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Identify the multiplicities of the linear factors of f(x)=(x−1)2(x+4)(2x+5)

.(2 points)
The multiplicity of the factor (x−1)
is
. The multiplicity of the factor (x+4)
is
. The multiplicity of the factor ​(2x+5)
is

1 answer

To identify the multiplicities of the linear factors in the function \( f(x)=(x−1)^2(x+4)(2x+5) \):

  1. For the factor \( (x - 1) \):

    • The power of this factor is 2 (it is squared).
    • Therefore, the multiplicity of the factor \( (x - 1) \) is 2.

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

    • This factor appears with no exponent written (which implies it is raised to the power of 1).
    • Therefore, the multiplicity of the factor \( (x + 4) \) is 1.

  3. For the factor \( (2x + 5) \):

    • This factor also appears with no exponent written (implying it is raised to the power of 1).
    • Therefore, the multiplicity of the factor \( (2x + 5) \) is 1.

In summary:

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