Identify the multiplicities of the linear factors of f(x)=(x−1)2(x+4)(2x+5)

Question

.(2 points)
The multiplicity of the factor (x−1)
is
. The multiplicity of the factor (x+4)
is
. The multiplicity of the factor ​(2x+5)
is
.

Answer 1

To identify the multiplicities of the linear factors of the polynomial \( f(x) = (x-1)^2 (x+4)(2x+5) \), we will look at each factor:

1. The factor \( (x - 1) \):

   • The exponent of this factor is 2, so the multiplicity of the factor \( (x - 1) \) is 2.

2. The factor \( (x + 4) \):

   • The exponent of this factor is 1, so the multiplicity of the factor \( (x + 4) \) is 1.

3. The factor \( (2x + 5) \):

   • The exponent of this factor is 1, so the multiplicity of the factor \( (2x + 5) \) is 1.

To summarize:

• The multiplicity of the factor \( (x - 1) \) is 2.
• The multiplicity of the factor \( (x + 4) \) is 1.
• The multiplicity of the factor \( (2x + 5) \) is 1.