Identify the multiplicities of the linear factors of h(x)=(x+4)(x-1)^2(x+3)^5

Answer 1

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 in the expression:

1. The factor \( (x + 4) \) has a multiplicity of 1 because it appears only once.
2. The factor \( (x - 1)^2 \) has a multiplicity of 2 because it appears with an exponent of 2.
3. The factor \( (x + 3)^5 \) has a multiplicity of 5 because it appears with an exponent of 5.

In summary, the multiplicities of the linear factors are:

• \( x + 4 \): multiplicity 1
• \( x - 1 \): multiplicity 2
• \( x + 3 \): multiplicity 5