Given the roots of the polynomial function [tex]\( f(x) \)[/tex] are [tex]\( -8 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 6i \)[/tex], we need to determine another root of the function.
### Step-by-Step Solution:
1. List the Given Roots:
- The roots given are [tex]\( -8 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( 6i \)[/tex].
2. Understanding Polynomial Roots:
- For polynomials with real coefficients, if a complex number is a root, its complex conjugate must also be a root.
- A complex number and its conjugate have the same real part and opposite imaginary parts.
- Here, [tex]\( 6i \)[/tex] is a complex number.
3. Identify the Complex Conjugate:
- The complex conjugate of [tex]\( 6i \)[/tex] is [tex]\( -6i \)[/tex].
4. Verify the Options:
- [tex]\( -6 \)[/tex]: Not a conjugate of [tex]\( 6i \)[/tex]; this is a real number.
- [tex]\( -6i \)[/tex]: This is the conjugate of [tex]\( 6i \)[/tex].
- [tex]\( 6-i \)[/tex]: This has a complex part but is not the conjugate of [tex]\( 6i \)[/tex].
- [tex]\( 6 \)[/tex]: This is a real number but not the conjugate.
5. Conclusion:
- Since [tex]\( -6i \)[/tex] is the conjugate of [tex]\( 6i \)[/tex], and for polynomials with real coefficients, complex roots occur in conjugate pairs. Therefore, [tex]\( -6i \)[/tex] must also be a root of [tex]\( f(x) \)[/tex].
Thus, the missing root for the polynomial function [tex]\( f(x) \)[/tex] must be [tex]\( -6i \)[/tex].
