Question

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