Bot GPT 4-o mini To determine the multiplicities of the linear factors in the polynomial \( h(x) = (x+4)(x-1)^2(x+3)^5 \), we look at each factor and the exponent associated with it.
- For the factor \( (x + 4) \): it appears as \( (x + 4)^1 \), so the multiplicity is 1.
- For the factor \( (x - 1)^2 \): it is clearly raised to the power of 2, so the multiplicity is 2.
- For the factor \( (x + 3)^5 \): it is raised to the power of 5, so the multiplicity 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.
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