Question

According to the Rational Root Theorem, which number is a potential root of f(x) = 9x8 + 9x6 – 12x + 7?

Answer 1

Answer-

$\boxed{\boxed{\frac{7}{3}}}$

Solution-

Rational Root Theorem-

$f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.......+a_1x+a_0\ \ \ and\ a_n\neq 0$

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.

Answer 2

It is D. 7/3 because the rational root theorem states that when a root 'x' is written as a fraction x = p/q in lowest terms, p is an integer factor of the constant term (i.e. 7), and q is an integer factor of the coefficient of the first monomial (i.e. 9x^8).