Great question! Let's analyze the polynomial function step-by-step to identify which one fits the given criteria:
We need a polynomial function that has:
1. A leading coefficient of 1.
2. Roots at [tex]\( x = -2 \)[/tex] and [tex]\( x = 7 \)[/tex] with multiplicity 1.
3. A root at [tex]\( x = 5 \)[/tex] with multiplicity 2.
### Breakdown
1. Leading Coefficient of 1: This means the polynomial should not have any leading constants other than 1.
2. Root at [tex]\( x = -2 \)[/tex] with multiplicity 1: This means [tex]\( (x + 2) \)[/tex] should be a factor of the polynomial.
3. Root at [tex]\( x = 7 \)[/tex] with multiplicity 1: This means [tex]\( (x - 7) \)[/tex] should be a factor of the polynomial.
4. Root at [tex]\( x = 5 \)[/tex] with multiplicity 2: This means [tex]\( (x - 5) \)[/tex] should be a factor of the polynomial twice, i.e., [tex]\( (x - 5)^2 \)[/tex].
### Forming the Polynomial
Combining these factors, the polynomial should look like:
[tex]\[ f(x) = (x + 2)(x - 7)(x - 5)(x - 5) \][/tex]
### Evaluating the Options:
Let's evaluate the provided options:
1. Option A: [tex]\( f(x) = 2(x + 7)(x + 5)(x - 2) \)[/tex]
- This option has a leading coefficient of 2, not 1. It doesn’t match.
2. Option B: [tex]\( f(x) = 2(x - 7)(x - 5)(x + 2) \)[/tex]
- This option has a leading coefficient of 2, not 1. It doesn’t match.
3. Option C: [tex]\( f(x) = (x + 7)(x + 5)(x + 5)(x - 2) \)[/tex]
- This option does not have the correct factors. Particularly, it does not have [tex]\( (x - 7) \)[/tex] and has extra positive roots.
4. Option D: [tex]\( f(x) = (x - 7)(x - 5)(x - 5)(x + 2) \)[/tex]
- This option indeed has all the required factors: [tex]\( (x + 2) \)[/tex], [tex]\( (x - 7) \)[/tex], [tex]\( (x - 5)^2 \)[/tex].
- Leading coefficient is 1.
Given all these considerations, the correct polynomial function that meets all the criteria is:
[tex]\[ f(x) = (x - 7)(x - 5)(x - 5)(x + 2) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
