To find the correct factor of the polynomial [tex]\( f(x) \)[/tex], we begin by considering the given roots of the polynomial: [tex]\(-9\)[/tex] and [tex]\(7-i\)[/tex].
First, we should understand that if a polynomial has complex roots, the complex conjugate of those roots must also be roots of the polynomial, assuming the polynomial has real coefficients. Therefore, the conjugate of [tex]\(7 - i\)[/tex], which is [tex]\(7 + i\)[/tex], must also be a root of the polynomial [tex]\( f(x) \)[/tex].
Given these roots, we can determine the corresponding factors of the polynomial:
1. The root [tex]\(-9\)[/tex] implies that [tex]\((x - (-9))\)[/tex] or [tex]\((x + 9)\)[/tex] is a factor.
2. The root [tex]\(7 - i\)[/tex] implies that [tex]\((x - (7 - i))\)[/tex] is a factor.
3. The conjugate root [tex]\(7 + i\)[/tex] implies that [tex]\((x - (7 + i))\)[/tex] is also a factor.
Out of the options given, we need to identify the correct factor corresponding to the root [tex]\(7 - i\)[/tex]:
- The factor corresponding to [tex]\((7 - i)\)[/tex] as a root is [tex]\((x - (7 - i))\)[/tex].
- Simplifying [tex]\((x - (7 - i))\)[/tex], we get:
[tex]\[
x - (7 - i) = x - 7 + i
\][/tex]
Let's review the options and see which one matches this form:
- [tex]\((x - (7 + i))\)[/tex] does not match. It corresponds to the conjugate root [tex]\((7 + i)\)[/tex].
- [tex]\((x - (-7 - i))\)[/tex] is not relevant as it is not tied to any given or deduced root.
- [tex]\((x + (7 + i))\)[/tex] is also not relevant since it’s not linked to any of the roots either.
- [tex]\((x + (7 - i))\)[/tex] simplifies to [tex]\(x - 7 + i\)[/tex], again not matching our required form.
So the correct factor should be:
[tex]\[
(x - (7 - i)) = x - 7 + i
\][/tex]
Thus, the specific factor of the polynomial [tex]\( f(x) \)[/tex] given that [tex]\(7 - i\)[/tex] is a root, is:
[tex]\[
\boxed{x - (7 - i)}
\][/tex]
This matches with the given first option [tex]\((x - (7 - i))\)[/tex] as highlighted by the analysis.
