To solve this problem, we need to determine the multiplicities of the zeros (roots) of the given function [tex]\( f(x) = (x-3)^2 (x+2)^2 (x-1) \)[/tex].
### Step-by-Step Solution:
1. Identify the factors of the function:
The function [tex]\( f(x) \)[/tex] is already factored as:
[tex]\[
f(x) = (x-3)^2 (x+2)^2 (x-1)
\][/tex]
2. Determine the zeros from the factors:
Each factor of the function [tex]\( f(x) \)[/tex] will provide a zero. The zeros are the values of [tex]\( x \)[/tex] that make each factor zero:
- From [tex]\( (x-3)^2 \)[/tex], the zero is [tex]\( x = 3 \)[/tex].
- From [tex]\( (x+2)^2 \)[/tex], the zero is [tex]\( x = -2 \)[/tex].
- From [tex]\( (x-1) \)[/tex], the zero is [tex]\( x = 1 \)[/tex].
3. Identify the multiplicity of each zero:
The multiplicity of a zero is determined by the exponent of the corresponding factor:
- For [tex]\( x = 3 \)[/tex], the factor is [tex]\( (x-3)^2 \)[/tex], and the exponent is 2. Thus, [tex]\( x = 3 \)[/tex] has multiplicity 2.
- For [tex]\( x = -2 \)[/tex], the factor is [tex]\( (x+2)^2 \)[/tex], and the exponent is 2. Thus, [tex]\( x = -2 \)[/tex] has multiplicity 2.
- For [tex]\( x = 1 \)[/tex], the factor is [tex]\( (x-1) \)[/tex], and the exponent is 1. Thus, [tex]\( x = 1 \)[/tex] has multiplicity 1.
4. Determine the zero with a multiplicity of 1:
Among the zeros [tex]\( 3, -2, \)[/tex] and [tex]\( 1 \)[/tex], the zero with multiplicity 1 is [tex]\( x = 1 \)[/tex].
5. Find the multiplicity of zero [tex]\(-2\)[/tex]:
As determined in step 3, the zero [tex]\( x = -2 \)[/tex] has a multiplicity of 2.
### Final Answer:
The zero [tex]\( 1 \)[/tex] has a multiplicity of 1.
The zero [tex]\(-2\)[/tex] has a multiplicity of 2.
