To determine a factor of a polynomial given its roots, let's analyze the following:
1. Given Roots:
- One root is [tex]\(-9\)[/tex].
- Another root is [tex]\(7 - i\)[/tex].
2. Complex Conjugate Root Theorem:
- If the coefficients of the polynomial are real, then complex roots occur in conjugate pairs. Therefore, if [tex]\(7 - i\)[/tex] is a root, then [tex]\(7 + i\)[/tex] must also be a root.
3. Constructing Factors:
- For a root [tex]\(a\)[/tex], the corresponding factor of the polynomial is [tex]\((x - a)\)[/tex].
4. Factors Based on Given Roots:
- The root [tex]\(-9\)[/tex] corresponds to the factor [tex]\((x - (-9))\)[/tex] or [tex]\((x + 9)\)[/tex].
- The root [tex]\(7 - i\)[/tex] corresponds to the factor [tex]\((x - (7 - i))\)[/tex].
- The root [tex]\(7 + i\)[/tex] corresponds to the factor [tex]\((x - (7 + i))\)[/tex].
Given the possible answers, we need to identify which explicitly matches the form of one of our derived factors for these roots:
- [tex]\((x - (7 + i))\)[/tex]: This matches the factor corresponding to the root [tex]\(7 + i\)[/tex].
- [tex]\((x - (-7 - i))\)[/tex]: This does not match any factor related to our given roots.
- [tex]\((x + (7 + i))\)[/tex]: This is not correctly formed, as it does not match the factor form [tex]\((x - (7 + i))\)[/tex] corresponding to the root [tex]\(7 + i\)[/tex].
- [tex]\((x + (7 - i))\)[/tex]: This is not correctly formed, as it does not match the factor form [tex]\((x - (7 - i))\)[/tex] corresponding to the root [tex]\(7 - i\)[/tex].
Therefore, the correct factor of [tex]\(f(x)\)[/tex] must be [tex]\((x - (7 + i))\)[/tex].
Thus, the correct factor is [tex]\((x - (7 + i))\)[/tex].
