Question

Greg was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write? Zero Multiplicity x=−4 2 x=2 3 Function #1: f(x)=(x+4)(x−2) Function #2: f(x)=(x+4)2(x−2)3 Function #3: f(x)=(x−4)2(x+2)3

Answer 1

To construct a polynomial function based on the given zeros and their multiplicities, we need to consider the information given:

- The zero \( x = -4 \) has a multiplicity of \( 2 \), which means the corresponding factor will be \( (x + 4)^2 \).
- The zero \( x = 2 \) has a multiplicity of \( 3 \), which means the corresponding factor will be \( (x - 2)^3 \).

Therefore, the polynomial function \( f(x) \) can be constructed as follows:
\[
f(x) = (x + 4)^2 (x - 2)^3
\]

Now, let's evaluate the provided function options:

- **Function #1**: \( f(x)=(x+4)(x−2) \)
- This does not match since it has only the first powers (no multiplicities).

- **Function #2**: \( f(x)=(x+4)^2(x−2)^3 \)
- This matches our constructed function since it includes the correct factors with the proper multiplicities.

- **Function #3**: \( f(x)=(x−4)^2(x+2)^3 \)
- This does not match because the zeros are incorrect.

Thus, the correct choice for Greg to write is:

### Function #2: \( f(x)=(x+4)^2(x−2)^3 \)