To determine the potential rational roots of the polynomial [tex]\( f(x) = 15x^{11} - 6x^8 + x^3 - 4x + 3 \)[/tex] using the Rational Root Theorem, we follow these steps:
1. Identify the constant term ([tex]\( p \)[/tex]) and the leading coefficient ([tex]\( q \)[/tex]):
- The constant term ([tex]\( p \)[/tex]) is 3.
- The leading coefficient ([tex]\( q \)[/tex]) is 15.
2. List the factors of each:
- Factors of [tex]\( p = 3 \)[/tex]: [tex]\( \pm 1, \pm 3 \)[/tex]
- Factors of [tex]\( q = 15 \)[/tex]: [tex]\( \pm 1, \pm 3, \pm 5, \pm 15 \)[/tex]
3. Form all possible fractions [tex]\( \frac{p}{q} \)[/tex]:
- Using the factors of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], we create fractions [tex]\( \frac{p}{q} \)[/tex] which are the possible rational roots of the polynomial.
The possible combinations are:
- [tex]\( \frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \frac{1}{15}, \frac{3}{1}, \frac{3}{3}, \frac{3}{5}, \frac{3}{15} \)[/tex]
- Including both positive and negative values for each combination.
Simplifying these fractions, we get:
- [tex]\( \pm 1, \pm \frac{1}{3}, \pm \frac{1}{5}, \pm \frac{1}{15}, \pm 3, \pm \frac{3}{5} \)[/tex]
These are all possible rational roots of the given polynomial.
Given the provided choices, the correct list of all potential rational roots is:
[tex]\( \pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3 \)[/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3}
\][/tex]
