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