Final answer:
The multiplicity of the given polynomial is 3, 4, and 5 for the factors (x-2), (x+3), and 4x respectively.
Step-by-step explanation:
The multiplicity of a given polynomial term is the number of times its corresponding factor appears in the polynomial. In this case, the given polynomial is f(x) = (x-2)³ (x+3)⁴ *4x⁵. To find the multiplicity of each term, we look at the powers of each factor.
The term (x-2)³ has a multiplicity of 3 because it is raised to the power of 3.
The term (x+3)⁴ has a multiplicity of 4 because it is raised to the power of 4.
The term 4x⁵ has a multiplicity of 5 because it is raised to the power of 5.
Therefore, the multiplicity of the given polynomial is 3, 4, and 5 for the factors (x-2), (x+3), and 4x respectively.