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

Question

Responses

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

Answer 1

To determine the multiplicities of the linear factors of the given polynomial \( h(x) = (x+4)(x−1)^2(x+3)^5 \), we can analyze each factor:

1. For the factor \( (x+4) \):

   • This factor appears with no exponent written next to it, which implies that it has a multiplicity of 1.

2. For the factor \( (x−1)^2 \):

   • This factor has an exponent of 2, indicating that it has a multiplicity of 2.

3. For the factor \( (x+3)^5 \):

   • This factor has an exponent of 5, indicating that it has a multiplicity of 5.

Putting this information together:

• The multiplicity of \( (x+4) \) is 1,
• The multiplicity of \( (x−1) \) is 2,
• The multiplicity of \( (x+3) \) is 5.

Thus, 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.