Given that [tex]\(-3+i\)[/tex] is a root of the polynomial function [tex]\(f(x)\)[/tex], we need to identify which other number must also be a root of [tex]\(f(x)\)[/tex].
For polynomials with real coefficients, if a complex number [tex]\(a+bi\)[/tex] (where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers and [tex]\(b \neq 0\)[/tex]) is a root, then its complex conjugate [tex]\(a-bi\)[/tex] must also be a root. This happens because the coefficients of the polynomial are real.
Given that the root is [tex]\(-3+i\)[/tex]:
1. The real part of the root is [tex]\(-3\)[/tex].
2. The imaginary part of the root is [tex]\(i\)[/tex].
The complex conjugate of [tex]\(-3+i\)[/tex] is obtained by changing the sign of the imaginary part:
[tex]\[
-3+i \rightarrow -3-i
\][/tex]
Therefore, [tex]\(-3-i\)[/tex] must also be a root of the polynomial [tex]\(f(x)\)[/tex].
So the correct answer is:
[tex]\(-3-i\)[/tex]
