To determine which of the given options must also be a root of the polynomial function [tex]\( f(x) \)[/tex], we need to consider the properties of polynomial functions with real coefficients.
1. Polynomials with real coefficients have complex roots that occur in conjugate pairs. This means that if the polynomial [tex]\( f(x) \)[/tex] has a complex root [tex]\( a + bi \)[/tex], the conjugate [tex]\( a - bi \)[/tex] must also be a root.
2. In this case, we are given that [tex]\( -3 + i \)[/tex] is a root of [tex]\( f(x) \)[/tex].
3. To find the conjugate of [tex]\( -3 + i \)[/tex]:
- The real part of [tex]\( -3 + i \)[/tex] is [tex]\( -3 \)[/tex].
- The imaginary part of [tex]\( -3 + i \)[/tex] is [tex]\( i \)[/tex].
The conjugate is found by changing the sign of the imaginary part while keeping the real part the same:
- Therefore, the conjugate of [tex]\( -3 + i \)[/tex] is [tex]\( -3 - i \)[/tex].
4. Given that polynomials with real coefficients have roots in conjugate pairs, if [tex]\( -3 + i \)[/tex] is a root, [tex]\( -3 - i \)[/tex] must also be a root.
Therefore, the correct answer is:
[tex]\[ -3 - i \][/tex]
