Question

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

any rational root of \\(f(x)\\) is a multiple of 12 divided by a multiple of 9.
any rational root of \\(f(x)\\) is a multiple of 9 divided by a multiple of 12.
any rational root of \\(f(x)\\) is a factor of 12 divided by a factor of 9.
any rational root of \\(f(x)\\) is a factor of 9 divided by a factor of 12.

Explanation:

Identify the coefficients of the polynomial

The given polynomial is:

$$f(x) = 12x^3 - 5x^2 + 6x + 9$$

The leading coefficient is \(a_n = 12\) and the constant term is \(a_0 = 9\).

Apply the Rational Root Theorem

According to the Rational Root Theorem, any rational root of a polynomial with integer coefficients must be of the form \(\frac{p}{q}\), where:

• \(p\) is a factor of the constant term \(a_0 = 9\).
• \(q\) is a factor of the leading coefficient \(a_n = 12\).

Match with the given statements

The rational root is a factor of 9 divided by a factor of 12.
This matches the fourth option.

Answer:

• Any rational root of \(f(x)\) is a multiple of 12 divided by a multiple of 9.
• Any rational root of \(f(x)\) is a multiple of 9 divided by a multiple of 12.
• Any rational root of \(f(x)\) is a factor of 12 divided by a factor of 9.
• Any rational root of \(f(x)\) is a factor of 9 divided by a factor of 12. (Correct answer)