A degree 3 polynomial with zeros of $1$, $-2$, and $3$ can be written as:

$f(x) = a(x - 1)(x + 2)(x - 3)$

Here, $a$ is a constant that can be any non-zero real number. For simplicity, let's assume $a = 1$. Expanding the polynomial:

1. Multiply $(x - 1)(x + 2)$: $(x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2$

2. Now multiply this by $(x - 3)$: $(x^2 + x - 2)(x - 3) = x^3 - 3x^2 + x^2 - 3x - 2x + 6$

3. Combine like terms: $f(x) = x^3 - 2x^2 - 5x + 6$

So the polynomial is:

$f(x) = x^3 - 2x^2 - 5x + 6$

If you want to scale the polynomial by a constant $a$, the general form would be:

$f(x) = a(x^3 - 2x^2 - 5x + 6)$

Would you like to explore any more details about this?

Here are 5 related questions:

1. How do you find the leading coefficient of a polynomial?
2. Can a polynomial have complex roots as well?
3. What are the implications of different values of $a$ in the polynomial?
4. How does the degree of the polynomial relate to the number of roots?
5. How do we find the derivative of this cubic polynomial?

Tip: The Fundamental Theorem of Algebra states that a degree $n$ polynomial has exactly $n$ roots (including complex and repeated roots).