Jaime was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?
1. f(x)=(x-4)(x+2)^3
2. f(x)=(x+4)(x-2)
3. f(x)=(x-4)(x+2)
4. f(x)=(x+4)(x-2)^3
3. f(x)=(x-4)(x+2)
Jaime should write the function option 1: f(x) = (x-4)(x+2)^3. This function has zeros at x = 4 and x = -2, with multiplicities 1 and 3, respectively.
To construct a polynomial function with given zeros and multiplicities, we need to use the factors of the zero values and their corresponding multiplicities.
In this case, the given zeros and multiplicities are as follows:
Zero: -4, Multiplicity: 1
Zero: 2, Multiplicity: 1
Now, let's analyze the given options:
1. f(x) = (x-4)(x+2)^3
This option includes the zero -4 with a multiplicity of 1 and the zero 2 with a multiplicity of 3. It matches the given zeros and multiplicities, so it could be the correct function.
2. f(x) = (x+4)(x-2)
This option includes the zero -4 with a multiplicity of 1 and the zero 2 with a multiplicity of 1. It doesn't match the given multiplicities, so it is not the correct function.
3. f(x) = (x-4)(x+2)
This option includes the zero -4 with a multiplicity of 1 and the zero 2 with a multiplicity of 1. It doesn't match the given multiplicities, so it is not the correct function.
4. f(x) = (x+4)(x-2)^3
This option includes the zero -4 with a multiplicity of 1 and the zero 2 with a multiplicity of 3. It doesn't match the given multiplicities, so it is not the correct function.
Based on the analysis, the correct function that Jaime should write is:
f(x) = (x-4)(x+2)^3 (Option 1)