- Identify the leading coefficient \(a_n=3\) and constant term \(a_0=12\) for the original function \(q(x)\).
- Determine the sets of potential rational roots based on the divisors of these two terms.
- Compare these sets with the coefficients of the provided options (a, b, c, and d).
- Select the option whose coefficients result in the same set of divisor ratios as the original function, which is option a.
- Final answer: \(\boxed{a}\)
Explanation
- Analyzing the Rational Root Theorem for q(x)
The Rational Root Theorem states that for any polynomial \(P(x) = a_n x^n + \dots + a_1 x + a_0\) with integer coefficients, any rational root \(p/q\) (in lowest terms) must satisfy \(p | a_0\) and \(q | a_n\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\).
For the given function \(q(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12\), the leading coefficient is \(a_n = 3\) and the constant term is \(a_0 = 12\). The set of possible values for \(q\) (divisors of 3) is \(\{1, 3\}\) and the set of possible values for \(p\) (divisors of 12) is \(\{1, 2, 3, 4, 6, 12\}\). The set of potential rational roots is the set of all possible ratios \(\pm p/q\).
- Comparing the Options
We analyze each provided function to see if it results in the same set of potential rational roots.
- For option (a), \(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\):
- The leading coefficient is \(a_n = 3\). Its divisors are \(\{1, 3\}\).
- The constant term is \(a_0 = -12\). Its divisors are \(\{1, 2, 3, 4, 6, 12\}\).
- The set of potential rational roots \(\pm p/q\) is formed by ratios of these sets of divisors. Since the sets of divisors for the leading coefficient and the constant term are identical to those of \(q(x)\), the set of potential rational roots for \(f(x)\) is the same as for \(q(x)\).
- For option (b), \(f(x) = 12x^5 - 8x^4 + 36x^3 - 4x^2 + 48\):
- The leading coefficient is \(a_n = 12\). The set of divisors is \(\{1, 2, 3, 4, 6, 12\}\).
- The constant term is \(a_0 = 48\). The set of divisors is \(\{1, 2, 3, 4, 6, 8, 12, 16, 24, 48\}\).
- This set of divisors is much larger than the original set, leading to a different set of potential rational roots.
- For option (c), \(f(x) = 12x^5 - 2x^4 + 9x^3 - x^2 + 3\):
- Here, the leading coefficient is 12 and the constant term is 3, reversing the roles of the original coefficients, which leads to a different set of potential roots.
- For option (d), \(f(x) = 3x^6 - 2x^5 + 9x^4 - x^3 + 12x\):
- The constant term is 0, which means \(x=0\) is a root and the polynomial can be written as \(x(3x^5 - 2x^4 + 9x^3 - x^2 + 12)\). This changes the degree and the constant term analysis of the polynomial itself.
- Conclusion
Based on the analysis, option (a) provides the same leading coefficient and constant term (up to a sign), meaning the set of divisors for \(p\) and \(q\) remains unchanged. Therefore, the set of potential rational roots for \(f(x) = 3x^5 - 2x^4 - 9x^3 + x^2 - 12\) is identical to that of the original function \(q(x) = 3x^5 - 2x^4 + 9x^3 - x^2 + 12\). The correct option is a.
Examples
The Rational Root Theorem is a powerful tool in algebra used to simplify the process of solving higher-degree polynomial equations. For instance, an engineer designing the shape of a structural support might end up with a polynomial equation representing the stress distribution across the beam. Instead of guessing roots blindly, the engineer can use this theorem to list a finite set of candidates that could potentially be the solution, effectively turning an infinite search into a systematic check of a small, finite list of rational numbers.