Final answer:
The Rational Root Theorem relates to the polynomial f(x) to establish that any rational roots must be a ratio of factors of the constant term to factors of the leading coefficient. For f(x) = 12x³ - 5x² + 6x + 9, this means the potential rational roots can be ±1, ±1/2, ±3/4, and other similar ratios.
Step-by-step explanation:
The question you're asking relates to the Rational Root Theorem, which is a principle in algebra that helps determine possible rational roots of a polynomial. The theorem states that if a polynomial equation with integer coefficients has any rational solutions (or roots), then each solution 'p/q' must be such that 'p' is a factor of the constant term and 'q' is a factor of the leading coefficient. This applies to the polynomial f(x) = 12x³ - 5x² + 6x + 9 you've provided.
Using the Rational Root Theorem, we first identify the factors of the constant term (9) and the leading coefficient (12). The factors of 9 are ±1, ±3, and ±9, and the factors of 12 are ±1, ±2, ±3, ±4, ±6, and ±12. The possible rational roots are the ratios of the factors of the constant term to the factors of the leading coefficient, which means we can have ±1, ±1/2, ±3/4, etc., as potential roots of the polynomial.