The polynomial function given is:

$f(x) = 7x(x + 6)(x - 3)^2(x - 9)$

To find the zeros of $f(x)$ and their multiplicities, let's analyze each factor:

1. Zero from $x$:

   • $x = 0$
   • Multiplicity: 1 (since $x$ appears as a single factor)

2. Zero from $x + 6$:

   • $x = -6$
   • Multiplicity: 1 (since $x + 6$ appears as a single factor)

3. Zero from $(x - 3)^2$:

   • $x = 3$
   • Multiplicity: 2 (since $(x - 3)$ is squared)

4. Zero from $x - 9$:

   • $x = 9$
   • Multiplicity: 1 (since $x - 9$ appears as a single factor)

Summary of Zeros and Their Multiplicities:

• Multiplicity one: $0, -6, 9$
• Multiplicity two: $3$
• Multiplicity three: None

So, fill in the answers as follows:

• Zero(s) of multiplicity one: 0, -6, 9
• Zero(s) of multiplicity two: 3
• Zero(s) of multiplicity three: None

Would you like more details on the solution, or do you have any questions?

Related Questions

1. How do you determine the multiplicity of a zero in a polynomial?
2. What effect does a zero with even multiplicity have on the graph of a polynomial?
3. How can we factor a polynomial to determine its zeros?
4. What is the significance of each factor's exponent in a polynomial function?
5. How does changing the multiplicity of a zero affect the shape of the graph near that zero?

Tip

When analyzing a polynomial, focus on the powers of each factor to quickly identify the multiplicities of its zeros.