Answer-
[tex]\boxed{\boxed{\frac{7}{3}}}[/tex]
Solution-
Rational Root Theorem-
[tex]f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.......+a_1x+a_0\ \ \ and\ a_n\neq 0[/tex]
All the potential rational roots are,
[tex]=\pm (\dfrac{\text{factors of}\ a_0}{\text{factors of}\ a_n})[/tex]
The given polynomial is,
[tex]f(x) = 9x^8 + 9x^6-12x + 7[/tex]
Here,
[tex]a_n=9,\ a_0=7\\\\\text{factors of}\ 9=1,3,9\\\\\text{factors of}\ 7=1,7[/tex]
The potential rational roots are,
[tex]=\pm \frac{1}{1},\pm \frac{1}{3}, \pm \frac{1}{9}, \pm \frac{7}{1}, \pm \frac{7}{3}, \pm \frac{7}{9}[/tex]
[tex]=\pm 1,\pm \frac{1}{3}, \pm \frac{1}{9}, \pm 7, \pm \frac{7}{3}, \pm \frac{7}{9}[/tex]
From, the given options only [tex]\frac{7}{3}[/tex] satisfies.
