Verified Answer :
- Use the Rational Root Theorem, which states that any rational root of a polynomial with integer coefficients must be a divisor of the constant term divided by a divisor of the leading coefficient.
- With the polynomial \(p(x) = x^4 - 9x^2 - 4x + 12\), the leading coefficient is 1 and the constant term is 12, meaning all rational roots must be divisors of 12.
- Evaluate the options \(\pm 2, \pm 6, \pm 9, \pm 4, \pm 12, 0, \pm 3, \frac{1}{2}\) to see which are divisors of 12.
- Five valid potential roots are \(\boxed{\pm 2, \pm 3, \pm 4, \pm 6, \pm 12}\).
Explanation
- Analyzing the Rational Root Theorem
We are given the polynomial \(p(x) = x^4 - 9x^2 - 4x + 12\). According to the Rational Root Theorem, for a polynomial with integer coefficients \(P(x) = a_n x^n + \dots + a_0\), any rational root \(p/q\) must have \(p\) as a divisor of the constant term \(a_0\) and \(q\) as a divisor of the leading coefficient \(a_n\). In our case, the leading coefficient is \(a_n = 1\) and the constant term is \(a_0 = 12\). Therefore, any rational root must be an integer that divides 12. The divisors of 12 are \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\). Any rational root must be one of these.
- Evaluating the Options
We need to check which of the provided options: \(\pm 2, \pm 6, \pm 9, \pm 4, \pm 12, 0, \pm 3, \frac{1}{2}\) are potential roots based on the Rational Root Theorem.
- \(\pm 2\): Since 2 and -2 are divisors of 12, they are potential roots.
- \(\pm 6\): Since 6 and -6 are divisors of 12, they are potential roots.
- \(\pm 9\): 9 and -9 are not divisors of 12, so they are not potential roots.
- \(\pm 4\): Since 4 and -4 are divisors of 12, they are potential roots.
- \(\pm 12\): Since 12 and -12 are divisors of 12, they are potential roots.
- \(0\): \(p(0) = 12 eq 0\), and 0 is not a divisor, so it is not a root.
- \(\pm 3\): Since 3 and -3 are divisors of 12, they are potential roots.
- \(\frac{1}{2}\): \(\frac{1}{2}\) is not an integer, and the leading coefficient is 1, so the only rational roots must be integers. Thus, \(\frac{1}{2}\) is not a potential root.
- Selecting Five Potential Roots
Based on the criteria, the potential roots from the given list are \(\pm 2, \pm 6, \pm 4, \pm 12, \pm 3\). We can choose any five from this set to answer the question. A valid set of five potential roots is \(\boxed{\pm 2, \pm 4, \pm 3, \pm 6, \pm 12}\).
Examples
The Rational Root Theorem is a powerful tool in algebra used to narrow down the search for roots of polynomial equations. For instance, if an engineer is designing a mechanism whose motion is described by a high-degree polynomial, identifying its roots can tell the engineer where the system reaches equilibrium or crosses zero values. By checking only the divisors of the constant and leading terms, the engineer saves significant time and effort compared to testing arbitrary numbers.