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

The Rational Zero Theorem tells us that any rational zero of the polynomial \[ f(x)=10x^{4}-x^{2}+25 \] must be of the form \[ \frac{p}{q}, \] where \( p \) is a factor of the constant term (25) and \( q \) is a factor of the leading coefficient (10). **Step 1. Factors of the constant term (25):** The factors are \[ \pm 1, \quad \pm 5, \quad \pm 25. \] **Step 2. Factors of the leading coefficient (10):** The factors are \[ \pm 1, \quad \pm 2, \quad \pm 5, \quad \pm 10. \] **Step 3. List the potential rational zeros:** The possible zeros are all fractions of the form \(\frac{p}{q}\) where \( p \) is one of \(\pm 1, \pm 5, \pm 25\) and \( q \) is one of \(\pm 1, \pm 2, \pm 5, \pm 10\). Listing them out: 1. For \( p=\pm 1 \): - \(\frac{\pm 1}{1} = \pm 1\) - \(\frac{\pm 1}{2} = \pm \frac{1}{2}\) - \(\frac{\pm 1}{5} = \pm \frac{1}{5}\) - \(\frac{\pm 1}{10} = \pm \frac{1}{10}\) 2. For \( p=\pm 5 \): - \(\frac{\pm 5}{1} = \pm 5\) - \(\frac{\pm 5}{2} = \pm \frac{5}{2}\) - \(\frac{\pm 5}{5} = \pm 1\) (already listed) - \(\frac{\pm 5}{10} = \pm \frac{1}{2}\) (already listed) 3. For \( p=\pm 25 \): - \(\frac{\pm 25}{1} = \pm 25\) - \(\frac{\pm 25}{2} = \pm \frac{25}{2}\) - \(\frac{\pm 25}{5} = \pm 5\) (already listed) - \(\frac{\pm 25}{10} = \pm \frac{5}{2}\) (already listed) **Step 4. Unique potential rational zeros:** Collecting all unique values, we have: \[ \pm 1, \quad \pm \frac{1}{2}, \quad \pm \frac{1}{5}, \quad \pm \frac{1}{10}, \quad \pm 5, \quad \pm \frac{5}{2}, \quad \pm 25, \quad \pm \frac{25}{2}. \] These are the potential rational zeros for the given polynomial.