To approach this problem, we need to use the Rational Root Theorem. The Rational Root Theorem states that for a polynomial [tex]\( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \)[/tex], any potential rational root, [tex]\( \frac{p}{q} \)[/tex], must satisfy the following conditions:
- [tex]\( p \)[/tex] is a factor of the constant term [tex]\( a_0 \)[/tex].
- [tex]\( q \)[/tex] is a factor of the leading coefficient [tex]\( a_n \)[/tex].
For the given polynomial [tex]\( f(x) = 12x^3 - 5x^2 + 6x + 9 \)[/tex]:
1. Identify the constant term [tex]\( a_0 \)[/tex] and its factors:
- The constant term [tex]\( a_0 \)[/tex] is 9.
- The factors of 9 are [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
2. Identify the leading coefficient [tex]\( a_n \)[/tex] and its factors:
- The leading coefficient [tex]\( a_n \)[/tex] is 12.
- The factors of 12 are [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
According to the Rational Root Theorem, any rational root of [tex]\( f(x) \)[/tex] must be a ratio [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term (9) and [tex]\( q \)[/tex] is a factor of the leading coefficient (12).
Thus, any rational root of [tex]\( f(x) \)[/tex] is a factor of 9 divided by a factor of 12. This matches the fourth statement given in the problem.
Therefore, the correct statement is:
Any rational root of [tex]\( f(x) \)[/tex] is a factor of 9 divided by a factor of 12.
