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

Question

Zero Multiplicity
x=16
2
x=−4
3
(1 point)
Responses

f(x)=(x−16)2(x+4)3
f left parenthesis x right parenthesis equals left parenthesis x minus 16 right parenthesis squared left parenthesis x plus 4 right parenthesis cubed

f(x)=(x+16)2(x−4)3
f left parenthesis x right parenthesis equals left parenthesis x plus 16 right parenthesis squared left parenthesis x minus 4 right parenthesis cubed

f(x)=(x+16)3(x−4)2
f left parenthesis x right parenthesis equals left parenthesis x plus 16 right parenthesis cubed left parenthesis x minus 4 right parenthesis squared

f(x)=(x−16)3(x+4)2

Answer 1

To construct a polynomial function given the zeros and their multiplicities, we use the fact that if \( c \) is a zero of the polynomial with multiplicity \( m \), then the factor corresponding to \( c \) is \( (x - c)^m \).

Given the zeros and multiplicities:

• Zero \( x = 16 \) with multiplicity \( 2 \) contributes the factor \( (x - 16)^2 \).
• Zero \( x = -4 \) with multiplicity \( 3 \) contributes the factor \( (x + 4)^3 \).

Putting these factors together, Vicente should write the polynomial function as:

\[ f(x) = (x - 16)^2 (x + 4)^3 \]

Thus the correct response is:

f(x) = (x - 16)² (x + 4)³