identify the ��the multiplicities of the linear factors of f(x)=(x-1)^2(x+4)(2x+5)

Answer 1

To identify the multiplicities of the linear factors of the polynomial function \( f(x) = (x-1)^2(x+4)(2x+5) \), we can analyze the factors:

1. The factor \( (x-1)^2 \) has a multiplicity of 2 because it appears squared.
2. The factor \( (x+4) \) has a multiplicity of 1 because it appears once.
3. The factor \( (2x+5) \) also has a multiplicity of 1 because it appears once.

So, the linear factors and their multiplicities are:

• \( x - 1 \): multiplicity 2
• \( x + 4 \): multiplicity 1
• \( 2x + 5 \): multiplicity 1

In summary:

• Multiplicity of \( x-1 \): 2
• Multiplicity of \( x+4 \): 1
• Multiplicity of \( 2x+5 \): 1