Question

According to the Rational Root Theorem, which statement about f(x)=12x³-5x²+6x+9 is true?

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

Answer 1

The Rational Root Theorem suggests that for the polynomial f(x)=12x³-5x²+6x+9, any rational root will be a factor of the constant term 9 divided by a factor of the leading coefficient 12, which corresponds to option (d).

Step-by-step explanation:

The question focuses on determining which statement is true according to the Rational Root Theorem for the given polynomial f(x)=12x³-5x²+6x+9. The Rational Root Theorem states that any rational root of the polynomial equation, in the form of a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. In the case of f(x), the constant term is 9 and the leading coefficient is 12.

Therefore, any rational root of f(x) would be in the form of a factor of 9 divided by a factor of 12. This aligns with option (d): Any rational root of f(x) is a factor of 9 divided by a factor of 12.