According to the Rational Root Theorem, what are all the potential rational roots of [tex]f(x)=15x^{11}-6x^8+x^3-4x+3[/tex]?

A. [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]

B. [tex]\pm \frac{1}{3}, \pm 1, \pm \frac{5}{3}, \pm 3, \pm 5, \pm 15[/tex]

C. [tex]\pm \frac{1}{15}, \pm \frac{1}{5}, \pm \frac{1}{3}, \pm \frac{3}{5}, \pm 1, \pm 3[/tex]

D. [tex]\pm \frac{1}{3}, \pm 1, \pm 3[/tex]



Answer :

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]