To find the correct statement about the rational roots of the polynomial [tex]\( f(x) = 12x^3 - 5x^2 + 6x + 9 \)[/tex], we will use the Rational Root Theorem. Let's go through this step-by-step:
1. Identify the coefficients:
- The polynomial [tex]\( f(x) \)[/tex] is given as [tex]\( 12x^3 - 5x^2 + 6x + 9 \)[/tex].
- The leading coefficient [tex]\( a_n \)[/tex] is the coefficient of the highest degree term, which is 12.
- The constant term [tex]\( a_0 \)[/tex] is the term without [tex]\( x \)[/tex], which is 9.
2. List the factors:
- The factors of the constant term (9) are: [tex]\( \pm 1, \pm 3, \pm 9 \)[/tex].
- The factors of the leading coefficient (12) are: [tex]\( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)[/tex].
3. Rational Root Theorem:
- According to the Rational Root Theorem, any rational root of the polynomial, in the form [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient.
4. Analyze the statements:
- The statement "Any rational root of [tex]\( f(x) \)[/tex] is a multiple of 12 divided by a multiple of 9" is incorrect because it talks about multiples rather than factors.
- The statement "Any rational root of [tex]\( f(x) \)[/tex] is a multiple of 9 divided by a multiple of 12" is again incorrect because it's about multiples.
- The statement "Any rational root of [tex]\( f(x) \)[/tex] is a factor of 12 divided by a factor of 9" is incorrect because it reverses the roles of [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
- The correct statement is "Any rational root of [tex]\( f(x) \)[/tex] is a factor of 9 divided by a factor of 12." This is precisely what the Rational Root Theorem states.
Thus, the correct statement is:
Any rational root of [tex]\( f(x) \)[/tex] is a factor of 9 divided by a factor of 12.
