Answer :
To solve this problem, we need to find which of the given equations can represent the polynomial function [tex]\( f \)[/tex] that has zeros at [tex]\(-6\)[/tex], [tex]\(4\)[/tex], and [tex]\(-1\)[/tex] with a multiplicity of 2.
### Understanding Zeros and Multiplicity
1. Zeros of the Polynomial:
- A zero of [tex]\(-6\)[/tex] means the factor is [tex]\((x + 6)\)[/tex].
- A zero of [tex]\(4\)[/tex] means the factor is [tex]\((x - 4)\)[/tex].
- A zero of [tex]\(-1\)[/tex] with a multiplicity of 2 means the factor is [tex]\((x + 1)^2\)[/tex].
2. Constructing the Polynomial:
- The polynomial [tex]\( f(x) \)[/tex] can be constructed using the above factors.
- [tex]\( f(x) = a(x + 6)(x - 4)(x + 1)^2 \)[/tex] where [tex]\( a \)[/tex] is a non-zero constant.
### Expanding the Polynomial
The polynomial in its expanded form will include coefficients that we need to match with the given options:
3. Expanded Form:
- The task is to expand [tex]\((x + 6)(x - 4)(x + 1)^2\)[/tex] to obtain the standard polynomial form: [tex]\( ax^4 + bx^3 + cx^2 + dx + e \)[/tex].
### Comparing with Given Options
We need to compare coefficients from the expanded polynomial with the given options:
4. Matching with Options:
- Coefficient [tex]\( a = 1 \)[/tex]: The expanded polynomial is:
[tex]\[
f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24
\][/tex]
- Another possible coefficient [tex]\( a = -7 \)[/tex]: The polynomial becomes:
[tex]\[
f(x) = -7x^4 - 28x^3 + 133x^2 + 322x + 168
\][/tex]
### Conclusion
There are two options among the given that match the derived polynomials based on the zeros and their multiplicities:
- Option A: [tex]\( f(x) = x^4 - 4x^3 - 19x^2 + 46x - 24 \)[/tex]
- This matches the form when [tex]\( a = 1 \)[/tex].
- Option B: [tex]\( f(x) = -7x^5 - 28x^3 + 133x^2 + 322x + 168 \)[/tex]
- This matches the form when [tex]\( a = -7 \)[/tex].
Thus, the equations that could represent the polynomial function [tex]\( f \)[/tex] are those matching these expanded forms.
### Understanding Zeros and Multiplicity
1. Zeros of the Polynomial:
- A zero of [tex]\(-6\)[/tex] means the factor is [tex]\((x + 6)\)[/tex].
- A zero of [tex]\(4\)[/tex] means the factor is [tex]\((x - 4)\)[/tex].
- A zero of [tex]\(-1\)[/tex] with a multiplicity of 2 means the factor is [tex]\((x + 1)^2\)[/tex].
2. Constructing the Polynomial:
- The polynomial [tex]\( f(x) \)[/tex] can be constructed using the above factors.
- [tex]\( f(x) = a(x + 6)(x - 4)(x + 1)^2 \)[/tex] where [tex]\( a \)[/tex] is a non-zero constant.
### Expanding the Polynomial
The polynomial in its expanded form will include coefficients that we need to match with the given options:
3. Expanded Form:
- The task is to expand [tex]\((x + 6)(x - 4)(x + 1)^2\)[/tex] to obtain the standard polynomial form: [tex]\( ax^4 + bx^3 + cx^2 + dx + e \)[/tex].
### Comparing with Given Options
We need to compare coefficients from the expanded polynomial with the given options:
4. Matching with Options:
- Coefficient [tex]\( a = 1 \)[/tex]: The expanded polynomial is:
[tex]\[
f(x) = x^4 + 4x^3 - 19x^2 - 46x - 24
\][/tex]
- Another possible coefficient [tex]\( a = -7 \)[/tex]: The polynomial becomes:
[tex]\[
f(x) = -7x^4 - 28x^3 + 133x^2 + 322x + 168
\][/tex]
### Conclusion
There are two options among the given that match the derived polynomials based on the zeros and their multiplicities:
- Option A: [tex]\( f(x) = x^4 - 4x^3 - 19x^2 + 46x - 24 \)[/tex]
- This matches the form when [tex]\( a = 1 \)[/tex].
- Option B: [tex]\( f(x) = -7x^5 - 28x^3 + 133x^2 + 322x + 168 \)[/tex]
- This matches the form when [tex]\( a = -7 \)[/tex].
Thus, the equations that could represent the polynomial function [tex]\( f \)[/tex] are those matching these expanded forms.
