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

Question

Zero Multiplicity
x=4 1
x=−2 3
(1 point)
Responses

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
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)3
f left parenthesis x right parenthesis equals left parenthesis x plus 4 right parenthesis left parenthesis x minus 2 right parenthesis cubed

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

Answer 1

To construct a polynomial function given the zeros and their multiplicities:

1. The zero \( x = 4 \) has a multiplicity of 1, which means it contributes a factor of \( (x - 4) \).
2. The zero \( x = -2 \) has a multiplicity of 3, which means it contributes a factor of \( (x + 2)^3 \).

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

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

Based on the provided options, the correct response is:

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