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

Question

Zero Multiplicity
x=1
3
x=81
5
(1 point)
Responses

f(x)=(x−1)5(x−81)3
f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis superscript 5 baseline left parenthesis x minus 81 right parenthesis cubed

f(x)=(x+1)3(x+81)5
f left parenthesis x right parenthesis equals left parenthesis x plus 1 right parenthesis cubed left parenthesis x plus 81 right parenthesis superscript 5 baseline

f(x)=(x−1)3(x−81)5
f left parenthesis x right parenthesis equals left parenthesis x minus 1 right parenthesis cubed left parenthesis x minus 81 right parenthesis superscript 5 baseline

f(x)=(x+1)5(x+81)3

Answer 1

To construct a polynomial function given the zeros and their multiplicities, we need to use the provided values correctly.

The zeros and their multiplicities given are:

• Zero \( x = 1 \) with multiplicity \( 3 \)
• Zero \( x = 81 \) with multiplicity \( 5 \)

When writing the polynomial in factored form, you use the zero and its multiplicity to create the factors:

• For the zero \( x = 1 \) with multiplicity 3, the factor is \( (x - 1)^3 \).
• For the zero \( x = 81 \) with multiplicity 5, the factor is \( (x - 81)^5 \).

Thus, the polynomial function can be written as: \[ f(x) = (x - 1)^3 (x - 81)^5 \]

Looking at the response options provided, the correct one is: \[ f(x) = (x - 1)^3 (x - 81)^5 \]

So, the correct response is: f(x) = (x - 1)³(x - 81)⁵.