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

To find the potential rational zeros for the polynomial \( f(x)=10 x^{4}-x^{2}+25 \), we can use the Rational Root Theorem. This theorem suggests that any potential rational zero, \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Here, the constant term is \( 25 \) (with factors \( \pm 1, \pm 5, \pm 25 \)) and the leading coefficient is \( 10 \) (with factors \( \pm 1, \pm 2, \pm 5, \pm 10 \)). Therefore, the potential rational zeros are: \[ \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{5}, \pm \frac{5}{5} (which simplifies to \pm 1) , \pm \frac{1}{10}, \pm \frac{5}{10} (which simplifies to \pm \frac{1}{2}). \] This gives us the complete set of potential rational zeros: \( \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}. \)