- A polynomial with real coefficients must have complex roots that occur in conjugate pairs.
- Given the root \(-2 + \sqrt{3}i\), the complex conjugate theorem dictates that its conjugate, \(-2 - \sqrt{3}i\), must also be a root.
- Comparing this to the given choices, the statement that \(-2 - \sqrt{3}i\) is a known root is correct.
- Final answer: \(\boxed{c}\)
Explanation
- Understanding the Problem and Data Analysis
We are given a function with real coefficients that has a root \(-2 + \sqrt{3}i\). Our objective is to identify the correct statement about other potential roots of this function based on the provided options. The fundamental theorem of algebra and properties of polynomials with real coefficients will guide our reasoning.
- Applying the Complex Conjugate Root Theorem
For any polynomial function with real coefficients, complex roots must occur in conjugate pairs. This is known as the Complex Conjugate Root Theorem. Specifically, if a complex number \(a + bi\) (where \(a\) and \(b\) are real numbers and \(b
eq 0\)) is a root of a polynomial with real coefficients, then its complex conjugate, \(a - bi\), is also a root of the same polynomial.
- Evaluating the Options and Conclusion
Given the complex root \(z = -2 + \sqrt{3}i\), we can identify its real and imaginary parts as \(a = -2\) and \(b = \sqrt{3}\). According to the Complex Conjugate Root Theorem, the conjugate of \(z\), denoted as \(\bar{z} = a - bi\), must also be a root. Calculating the conjugate, we get \(\bar{z} = -2 - \sqrt{3}i\). Therefore, \(-2 - \sqrt{3}i\) is a known root of the function. Comparing this with the provided choices, option (c) correctly states this conclusion.
Examples
The concept of complex conjugate roots is essential in engineering, particularly in control systems and electrical circuit analysis. For instance, when analyzing the stability of a system, the roots of the system's characteristic equation represent its natural frequencies and damping behavior. If a system has real coefficients, its complex poles must come in conjugate pairs to ensure the resulting physical response (like voltage or displacement) is expressed purely in real-valued terms, preventing impossible physical outputs like imaginary time-domain oscillations.