The zeros of the polynomial are given as $-4$, $4$, and $5$, and the degree of the polynomial is 3. To form a polynomial, we use the fact that if $c$ is a zero of a polynomial, then $(x - c)$ is a factor.

So, for the given zeros, the polynomial can be written as:

$f(x) = (x - (-4))(x - 4)(x - 5)$

Simplify the factors:

$f(x) = (x + 4)(x - 4)(x - 5)$

Now, expand the factors. First, use the difference of squares on $(x + 4)(x - 4)$:

$f(x) = (x^2 - 16)(x - 5)$

Next, distribute $(x - 5)$ over $(x^2 - 16)$:

$f(x) = x^2(x - 5) - 16(x - 5)$

$f(x) = x^3 - 5x^2 - 16x + 80$

Thus, the polynomial is:

$f(x) = x^3 - 5x^2 - 16x + 80$

This is a polynomial with integer coefficients and a leading coefficient of 1.

Would you like more details or explanations on any of the steps?

Here are 5 related questions you might find interesting:

1. How do you find the degree of a polynomial given its expression?
2. What is the difference of squares, and how does it apply in polynomial expansion?
3. How can you verify if a polynomial has certain zeros?
4. How do the coefficients of a polynomial relate to its graph's shape?
5. What is synthetic division and how does it help with polynomial roots?

Tip: Always check your work by substituting the zeros back into the polynomial to confirm they yield 0!