To find the roots of the function [tex]\( f(x) = (x-6)^2 (x+2)^2 \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make the function equal to zero. Let's analyze the given function step-by-step.
1. Identify the factors of the function:
The function [tex]\( f(x) \)[/tex] can be expressed as the product of its factors:
[tex]\[
f(x) = (x-6)^2 (x+2)^2
\][/tex]
2. Find the values of [tex]\( x \)[/tex] that make each factor zero:
- For the factor [tex]\( (x-6)^2 \)[/tex]:
[tex]\[
(x-6)^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 6 = 0 \implies x = 6
\][/tex]
The root here is [tex]\( x = 6 \)[/tex]. Since the factor [tex]\( (x-6) \)[/tex] is squared, the root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- For the factor [tex]\( (x+2)^2 \)[/tex]:
[tex]\[
(x+2)^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
The root here is [tex]\( x = -2 \)[/tex]. Since the factor [tex]\( (x+2) \)[/tex] is squared, the root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
3. Summarize the roots and their multiplicities:
- The root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- The root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
Therefore, the correct roots and their multiplicities are:
- [tex]\( -2 \)[/tex] with multiplicity 2
- [tex]\( 6 \)[/tex] with multiplicity 2
Hence, the answer is:
- [tex]\( -2 \)[/tex] with multiplicity 2
- [tex]\( 6 \)[/tex] with multiplicity 2
