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

Ask by Bartlett Guerrero. in the United States
Mar 14,2025

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

Solution

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: \[ \pm 1, \quad \pm \frac{1}{2}, \quad \pm \frac{1}{5}, \quad \pm \frac{1}{10}, \quad \pm 5, \quad \pm \frac{5}{2}, \quad \pm 25, \quad \pm \frac{25}{2}. \]

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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: \( \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}. \)

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