Certainly! Let's solve this step-by-step.
When you are given the roots of a polynomial, you know that the factors of the polynomial correspond to [tex]\((x - \text{root})\)[/tex].
Given roots:
1. [tex]\(-9\)[/tex]
2. [tex]\(7 - i\)[/tex]
### Step 1: Discovering All Roots
Since the function has complex roots, and complex roots of polynomials with real coefficients come in conjugate pairs, the third root must be the conjugate of [tex]\(7 - i\)[/tex], which is [tex]\(7 + i\)[/tex].
So, the roots are:
- [tex]\(-9\)[/tex]
- [tex]\(7 - i\)[/tex]
- [tex]\(7 + i\)[/tex]
### Step 2: Writing Corresponding Factors
Each root [tex]\(r\)[/tex] corresponds to a factor of the form [tex]\((x - r)\)[/tex]. Therefore, the factors of the polynomial are:
- [tex]\((x - (-9)) = (x + 9)\)[/tex]
- [tex]\((x - (7 - i)) = (x - 7 + i)\)[/tex]
- [tex]\((x - (7 + i)) = (x - 7 - i)\)[/tex]
### Step 3: Identifying the Given Options
Now, let's match the options with the factors we have derived:
1. [tex]\((x - (7 + i))\)[/tex]: This directly matches one of the factors we identified.
2. [tex]\((x - (-7-1))\)[/tex]: This doesn't match any of our identified factors.
3. [tex]\((x + (7 + i))\)[/tex]: This also doesn't match any of our identified factors.
4. [tex]\((x + (7 - i))\)[/tex]: This isn't a match either.
### Conclusion
The correct factor of the polynomial [tex]\(f(x)\)[/tex] that matches one of the given roots must be:
[tex]\[
(x - (7 + i))
\][/tex]
Hence the correct factor is:
[tex]\[ (x - (7 + i)) \][/tex]
