Tatiana was asked to construct a polynomial function with the following zeros and multiplicities. Which function should she write? (1 point)

Question

Responses

f(x)=(x−1)5(x−81)3
f of x is equal to open paren x minus 1 close paren to the 5th power times open paren x minus 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 determine the correct polynomial function based on the given zeros and their multiplicities, we need to look at the provided options while focusing on the numerical values of the zeros and their respective multiplicities.

The first option has the zeros at \( 1 \) (with a multiplicity of \( 5 \)) and \( 81 \) (with a multiplicity of \( 3 \)).

The second option has the zeros at \( 1 \) (with a multiplicity of \( 3 \)) and \( 81 \) (with a multiplicity of \( 5 \)).

The third option has the zeros at \( -1 \) (with a multiplicity of \( 3 \)) and \( -81 \) (with a multiplicity of \( 5 \)).

The fourth option has the zeros at \( -1 \) (with a multiplicity of \( 5 \)) and \( -81 \) (with a multiplicity of \( 3 \)).

To construct the right polynomial, we must take into account that the zeros must be listed with the correct values and corresponding multiplicities. If we denote the zeros as \( 1 \) and \( 81 \) with multiplicities as \( 5 \) and \( 3 \), respectively, the polynomial is:

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

Thus, the correct function that Tatiana should write is:

f(x) = (x−1)^5 (x−81)^3