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Find the root(s) of [tex]f(x)=(x-6)^2(x+2)^2[/tex].

A. -6 with multiplicity 1
B. -6 with multiplicity 2
C. 6 with multiplicity 1
D. 6 with multiplicity 2
E. -2 with multiplicity 1
F. -2 with multiplicity 2
G. 2 with multiplicity 1
H. 2 with multiplicity 2

Sagot :

GinnyAnsweridn
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].
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