Question

Identify the multiplicities of the linear factors of h(x)=(x+4)(x−1)2(x+3)5 .(1 point)
Responses

The multiplicity of (x+4) is 4, the multiplicity of (x−1) is −1, and the multiplicity of (x+3) is 3.
The multiplicity of left parenthesis x plus 4 right parenthesis is 4, the multiplicity of left parenthesis x minus 1 right parenthesis is negative 1 , and the multiplicity of left parenthesis x plus 3 right parenthesis is 3.

The multiplicity of (x+4) is −4, the multiplicity of (x−1) is 1, and the multiplicity of (x+3) is −3.
The multiplicity of left parenthesis x plus 4 right parenthesis is negative 4 , the multiplicity of left parenthesis x minus 1 right parenthesis is 1, and the multiplicity of left parenthesis x plus 3 right parenthesis is negative 3 .

The multiplicity of (x+4) is 1, the multiplicity of (x−1) is 2, and the multiplicity of (x+3) is 5.
The multiplicity of left parenthesis x plus 4 right parenthesis is 1, the multiplicity of left parenthesis x minus 1 right parenthesis is 2, and the multiplicity of left parenthesis x plus 3 right parenthesis is 5.

The multiplicity of (x+4) is 0, the multiplicity of (x−1) is 2, and the multiplicity of (x+3) is 5.

Answer 1

To identify the multiplicities of the linear factors of the polynomial \( h(x) = (x + 4)(x - 1)^2(x + 3)^5 \), we look at the exponents of each factor:

1. The factor \( (x + 4) \) has an exponent of 1, so its multiplicity is **1**.
2. The factor \( (x - 1)^2 \) has an exponent of 2, so its multiplicity is **2**.
3. The factor \( (x + 3)^5 \) has an exponent of 5, so its multiplicity is **5**.

Thus, the multiplicities are:
- The multiplicity of \( (x + 4) \) is **1**,
- The multiplicity of \( (x - 1) \) is **2**,
- The multiplicity of \( (x + 3) \) is **5**.

The correct response is:
**The multiplicity of \( (x + 4) \) is 1, the multiplicity of \( (x - 1) \) is 2, and the multiplicity of \( (x + 3) \) is 5.**