Question

Identify the multiplicities of the linear factors off (x)=(x-1)^(2)(x+4)(2x+5)

Answer 1

To identify the multiplicities of the linear factors of the function \( f(x) = (x-1)^2 (x+4)(2x+5) \), we start by recognizing the linear factors in the expression.

1. The first factor is \( (x-1)^2 \):
- The linear factor is \( x - 1 \).
- The exponent is 2, which means the multiplicity is **2**.

2. The second factor is \( (x + 4) \):
- The linear factor is \( x + 4 \).
- The exponent is 1, which means the multiplicity is **1**.

3. The third factor is \( (2x + 5) \):
- The linear factor is \( 2x + 5 \).
- This can be rewritten as \( 2(x + \frac{5}{2}) \), but since we are interested in linear factors, we consider it as a factor of degree 1. The exponent is 1, which means the multiplicity is **1**.

To summarize, the linear factors and their corresponding multiplicities are:

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

So the final answer is:
- \( x - 1 \) has a multiplicity of **2**
- \( x + 4 \) has a multiplicity of **1**
- \( 2x + 5 \) has a multiplicity of **1**