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Question

shyFog36
11/07/2023
Mathematics
College

answered • expert verified
Identify the multiplicities of the linear factors of
h(x) = (x + 4) (x - 1)2(x+3)5 (1 point)
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 (x + 4) is 1, the multiplicity of (x - 1) is 2, and the
multiplicity of (x + 3) 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 (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 exponents of each factor in the expression.

1. The factor \( (x + 4) \) appears once (no exponent), which means 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 correct identification of the multiplicities is:

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

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