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


(1 point)
Greg should write Function #
.

Answer 1

To construct a polynomial function based on the given zeros and their multiplicities, we start with the zeros and multiplicities provided:

- For the zero \( x = -4 \) with multiplicity 2, the corresponding factor of the polynomial is \( (x + 4)^2 \).
- For the zero \( x = 2 \) with multiplicity 3, the corresponding factor of the polynomial is \( (x - 2)^3 \).

Putting these together, the polynomial function can be written as:

\[
f(x) = (x + 4)^2 (x - 2)^3
\]

Now let's compare this result with the provided function options:

- **Function #1:** \( f(x) = (x + 4)(x - 2) \) (This is incorrect, as it does not account for the correct multiplicities.)
- **Function #2:** \( f(x) = (x + 4)^2 (x - 2)^3 \) (This is correct.)
- **Function #3:** \( f(x) = (x - 4)^2 (x + 2)^3 \) (This is incorrect, as the zeros are not correct.)

Thus, Greg should write **Function #2**.