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If [tex]\(-3+i\)[/tex] is a root of the polynomial function [tex]\(f(x)\)[/tex], which of the following must also be a root of [tex]\(f(x)\)[/tex]?

A. [tex]\(-3-i\)[/tex]
B. [tex]\(-3i\)[/tex]
C. [tex]\(3-i\)[/tex]
D. [tex]\(3i\)[/tex]


Sagot :

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]