- Identify the divisors of the constant term \(4\) as \(p \in \{\pm 1, \pm 2, \pm 4\}\).
- Identify the divisors of the leading coefficient \(9\) as \(q \in \{1, 3, 9\}\).
- Use the Rational Root Theorem to form the set of potential roots \(\frac{p}{q}\).
- Match the resulting set \(\{\pm \frac{1}{9}, \pm \frac{2}{9}, \pm \frac{1}{3}, \pm \frac{4}{9}, \pm \frac{2}{3}, \pm 1, \pm \frac{4}{3}, \pm 2, \pm 4\}\) with option \(\boxed{c}\).
Explanation
- Applying the Rational Root Theorem
To find the potential rational roots of the polynomial \(f(x) = 9x^4 - 2x^2 - 3x + 4\), we use the Rational Root Theorem. This theorem states that if a polynomial with integer coefficients \(a_n x^n + \dots + a_1 x + a_0\) has a rational root \(\frac{p}{q}\) in lowest terms, then \(p\) must be a divisor of the constant term \(a_0\) and \(q\) must be a divisor of the leading coefficient \(a_n\). Here, the constant term is \(a_0 = 4\) and the leading coefficient is \(a_n = 9\).
- Identifying Divisors
First, identify all divisors of the constant term \(a_0 = 4\). These are \(p \in \{\pm 1, \pm 2, \pm 4\}\). Next, identify all positive divisors of the leading coefficient \(a_n = 9\). These are \(q \in \{1, 3, 9\}\). The set of potential rational roots \(\frac{p}{q}\) consists of all possible combinations of these divisors, considering both positive and negative values.
- Evaluating Potential Rational Roots and Concluding
By forming all combinations of \(\frac{p}{q}\) from the sets \(p \in \{\pm 1, \pm 2, \pm 4\}\) and \(q \in \{1, 3, 9\}\), we get:
- For \(q=1\): \(\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{4}{1}\), which are \(\pm 1, \pm 2, \pm 4\).
- For \(q=3\): \(\pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}\).
- For \(q=9\): \(\pm \frac{1}{9}, \pm \frac{2}{9}, \pm \frac{4}{9}\).
Combining all these, we obtain the list: \(\pm \frac{1}{9}, \pm \frac{2}{9}, \pm \frac{1}{3}, \pm \frac{4}{9}, \pm \frac{2}{3}, \pm 1, \pm \frac{4}{3}, \pm 2, \pm 4\). This set perfectly matches option (c).
Examples
The Rational Root Theorem is a powerful tool in algebra for identifying potential rational solutions to polynomial equations, which is essential in fields like engineering and physics when modeling system behavior. For example, if you are designing a control system and need to find the specific time values where the system output crosses a reference point (represented as roots of a high-degree polynomial), this theorem helps narrow down the search space for exact solutions. By systematically testing these potential rational roots, you can solve complex design problems more efficiently.