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The given polynomial is \( f(x) = (x - 6)^4(x^2 - 36)^9 \).

### Step-by-step solution:

1. **Factoring the Polynomial**:
   - The expression \( x^2 - 36 \) can be factored as a difference of squares:
     \[
     x^2 - 36 = (x - 6)(x + 6)
     \]
   - Therefore, the polynomial becomes:
     \[
     f(x) = (x - 6)^4 \cdot [(x - 6)(x + 6)]^9
     \]
   - Simplifying further:
     \[
     f(x) = (x - 6)^4 \cdot (x - 6)^9 \cdot (x + 6)^9
     \]
   - Combining the terms with \( (x - 6) \):
     \[
     f(x) = (x - 6)^{13}(x + 6)^9
     \]

2. **Identifying the Zeroes**:
   - The zeroes are the values of \( x \) that make the factors zero.
   - \( (x - 6)^{13} = 0 \) when \( x = 6 \).
   - \( (x + 6)^9 = 0 \) when \( x = -6 \).

3. **Multiplicity of Zeroes**:
   - The multiplicity of a zero is the exponent of the factor corresponding to that zero.
   - The zero \( x = 6 \) has multiplicity 13 (from \( (x - 6)^{13} \)).
   - The zero \( x = -6 \) has multiplicity 9 (from \( (x + 6)^9 \)).

### Final Answer:
- The zeroes are \( x = 6 \) and \( x = -6 \).
- The multiplicity of \( x = 6 \) is 13.
- The multiplicity of \( x = -6 \) is 9.

It seems the provided answers in the image are incorrect. The correct multiplicities should be 13 and 9, not 18 and 4.

Would you like more details or have any questions about this solution?

Here are some related questions to deepen your understanding:
1. How is the multiplicity of a zero defined?
2. What happens graphically at a zero with even versus odd multiplicity?
3. How would the graph of \( f(x) \) look near \( x = 6 \) and \( x = -6 \)?
4. How do you factor polynomials with higher powers, like \( (x^2 - 36)^9 \)?
5. Can a polynomial have a zero with multiplicity 1?

**Tip**: A zero with an even multiplicity touches the x-axis without crossing it, while a zero with odd multiplicity crosses the x-axis.

Find the zeroes and multiplicities of the polynomial f(x) = (x-6)^4 (x^2 - 36)^9.

Learn how to find the zeroes and multiplicities of the polynomial f(x) = (x-6)^4 (x^2 - 36)^9. This solution includes factoring and understanding the multiplicity of zeroes, suitable for high school students in Grades 10-12.

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