To determine the second-degree polynomial function with a leading coefficient of -1 and a root 4 with multiplicity 2, we need to follow these steps:
1. Understand what a root with multiplicity means:
- A root [tex]\( r \)[/tex] with multiplicity [tex]\( m \)[/tex] means that the factor [tex]\( (x - r)^m \)[/tex] appears in the polynomial.
- In this case, the root is 4, and it has a multiplicity of 2. This implies the factor [tex]\( (x - 4)^2 \)[/tex].
2. Form the polynomial using the root and given multiplicity:
- The polynomial factor can be written as [tex]\( (x - 4)^2 \)[/tex].
- Since the polynomial has a leading coefficient of -1, we multiply the entire factor by -1.
- Thus, the polynomial function [tex]\( f(x) \)[/tex] would be [tex]\( -1 \times (x - 4)^2 \)[/tex].
3. Expand the polynomial:
- First, expand [tex]\( (x - 4)^2 \)[/tex]:
[tex]\[
(x - 4)^2 = x^2 - 8x + 16
\][/tex]
- Now multiply this by the leading coefficient -1:
[tex]\[
-1 \times (x^2 - 8x + 16) = -x^2 + 8x - 16
\][/tex]
4. Compare it to the given options:
- The expanded form yields the polynomial [tex]\( -x^2 + 8x - 16 \)[/tex].
- Compare this with the options provided:
[tex]\[
f(x) = -x^2 - 8x - 16 \\
f(x) = -x^2 + 8x - 16 \\
f(x) = -x^2 - 8x + 16 \\
f(x) = -x^2 + 8x + 16
\][/tex]
- The correct polynomial [tex]\( -x^2 + 8x - 16 \)[/tex] matches the second option.
Hence, the second-degree polynomial function with a leading coefficient of -1 and a root 4 with multiplicity 2 is:
[tex]\[
\boxed{f(x) = -x^2 + 8x - 16}
\][/tex]
