To find the roots of the polynomial function [tex]\( f(x) = (x-6)^2 (x+2)^2 \)[/tex], we'll follow these steps:
### Step-by-Step Solution:
1. Identify the Factors:
The function can be factored as [tex]\( (x-6)^2 \)[/tex] and [tex]\( (x+2)^2 \)[/tex]. Each of these factors will help us find the roots of the equation.
2. Finding the Roots:
- For [tex]\( (x-6)^2 \)[/tex]:
- Set [tex]\( (x-6) = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex] gives [tex]\( x = 6 \)[/tex].
- For [tex]\( (x+2)^2 \)[/tex]:
- Set [tex]\( (x+2) = 0 \)[/tex].
- Solving for [tex]\( x \)[/tex] gives [tex]\( x = -2 \)[/tex].
3. Determining the Multiplicity of Each Root:
- The factor [tex]\( (x-6) \)[/tex] is raised to the power of 2. This indicates that the root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- Similarly, the factor [tex]\( (x+2) \)[/tex] is also raised to the power of 2. This indicates that the root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
### Conclusion:
- The root [tex]\( x = 6 \)[/tex] has a multiplicity of 2.
- The root [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
So, the correct detailed solution is:
- [tex]\( 6 \)[/tex] with multiplicity 2
- [tex]\(-2\)[/tex] with multiplicity 2
This matches the numerical result from the initial computation: [tex]\( (6, 2, -2, 2) \)[/tex].
