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

Ask by Olson Summers. in the United States
Mar 15,2025

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

Solución

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.

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Beyond the Answer

To discover potential rational zeros for the polynomial \( f(x) = 10x^4 - x^2 + 25 \), we can use the Rational Root Theorem, which suggests that any rational solution \( \frac{p}{q} \) (where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient) should be explored. Here, the constant term is \( 25 \) (factors are \( \pm 1, \pm 5, \pm 25 \)), and the leading coefficient is \( 10 \) (factors are \( \pm 1, \pm 2, \pm 5, \pm 10 \)). Therefore, the potential rational zeros are: \[ \text{Potential rational zeros: } \pm 1, \pm 5, \pm 25, \pm \frac{1}{1}, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{5}{5}, \pm \frac{25}{1}, \pm \frac{25}{2}, \pm \frac{25}{5}, \pm \frac{25}{10} \] In simpler terms, that's \( \pm 1, \pm 5, \pm 25, \pm \frac{1}{2}, \pm \frac{5}{2}, \pm \frac{25}{2}, \pm \frac{5}{10}, \pm \frac{5}{5}, \pm \frac{25}{10} \). Now you've got a fun little list to work with! Happy zero hunting!

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