- Identify the constant term \(a_0 = 3\) and the leading coefficient \(a_{11} = 15\).
- Determine all integer factors of \(a_0\): \(\{\pm 1, \pm 3\}\) and all positive factors of \(a_{11}\): \(\{1, 3, 5, 15\}\).
- Form all possible ratios \(\frac{p}{q}\) to find the potential rational roots: \(\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3\).
- Select option (d) as it corresponds to the calculated set. The final answer is \(\boxed{d}\).
Explanation
- Problem Analysis and Rational Root Theorem Application
To identify the potential rational roots of the polynomial \(f(x) = 15x^{11} - 6x^8 + x^3 - 4x + 3\) using the Rational Root Theorem, we focus on the relationship between the leading coefficient and the constant term. The theorem states that for a polynomial with integer coefficients, any rational root \(\frac{p}{q}\) (where \(\gcd(p, q) = 1\)) must satisfy the condition that \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. In our case, the constant term is \(a_0 = 3\) and the leading coefficient is \(a_{11} = 15\).
- Listing Potential Rational Roots
Next, we list the potential factors \(p\) and \(q\). The integer factors of the constant term \(a_0 = 3\) are \(p \in \{\pm 1, \pm 3\}\). The positive integer factors of the leading coefficient \(a_{11} = 15\) are \(q \in \{1, 3, 5, 15\}\). By taking all possible ratios \(\frac{p}{q}\), we generate a list of candidates for the roots:
- For \(q=1\): \(\frac{\pm 1}{1} = \pm 1, \frac{\pm 3}{1} = \pm 3\)
- For \(q=3\): \(\frac{\pm 1}{3} = \pm \frac{1}{3}, \frac{\pm 3}{3} = \pm 1\) (duplicate)
- For \(q=5\): \(\frac{\pm 1}{5} = \pm \frac{1}{5}, \frac{\pm 3}{5} = \pm \frac{3}{5}\)
- For \(q=15\): \(\frac{\pm 1}{15} = \pm \frac{1}{15}, \frac{\pm 3}{15} = \pm \frac{1}{5}\) (duplicate)
- Comparison with Options and Conclusion
Collecting all unique values identified in the previous step, we obtain the complete set of potential rational roots: \(\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3\). Comparing this list with the provided options, we see that it exactly matches option (d). Therefore, the potential rational roots are \(\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3\).
Examples
Understanding the Rational Root Theorem is essential in algebra when you need to solve high-degree polynomial equations. For example, if an engineer is modeling the motion of a bridge under wind loads and derives a complex polynomial, testing these potential rational roots is often the first step in simplifying the equation into manageable factors. By narrowing down the search, you can efficiently identify the exact points where the function equals zero, which is crucial for structural integrity and safety analysis.