To determine the potential rational roots of the polynomial [tex]\( f(x) = 9x^8 + 9x^6 - 12x + 7 \)[/tex] using the Rational Root Theorem, we follow a series of steps.
1. Identify the constant term (p) and the leading coefficient (q):
- The constant term [tex]\( p \)[/tex] is 7.
- The leading coefficient [tex]\( q \)[/tex] is 9.
2. Find the factors of the constant term (p):
- The factors of 7 are [tex]\( \pm 1, \pm 7 \)[/tex].
3. Find the factors of the leading coefficient (q):
- The factors of 9 are [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
4. Form all possible fractions [tex]\( \frac{p}{q} \)[/tex]:
- Given the factors of [tex]\( p \)[/tex] and [tex]\( q \)[/tex], we form the following potential rational roots:
- [tex]\( \frac{1}{1} \)[/tex], [tex]\( \frac{1}{3} \)[/tex], [tex]\( \frac{1}{9} \)[/tex],
- [tex]\( \frac{7}{1} \)[/tex], [tex]\( \frac{7}{3} \)[/tex], [tex]\( \frac{7}{9} \)[/tex],
- Negative counterparts: [tex]\( -\frac{1}{1} \)[/tex], [tex]\( -\frac{1}{3} \)[/tex], [tex]\( -\frac{1}{9} \)[/tex],
[tex]\( -\frac{7}{1} \)[/tex], [tex]\( -\frac{7}{3} \)[/tex], [tex]\( -\frac{7}{9} \)[/tex].
This results in the set:
[tex]\[
\left\{\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 7, \pm \frac{7}{3}, \pm \frac{7}{9}\right\}
\][/tex]
Among these potential roots, we check our given options:
- [tex]\( 0 \)[/tex] (Zero)
- [tex]\( \frac{2}{7} \)[/tex]
- [tex]\( 2 \)[/tex]
- [tex]\( \frac{7}{3} \)[/tex]
From our set of potential rational roots, the fractional root [tex]\( \frac{7}{3} \)[/tex] is indeed present.
Thus, according to the Rational Root Theorem, the number which is a potential root of [tex]\( f(x) = 9x^8 + 9x^6 - 12x + 7 \)[/tex] is:
[tex]\[
\boxed{\frac{7}{3}}
\][/tex]
