Question
Identify the multiplicities of the linear factors of f(x)=(x−2)2(x+3)(3x−4). (1 point)
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GPT-4o mini
To identify the multiplicities of the linear factors of the polynomial function \( f(x) = (x-2)^2(x+3)(3x-4) \), we can break down the factors of \( f(x) \):
1. The factor \( (x-2)^2 \) has a multiplicity of 2 because it is squared.
2. The factor \( (x+3) \) has a multiplicity of 1 because it is not raised to any power other than 1.
3. The factor \( (3x-4) \) also has a multiplicity of 1, as it is similarly not raised to any power other than 1.
Thus, the multiplicities of the linear factors are as follows:
- For the factor \( (x-2) \): multiplicity 2
- For the factor \( (x+3) \): multiplicity 1
- For the factor \( (3x-4) \): multiplicity 1
In summary:
- \( x - 2 \): multiplicity 2
- \( x + 3 \): multiplicity 1
- \( 3x - 4 \): multiplicity 1
1. The factor \( (x-2)^2 \) has a multiplicity of 2 because it is squared.
2. The factor \( (x+3) \) has a multiplicity of 1 because it is not raised to any power other than 1.
3. The factor \( (3x-4) \) also has a multiplicity of 1, as it is similarly not raised to any power other than 1.
Thus, the multiplicities of the linear factors are as follows:
- For the factor \( (x-2) \): multiplicity 2
- For the factor \( (x+3) \): multiplicity 1
- For the factor \( (3x-4) \): multiplicity 1
In summary:
- \( x - 2 \): multiplicity 2
- \( x + 3 \): multiplicity 1
- \( 3x - 4 \): multiplicity 1
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