To determine which second-degree polynomial function has a leading coefficient of -1 and a root at 4 with multiplicity 2, we need to understand a few key concepts:
1. Leading Coefficient: This is the coefficient of the term with the highest degree. In this case, we need a leading coefficient of -1.
2. Root with Multiplicity: A root of a polynomial is a value for which the polynomial equals zero. A root with multiplicity 2 means the polynomial can be written in the form [tex]\( (x - r)^2 \)[/tex], where [tex]\( r \)[/tex] is the root.
Given these, we can find the polynomial by following these steps:
### Step-by-Step Solution:
1. Formulation:
Since the root is 4 with multiplicity 2, the polynomial will take the form:
[tex]\[
f(x) = a(x - 4)^2
\][/tex]
where [tex]\(a\)[/tex] is the leading coefficient. Given that the leading coefficient is -1, we have:
[tex]\[
f(x) = -1(x - 4)^2
\][/tex]
2. Expansion:
We need to expand [tex]\((x - 4)^2\)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 8x + 16
\][/tex]
Now, substituting back into the polynomial, we get:
[tex]\[
f(x) = -1(x^2 - 8x + 16)
\][/tex]
3. Simplification:
Distributing the -1, we get:
[tex]\[
f(x) = -x^2 + 8x - 16
\][/tex]
4. Matching with Given Options:
We compare our result with the given options:
- [tex]\( f(x) = -x^2 - 8x - 16 \)[/tex]
- [tex]\( f(x) = -x^2 + 8x - 16 \)[/tex]
- [tex]\( f(x) = -x^2 - 8x + 16 \)[/tex]
- [tex]\( f(x) = -x^2 + 8x + 16 \)[/tex]
The polynomial function [tex]\(-x^2 + 8x - 16\)[/tex] matches our derived polynomial.
Therefore, the correct polynomial function is:
[tex]\[ f(x) = -x^2 + 8x - 16 \][/tex]
And this corresponds to option 2.
