Question

Jaime was asked to construct a polynomial function with the following zeros and multiplicities. Which function should he write?

This table shows the polynomial function with given zeros and their multiplicities.
Zero Multiplicity
x=4
𝑥
=
4
1
x=−2
𝑥
=
−
2
3
(1 point)
Responses

f(x)=(x−4)(x+2)3
𝑓
(
𝑥
)
=
(
𝑥
−
4
)
(
𝑥
+
2
)
3
f left parenthesis x right parenthesis equals left parenthesis x minus 4 right parenthesis left parenthesis x plus 2 right parenthesis cubed

f(x)=(x−4)(x+2)
𝑓
(
𝑥
)
=
(
𝑥
−
4
)
(
𝑥
+
2
)
f left parenthesis x right parenthesis equals left parenthesis x minus 4 right parenthesis left parenthesis x plus 2 right parenthesis

f(x)=(x+4)(x−2)
𝑓
(
𝑥
)
=
(
𝑥
+
4
)
(
𝑥
−
2
)
f left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis left parenthesis x minus 2 right parenthesis

f(x)=(x+4)(x−2)3

Answer 1

To construct the polynomial function from the given zeros and their multiplicities, we need to follow these steps:

1. Identify the zeros and their multiplicities:
- Zero \( x = 4 \) with multiplicity 1
- Zero \( x = -2 \) with multiplicity 3

2. Write the factors of the polynomial corresponding to each zero:
- For \( x = 4 \) (multiplicity 1), the factor is \( (x - 4) \).
- For \( x = -2 \) (multiplicity 3), the factor is \( (x + 2)^3 \).

3. Combine the factors to form the polynomial:
\[
f(x) = (x - 4)(x + 2)^3
\]

Based on the provided options, the correct choice is:
\[
f(x) = (x - 4)(x + 2)^3
\]

This matches the first response listed:
- **Response:** \( f(x)=(x−4)(x+2)^3 \)