Question

according to the rational root theorem, which function has the same set of potential rational roots as the function \\(g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12\\)?

\\(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\\)
\\(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\\)
\\(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\\)
\\(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\\)

Explanation:

Identify the key parameters of the given function

The given function is:

$$g(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12$$

The leading coefficient is \(a_n = 3\).
The constant term is \(a_0 = 12\).

State the Rational Root Theorem

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

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

Thus, the set of potential rational roots depends solely on the factors of the constant term and the leading coefficient.

Analyze the options for matching coefficients

We need to find a function \(f(x)\) whose leading coefficient and constant term have the same set of factors (or ratio of factors) as \(g(x)\).

• For \(g(x)\): leading coefficient is \(3\), constant term is \(12\).
• Option 1: \(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\)

Leading coefficient is \(3\), constant term is \(-12\).
The factors of \(3\) are \(\pm 1, \pm 3\).
The factors of \(-12\) are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\).
This yields the exact same set of potential rational roots \(\pm \frac{p}{q}\).

Verify other options

• Option 2: \(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)

The constant term is \(0\), which does not yield the same set.

• Option 3: \(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)

Leading coefficient is \(12\), constant term is \(3\). This reverses the roles of \(p\) and \(q\), yielding a different set of potential roots.

• Option 4: \(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)

Leading coefficient is \(12\), constant term is \(48\). This yields a different set of factors.

Conclude the correct option

Option 1 has the leading coefficient \(3\) and constant term \(-12\). Since the signs do not affect the set of factors, the set of potential rational roots is identical to that of \(g(x)\).

Answer:

• (A) \(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\) (Correct answer)
• (B) \(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\)
• (C) \(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\)
• (D) \(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\)