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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 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. 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 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.

1 answer

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

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

Now we summarize the multiplicities:

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

Therefore, 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.