Answer :
Given that a polynomial function [tex]\( f(x) \)[/tex] has roots [tex]\( 4 - 13i \)[/tex] and [tex]\( 5 \)[/tex], let's go through the steps to determine what must be a factor of this function.
### Step 1: Given roots and their properties
1. The roots provided are [tex]\( 4 - 13i \)[/tex] and [tex]\( 5 \)[/tex].
2. Polynomials with real coefficients have complex roots in conjugate pairs. This means if [tex]\( 4 - 13i \)[/tex] is a root, its conjugate [tex]\( 4 + 13i \)[/tex] must also be a root.
Thus, the roots of the polynomial are:
- [tex]\( 4 - 13i \)[/tex]
- [tex]\( 5 \)[/tex]
- [tex]\( 4 + 13i \)[/tex]
### Step 2: Construct factors for each root
For any given root [tex]\( r \)[/tex], [tex]\( (x - r) \)[/tex] is a factor of the polynomial.
- For the root [tex]\( 4 - 13i \)[/tex], the factor is [tex]\( (x - (4 - 13i)) \)[/tex].
- For the root [tex]\( 5 \)[/tex], the factor is [tex]\( (x - 5) \)[/tex].
- For the root [tex]\( 4 + 13i \)[/tex], the factor is [tex]\( (x - (4 + 13i)) \)[/tex].
### Step 3: Given choices for the factor of [tex]\( f(x) \)[/tex]
The possible factors provided in the choices are:
1. [tex]\( x + (13 - 4) \)[/tex]
2. [tex]\( x - (13 + 4) \)[/tex]
3. [tex]\( x + (4 + 13i) \)[/tex]
4. [tex]\( x - (4 + 13i) \)[/tex]
### Step 4: Identify the correct factor from the choices
Comparing the possible factors with our derived factors:
- [tex]\( x + (13 - 4) \)[/tex]
- This simplifies to [tex]\( x + 9 \)[/tex], which does not match any of our derived factors.
- [tex]\( x - (13 + 4) \)[/tex]
- This simplifies to [tex]\( x - 17 \)[/tex], which does not match any of our derived factors.
- [tex]\( x + (4 + 13i) \)[/tex]
- This simplifies to [tex]\( x - (-4 - 13i) \)[/tex], which is not in our set of derived factors.
- [tex]\( x - (4 + 13i) \)[/tex]
- This exactly matches the derived factor [tex]\( x - (4 + 13i) \)[/tex].
### Conclusion
The correct factor of [tex]\( f(x) \)[/tex] from the given choices is:
[tex]\[ x - (4 + 13i) \][/tex]
This corresponds to choice 4.
### Step 1: Given roots and their properties
1. The roots provided are [tex]\( 4 - 13i \)[/tex] and [tex]\( 5 \)[/tex].
2. Polynomials with real coefficients have complex roots in conjugate pairs. This means if [tex]\( 4 - 13i \)[/tex] is a root, its conjugate [tex]\( 4 + 13i \)[/tex] must also be a root.
Thus, the roots of the polynomial are:
- [tex]\( 4 - 13i \)[/tex]
- [tex]\( 5 \)[/tex]
- [tex]\( 4 + 13i \)[/tex]
### Step 2: Construct factors for each root
For any given root [tex]\( r \)[/tex], [tex]\( (x - r) \)[/tex] is a factor of the polynomial.
- For the root [tex]\( 4 - 13i \)[/tex], the factor is [tex]\( (x - (4 - 13i)) \)[/tex].
- For the root [tex]\( 5 \)[/tex], the factor is [tex]\( (x - 5) \)[/tex].
- For the root [tex]\( 4 + 13i \)[/tex], the factor is [tex]\( (x - (4 + 13i)) \)[/tex].
### Step 3: Given choices for the factor of [tex]\( f(x) \)[/tex]
The possible factors provided in the choices are:
1. [tex]\( x + (13 - 4) \)[/tex]
2. [tex]\( x - (13 + 4) \)[/tex]
3. [tex]\( x + (4 + 13i) \)[/tex]
4. [tex]\( x - (4 + 13i) \)[/tex]
### Step 4: Identify the correct factor from the choices
Comparing the possible factors with our derived factors:
- [tex]\( x + (13 - 4) \)[/tex]
- This simplifies to [tex]\( x + 9 \)[/tex], which does not match any of our derived factors.
- [tex]\( x - (13 + 4) \)[/tex]
- This simplifies to [tex]\( x - 17 \)[/tex], which does not match any of our derived factors.
- [tex]\( x + (4 + 13i) \)[/tex]
- This simplifies to [tex]\( x - (-4 - 13i) \)[/tex], which is not in our set of derived factors.
- [tex]\( x - (4 + 13i) \)[/tex]
- This exactly matches the derived factor [tex]\( x - (4 + 13i) \)[/tex].
### Conclusion
The correct factor of [tex]\( f(x) \)[/tex] from the given choices is:
[tex]\[ x - (4 + 13i) \][/tex]
This corresponds to choice 4.
