To determine the potential rational roots of the polynomial [tex]\( f(x) = 15x^{11} - 6x^8 + x^3 - 4x + 3 \)[/tex], we can use the Rational Root Theorem. This theorem states that if a polynomial has any rational roots [tex]\(\frac{p}{q}\)[/tex], then [tex]\(p\)[/tex] must be a factor of the constant term, and [tex]\(q\)[/tex] must be a factor of the leading coefficient.
Here are the steps to find the potential rational roots:
1. Identify the constant term and the leading coefficient:
- The constant term of the polynomial is [tex]\(3\)[/tex].
- The leading coefficient of the polynomial is [tex]\(15\)[/tex].
2. List the factors of the constant term and the leading coefficient:
- Factors of [tex]\(3\)[/tex] (constant term): [tex]\(\pm 1, \pm 3\)[/tex]
- Factors of [tex]\(15\)[/tex] (leading coefficient): [tex]\(\pm 1, \pm 3, \pm 5, \pm 15\)[/tex]
3. Generate all possible fractions [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term and [tex]\(q\)[/tex] is a factor of the leading coefficient:
- [tex]\(\frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \frac{1}{15}\)[/tex]
- [tex]\(\frac{3}{1}, \frac{3}{3}, \frac{3}{5}, \frac{3}{15}\)[/tex]
- Include both the positive and negative versions of these fractions.
4. Simplify these fractions and list all distinct potential rational roots:
- [tex]\(\pm 1\)[/tex]
- [tex]\(\pm \frac{1}{3}\)[/tex]
- [tex]\(\pm \frac{1}{5}\)[/tex]
- [tex]\(\pm \frac{1}{15}\)[/tex]
- [tex]\(\pm 3\)[/tex]
- [tex]\(\pm \frac{3}{5}\)[/tex]
- [tex]\(\pm 5\)[/tex]
- [tex]\(\pm \frac{5}{3}\)[/tex]
- [tex]\(\pm 15\)[/tex]
Therefore, combining all these fractions and simplifying gives us the complete set of potential rational roots. The correct list of potential rational roots is:
[tex]\[
\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm \frac{5}{3}, \pm 3, \pm 5, \pm 15
\][/tex]
So, the answer is:
[tex]\[
\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm \frac{5}{3}, \pm 3, \pm 5, \pm 15
\][/tex]
