Question

bvTatiana 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 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 plus 1 close paren cubed times open paren x plus 81 close paren to the 5th power 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)5(x+81)3

Answer 1

To construct a polynomial function given the zeros and their multiplicities, we use the following rules:

1. For each zero, we create a factor in the polynomial.
2. The multiplicity tells us the exponent of the factor.

From the information given:

- The zero \( x = 1 \) has a multiplicity of 3, so the factor will be \( (x - 1)^3 \).
- The zero \( x = 81 \) has a multiplicity of 5, so the factor will be \( (x - 81)^5 \).

Putting these together, the polynomial function will be:

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

Therefore, the correct choice from the responses is:

**f(x) = (x - 1)^3 (x - 81)^5**.