To find a polynomial function $f(x)$ of degree 3 with the specified zeros and conditions, we start with the zeros:

1. A zero of $0$ with multiplicity 2 means the factor $x^2$.
2. A zero of $4$ with multiplicity 1 means the factor $(x - 4)$.

Thus, the polynomial can be expressed as:

$f(x) = k \cdot x^2 (x - 4)$

where $k$ is a constant to be determined.

Next, we expand the polynomial:

$f(x) = k \cdot x^2 (x - 4) = k \cdot (x^3 - 4x^2)$

Now we apply the condition $f(5) = 20$:

$f(5) = k \cdot (5^3 - 4 \cdot 5^2)$ $= k \cdot (125 - 100) = k \cdot 25$ Setting this equal to 20 gives us:

$k \cdot 25 = 20 \quad \Rightarrow \quad k = \frac{20}{25} = \frac{4}{5}$

Thus, the polynomial function is:

$f(x) = \frac{4}{5} \cdot (x^3 - 4x^2)$

We can rewrite it as:

$f(x) = \frac{4}{5}x^3 - \frac{16}{5}x^2$

Summary

The polynomial function is:

$f(x) = \frac{4}{5}x^3 - \frac{16}{5}x^2$

Would you like more details or have any questions? Here are some related questions you might consider:

1. What are the properties of polynomial functions of degree 3?
2. How do multiplicities of zeros affect the graph of a polynomial?
3. Can you find the derivative of this polynomial?
4. How would you determine the zeros of this polynomial?
5. What does it mean for a polynomial to have real coefficients?

Tip: When analyzing polynomials, always check how the multiplicities of the zeros influence the shape of the graph.