Let's dissect the given function [tex]\( f(x) = (x-3)^2(x+2)^2(x-1) \)[/tex] to identify its zeros and their multiplicities.
1. Zeros and Their Factors:
- The factor [tex]\( (x - 3)^2 \)[/tex] indicates that [tex]\( x = 3 \)[/tex] is a zero.
- The factor [tex]\( (x + 2)^2 \)[/tex] indicates that [tex]\( x = -2 \)[/tex] is a zero.
- The factor [tex]\( (x - 1) \)[/tex] indicates that [tex]\( x = 1 \)[/tex] is a zero.
2. Multiplicity Determination:
- For [tex]\( x = 3 \)[/tex]:
The factor involving [tex]\( x = 3 \)[/tex] is [tex]\((x - 3)^2\)[/tex]. Since the exponent is 2, the multiplicity of the zero [tex]\( 3 \)[/tex] is [tex]\( 2 \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
The factor involving [tex]\( x = -2 \)[/tex] is [tex]\((x + 2)^2\)[/tex]. Since the exponent is 2, the multiplicity of the zero [tex]\(-2 \)[/tex] is [tex]\( 2 \)[/tex].
- For [tex]\( x = 1 \)[/tex]:
The factor involving [tex]\( x = 1 \)[/tex] is [tex]\((x - 1)\)[/tex]. Since the exponent is 1, the multiplicity of the zero [tex]\( 1 \)[/tex] is [tex]\( 1 \)[/tex].
Given this analysis, the solutions to the problem are as follows:
The zero [tex]\(1\)[/tex] has a multiplicity of 1.
The zero [tex]\(-2\)[/tex] has a multiplicity of 2.
