The given polynomial function is:

$f(x) = 6x(x + 2)(x - 13)^3$

Step-by-step analysis:

1. Identify the zeros of the function:

   • From $6x$, the zero is $x = 0$.
   • From $x + 2$, the zero is $x = -2$.
   • From $(x - 13)^3$, the zero is $x = 13$.

2. Determine the multiplicity of each zero:

   • For $x = 0$, it appears in the term $6x$, which is a linear term. Hence, multiplicity = 1.
   • For $x = -2$, it appears in the term $x + 2$, which is also linear. Hence, multiplicity = 1.
   • For $x = 13$, it appears in the term $(x - 13)^3$, which is raised to the third power. Hence, multiplicity = 3.

Final Answer:

• Zeros of multiplicity 1: $0, -2$
• Zeros of multiplicity 2: None
• Zeros of multiplicity 3: $13$

Would you like further details or explanations?

Here are 5 related questions to deepen understanding:

1. What does the term "multiplicity of a zero" mean in a polynomial function?
2. How does the multiplicity of a zero affect the graph of a polynomial near that zero?
3. Can a polynomial have a zero with multiplicity greater than the degree of the polynomial?
4. How would the polynomial change if $(x - 13)^3$ was replaced with $(x - 13)^4$?
5. What are the implications of the leading coefficient ($6$) in the polynomial?

Tip: Zeros with odd multiplicities (e.g., 1 or 3) cause the graph to cross the x-axis, while those with even multiplicities only touch the x-axis.