To determine the possible rational zeros of the polynomial function [tex]\( n(x) = 9x^7 + 19x^3 - 6x^2 - 6 \)[/tex], we use the Rational Root Theorem. The Rational Root Theorem states that any rational root, expressed as [tex]\(\frac{p}{q}\)[/tex], must satisfy the following conditions:
- [tex]\(p\)[/tex] is a factor of the constant term of the polynomial.
- [tex]\(q\)[/tex] is a factor of the leading coefficient of the polynomial.
For the polynomial [tex]\( n(x) \)[/tex], the constant term ([tex]\(a_0\)[/tex]) is [tex]\(-6\)[/tex], and the leading coefficient ([tex]\(a_n\)[/tex]) is [tex]\(9\)[/tex].
1. Determine the factors of the constant term ([tex]\(a_0 = -6\)[/tex]):
The factors of [tex]\(-6\)[/tex] are:
[tex]\[
\pm 1, \pm 2, \pm 3, \pm 6
\][/tex]
2. Determine the factors of the leading coefficient ([tex]\(a_n = 9\)[/tex]):
The factors of [tex]\(9\)[/tex] are:
[tex]\[
\pm 1, \pm 3, \pm 9
\][/tex]
3. Form the possible rational zeros:
Possible rational zeros are given by [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. Therefore, the possible rational zeros are:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{9}, \pm 3, \pm \frac{3}{3}, \pm \frac{3}{9}, \pm 6, \pm \frac{6}{3}, \pm \frac{6}{9}
\][/tex]
4. Simplify the list:
Simplifying the fractions, we get:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{9}, \pm 3, \pm \frac{6}{3}=\pm 2, \pm \frac{6}{9}=\pm \frac{2}{3}, \pm 6
\][/tex]
Removing duplicates, we have:
[tex]\[
\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 2, \pm \frac{2}{3}, \pm \frac{2}{9}, \pm 3, \pm 6
\][/tex]
If no rational zeros exist among these candidates, we would denote this by "None". After analyzing the polynomial and determining which rational zeros are valid, we can assert the final answer.
The results indicate that none of these candidates are actual rational zeros for the polynomial [tex]\( n(x) \)[/tex].
Thus, the possible rational zeros of [tex]\( n(x) \)[/tex] are:
[tex]\[
\text{None}
\][/tex]
