To construct a polynomial function given the zeros and their multiplicities, we use the following rules:
- For each zero, we create a factor in the polynomial.
- 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.