A polynomial function $f(x)$ has roots $-8, 1$, and $6i$. What must also be a root of $f(x)$?
We are given a polynomial function \(f(x)\) with known roots \(-8\), \(1\), and \(6i\). Our objective is to determine another value that must also be a root of \(f(x)\). In mathematics, when we discuss polynomials, we typically work with polynomials that have real coefficients unless otherwise specified. A fundamental theorem, the Complex Conjugate Root Theorem, states that if a polynomial \(f(x)\) has real coefficients and a complex number \(z = a + bi\) is a root, then its complex conjugate \(\bar{z} = a - bi\) must also be a root of the polynomial.
The root \(6i\) can be written in the form \(a + bi\) where \(a = 0\) and \(b = 6\). Applying the Complex Conjugate Root Theorem, we calculate the conjugate of \(6i\). The complex conjugate \(\bar{z}\) of \(z = a + bi\) is given by \(a - bi\). For \(z = 0 + 6i\), the conjugate is \(\bar{z} = 0 - 6i\), which is simply \(-6i\). Therefore, if \(6i\) is a root of the polynomial \(f(x)\) with real coefficients, \(-6i\) must also be a root of \(f(x)\).
Comparing our result with the given choices, we see that \(-6i\) is one of the options. Based on our analysis using the Complex Conjugate Root Theorem, we conclude that \(-6i\) is a root of the polynomial function \(f(x)\). The final answer is \(-6i\).
Understanding the properties of roots is essential in fields like electrical engineering and control theory, where polynomials are used to model the stability of systems. For example, when analyzing a circuit, the roots of the characteristic equation (poles) determine how the voltage or current behaves over time. Knowing that complex poles always appear in conjugate pairs for physical systems allows engineers to simplify their calculations and ensure the system remains stable.