To determine which of the given options must also be a root of the polynomial function [tex]\( f(x) \)[/tex], we first need to understand an important property of polynomials with real coefficients:
Complex Conjugate Root Theorem: If a polynomial has real coefficients and a complex number [tex]\( a + bi \)[/tex] (where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are real numbers) is a root of the polynomial, then its complex conjugate [tex]\( a - bi \)[/tex] must also be a root.
Given that [tex]\( -3 + i \)[/tex] is a root of the polynomial function [tex]\( f(x) \)[/tex] and the coefficients of [tex]\( f(x) \)[/tex] are real, the complex conjugate of [tex]\( -3 + i \)[/tex], which is [tex]\( -3 - i \)[/tex], must also be a root of [tex]\( f(x) \)[/tex].
Hence, the correct answer is:
- [tex]\( -3 - i \)[/tex]
Therefore, among the provided options, the root that must also be a root of [tex]\( f(x) \)[/tex] is:
[tex]\[ -3 - i \][/tex]
