Suppose that $$ f\left(x\right)=2x^2\left(x+6\right)^3\left(x-7\right)^3 $$ list each zero of according to its multiplicity for multiplicity one, two, and three. If there is more than one zero for each multiplicity list all of them.

To find the zeros of the function $ f(x) = 2x^2(x+6)^3(x-7)^3 $ and determine their multiplicities, let's analyze each factor of the function:

1. **Factor 1: $ 2x^2 $**
   - The factor $ x^2 $ gives a zero at $ x = 0 $.
   - The multiplicity of this zero is 2 (since the exponent of $ x $ is 2).

2. **Factor 2: $ (x+6)^3 $**
   - The factor $ x + 6 $ gives a zero at $ x = -6 $.
   - The multiplicity of this zero is 3 (since the exponent of $ x + 6 $ is 3).

3. **Factor 3: $ (x-7)^3 $**
   - The factor $ x - 7 $ gives a zero at $ x = 7 $.
   - The multiplicity of this zero is 3 (since the exponent of $ x - 7 $ is 3).

### Summary of Zeros and Their Multiplicities:

- **Multiplicity 1:** None.
- **Multiplicity 2:** $ x = 0 $.
- **Multiplicity 3:** $ x = -6 $, $ x = 7 $.

So the zeros according to their multiplicities are:

- **Multiplicity 1:** (None)
- **Multiplicity 2:** $ x = 0 $
- **Multiplicity 3:** $ x = -6 $, $ x = 7 $

Would you like further details on this, or do you have any questions? 

Here are some related questions to expand on this topic:
1. What does the multiplicity of a zero tell us about the graph of the function at that point?
2. How does the function behave near a zero with even multiplicity?
3. How does the function behave near a zero with odd multiplicity?
4. What is the general effect of higher multiplicities on the shape of the graph?
5. How would the graph of $ f(x) = 2x^2(x+6)^3(x-7)^3 $ look near each zero?
6. How do you determine the degree of the polynomial from its factored form?
7. What happens to the graph of the function as $ x $ approaches infinity or negative infinity?
8. How does the leading coefficient (2 in this case) affect the graph of the polynomial?

**Tip:** The sum of the multiplicities of all zeros of a polynomial equals the degree of the polynomial. This is useful for checking your work!

Explore how to identify and analyze the zeroes of polynomial functions with varying multiplicities. This example focuses on the function f(x) = 2x^2(x+6)^3(x-7)^3 and provides a detailed breakdown of its zeroes and their multiplicities. Suitable for advanced high school students (Grades 10-12).

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