- Apply the Rational Root Theorem to determine potential rational roots of \(f(x) = 3x^3 + x^2 - 38x + 24\).
- Eliminate \(\frac{1}{2}\) and \(-\frac{1}{4}\) as they do not satisfy the condition that the denominator must divide the leading coefficient.
- Evaluate the remaining candidates \(-8, \frac{2}{3}, 4, -\frac{1}{3}\) by substituting them into the function \(f(x)\).
- Confirm that \(\boxed{\frac{2}{3}}\) is the only rational root from the list.
Explanation
- Problem Analysis and the Rational Root Theorem
The function provided is \(f(x) = 3x^3 + x^2 - 38x + 24\). To find its rational roots using the Rational Root Theorem, we identify that for any rational root \(\frac{p}{q}\) in simplest form, the numerator \(p\) must be a divisor of the constant term \(a_0 = 24\), and the denominator \(q\) must be a divisor of the leading coefficient \(a_n = 3\).
The divisors of \(24\) are \(\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\}\) and the divisors of \(3\) are \(\{1, 3\}\). Thus, all potential rational roots must be in the set formed by quotients \(\frac{p}{q}\).
- Identifying Potential Candidates
We test the given candidates against the Rational Root Theorem:
- \(-8\): \(p = -8, q = 1\). Here, \(-8\) is a divisor of \(24\) and \(1\) is a divisor of \(3\). It is a valid candidate.
- \(\frac{2}{3}\): \(p = 2, q = 3\). Here, \(2\) is a divisor of \(24\) and \(3\) is a divisor of \(3\). It is a valid candidate.
- \(4\): \(p = 4, q = 1\). Here, \(4\) is a divisor of \(24\) and \(1\) is a divisor of \(3\). It is a valid candidate.
- \(\frac{1}{2}\): \(p = 1, q = 2\). Here, \(2\) is not a divisor of \(3\). It is NOT a valid candidate.
- \(-\frac{1}{3}\): \(p = -1, q = 3\). Here, \(-1\) is a divisor of \(24\) and \(3\) is a divisor of \(3\). It is a valid candidate.
- \(-\frac{1}{4}\): \(p = -1, q = 4\). Here, \(4\) is not a divisor of \(3\). It is NOT a valid candidate.
- Evaluating Potential Candidates and Conclusion
Now we evaluate \(f(x) = 3x^3 + x^2 - 38x + 24\) at each of the valid candidates to see which are actual roots:
- For \(-8\): \(f(-8) = 3(-512) + 64 - 38(-8) + 24 = -1536 + 64 + 304 + 24 = -1144
eq 0\).
- For \(\frac{2}{3}\): \(f\left(\frac{2}{3}\right) = 3\left(\frac{8}{27}\right) + \left(\frac{4}{9}\right) - 38\left(\frac{2}{3}\right) + 24 = \frac{8}{9} + \frac{4}{9} - \frac{76}{3} + 24 = \frac{12}{9} - \frac{228}{9} + \frac{216}{9} = \frac{0}{9} = 0\).
- For \(4\): \(f(4) = 3(64) + 16 - 38(4) + 24 = 192 + 16 - 152 + 24 = 80
eq 0\).
- For \(-\frac{1}{3}\): \(f\left(-\frac{1}{3}\right) = 3\left(-\frac{1}{27}\right) + \frac{1}{9} - 38\left(-\frac{1}{3}\right) + 24 = -\frac{1}{9} + \frac{1}{9} + \frac{38}{3} + 24
eq 0\).
The only value from the provided list that is a rational root of the function is \(\frac{2}{3}\).
Examples
In engineering and physics, finding the roots of a polynomial is essential for understanding the stability of systems or the points where a physical quantity becomes zero, such as the equilibrium position of a mechanical system. The Rational Root Theorem provides an efficient way to narrow down the search space for these critical points, saving time and computational effort in complex design processes. By testing only the potential rational values, engineers can quickly identify candidates and verify them, facilitating more streamlined calculations in fields like control theory or structural analysis.