- The problem requires finding a missing root of a polynomial with rational coefficients given the root \(3+\sqrt{11}\).
- According to the Conjugate Root Theorem for polynomials with rational coefficients, if \(a+\sqrt{b}\) is a root, then \(a-\sqrt{b}\) must also be a root.
- The root \(3+\sqrt{11}\) has the conjugate \(3-\sqrt{11}\).
- Therefore, the required root is \(\boxed{3-\sqrt{11}}\).
Explanation
- Analysis of the Problem
We are given a polynomial function \(f(x)\) with rational coefficients. We are told that \(0\), \(4\), and \(3+\sqrt{11}\) are roots of this polynomial. Our goal is to determine another value that must also be a root of \(f(x)\) based on the properties of polynomials with rational coefficients.
- Applying the Conjugate Root Theorem
The fundamental theorem governing this problem is the Conjugate Root Theorem for polynomials with rational coefficients. It states that if a polynomial \(f(x)\) has rational coefficients and an irrational number of the form \(a + \sqrt{b}\) is a root (where \(a, b \in \mathbb{Q}\) and \(\sqrt{b}\) is irrational), then its conjugate \(a - \sqrt{b}\) must also be a root of the polynomial. In our given root \(3 + \sqrt{11}\), we have \(a = 3\) and \(b = 11\), both of which are rational numbers. Since \(\sqrt{11}\) is irrational, the conditions for the theorem are met.
- Identifying the Mandatory Root
Applying the theorem to the root \(3 + \sqrt{11}\), the conjugate is \(3 - \sqrt{11}\). Because the polynomial \(f(x)\) has rational coefficients, this conjugate must necessarily be a root of the polynomial. Comparing this to the given options, we find that \(3 - \sqrt{11}\) is listed as one of the choices. Thus, \(3 - \sqrt{11}\) must also be a root of \(f(x)\).
Examples
This concept is useful in various fields of engineering and science where modeling phenomena with polynomials is required. For example, in control theory, the stability of a system is determined by the roots of a characteristic polynomial. Knowing that irrational roots must occur in conjugate pairs allows engineers to quickly identify all roots of a system's model, ensuring complete and accurate analysis of the system's behavior.