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.