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

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 form of each factor:

• The factor \( (x + 4) \) appears once, so its multiplicity is 1.
• The factor \( (x - 1)^2 \) appears squared, so its multiplicity is 2.
• The factor \( (x + 3)^5 \) appears raised to the fifth power, so its multiplicity is 5.

Therefore, the multiplicities of the linear factors 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.