm \frac{1}{5} = \pm \frac{1}{5}\) - \(\pm \frac{1}{10} = \pm \frac{1}{10}\) - \(\pm \frac{5}{1} = \pm 5\) - \(\pm \frac{5}{2} = \pm \frac{5}{2}\) - \(\pm \frac{5}{5} = \pm 1\) (duplicate) - \(\pm \frac{5}{10} = \pm \frac{1}{2}\) (duplicate) - \(\pm \frac{25}{1} = \pm 25\) - \(\pm \frac{25}{2} = \pm \frac{25}{2}\) - \(\pm \frac{25}{5} = \pm 5\) (duplicate) - \(\pm \frac{25}{10} = \pm \frac{5}{2}\) (duplicate) Thus, the unique potential rational zeros are: \[ \pm 1, \quad \pm \frac{1}{2}, \quad \pm \frac{1}{5}, \quad \pm \frac{1}{10}, \quad \pm 5, \quad \pm \frac{5}{2}, \quad \pm 25, \quad \pm \frac{25}{2}. \]