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 \( f(x) = (x - 2)^2 (x + 3)(3x - 4) \), we need to look at each factor:

1. The factor \( (x - 2)^2 \) is a linear factor with a multiplicity of \( 2 \) (since it is squared).
2. The factor \( (x + 3) \) is a linear factor with a multiplicity of \( 1 \).
3. The factor \( (3x - 4) \) is a linear factor with a multiplicity of \( 1 \).

Now we summarize the multiplicities of the linear factors:

• \( x - 2 \): multiplicity \( 2 \)
• \( x + 3 \): multiplicity \( 1 \)
• \( 3x - 4 \): multiplicity \( 1 \)

So the multiplicities of the linear factors of \( f(x) \) are:

• \( (x - 2) \): \( 2 \)
• \( (x + 3) \): \( 1 \)
• \( (3x - 4) \): \( 1 \)