Question

Identify the multiplicities of the linear factors of f(x)=(x−2)2(x+3)(3x−4). (1 point)

Answer 1

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