Answer
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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