Tatiana was asked to construct a polynomial function with the following zeros and multiplicities. Which function should she write? Zero Multiplicity x=1 3 x=81 5 (1 point) Responses f(x)=(x+1)5(x+81)3 f of x is equal to open paren x plus 1 close paren to the 5th power times open paren x plus 81 close paren cubed f(x)=(x−1)3(x−81)5 f of x is equal to open paren x minus 1 close paren cubed times open paren x minus 81 close paren to the 5th power f(x)=(x+1)3(x+81)5 f of x is equal to open paren x plus 1 close paren cubed times open paren x plus 81 close paren to the 5th power f(x)=(x−1)5(x−81)3

Answer 1

To construct a polynomial function based on the provided zeros and their multiplicities, we need to translate the zeros and their associated multiplicities into factors of the polynomial.

1. The zero \( x = 1 \) has a multiplicity of 3, which means the factor is \( (x - 1)^3 \).
2. The zero \( x = 81 \) has a multiplicity of 5, which means the factor is \( (x - 81)^5 \).

Combining these factors, the polynomial function can be expressed as:

\[ f(x) = (x - 1)^3 (x - 81)^5 \]

Thus, the correct response is:

f(x)=(x−1)3(x−81)5

This means the function Tatiana should write is \( f(x) = (x - 1)^3 (x - 81)^5 \).