according to the rational root theorem, which statement about f(x) = 12x^3 - 5x^2 + 6x + 9 is true? any…

Explanation:

Step1: Recall Rational Root Theorem

For a polynomial (a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0), the rational roots are of the form (\frac{p}{q}), where (p) is a factor of the constant term (a_0) and (q) is a factor of the leading - coefficient (a_n).

Step2: Identify (a_0) and (a_n) in (f(x))

In the polynomial (f(x)=12x^{3}-5x^{2}+6x + 9), the leading - coefficient (a_n = 12) and the constant term (a_0=9).

Step3: Determine the form of rational roots

By the Rational Root Theorem, any rational root of (f(x)) is of the form (\frac{p}{q}), where (p) is a factor of (9) and (q) is a factor of (12), i.e., any rational root of (f(x)) is a factor of (9) divided by a factor of (12).

Answer:

Any rational root of (f(x)) is a factor of 9 divided by a factor of 12.