14. List the potential rational zeros for the polynomial: \( f(x)=10 x^{4}-x^{2}+25 \)
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To find the potential rational zeros of the polynomial \( f(x)=10x^{4}-x^{2}+25 \), we use the Rational Root Theorem. This states that any potential rational zero, in the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient. Here, the constant term is 25 (factors: ±1, ±5, ±25) and the leading coefficient is 10 (factors: ±1, ±2, ±5, ±10). Thus, the potential rational zeros are: \[ \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{5}, \pm \frac{5}{10} = \pm \frac{1}{2}, \pm \frac{5}{10} \] So the complete list of potential rational zeros is: \[ \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{5}, \pm \frac{1}{10} \]
