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Let's go through each part of the problem step by step:

### Questions 1-5: Determine which functions are polynomial functions. If the function is a polynomial, find its degree.

1. \( f(x) = 5x^2 + 6x^3 \)
   - This is a polynomial because it consists of terms with non-negative integer exponents.
   - **Degree**: The degree is the highest exponent, which is 3.

2. \( g(x) = 7x^5 - \pi x^3 + \frac{1}{5}x \)
   - This is also a polynomial, as all the terms have non-negative integer exponents.
   - **Degree**: The highest exponent is 5.

3. \( h(x) = 7x^3 + 2x^2 + \frac{1}{x} \)
   - This is **not** a polynomial because the term \( \frac{1}{x} \) is equivalent to \( x^{-1} \), which is a negative exponent.

4. \( F(x) = \frac{1}{x^2} - 3x^2 + 5 \)
   - This is **not** a polynomial because \( \frac{1}{x^2} \) is equivalent to \( x^{-2} \), which is a negative exponent.

5. \( G(x) = \frac{x^2 + 7}{x^3} \)
   - This is **not** a polynomial because dividing by \( x^3 \) gives terms with negative exponents.

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### Questions 6-13: Find the zeros for each polynomial function. For each zero, determine its multiplicity. Then use the multiplicity to determine whether the graph would cross or touch the x-axis there.

6. \( f(x) = 2(x - 5)(x + 4)^2 \)
   - **Zeros**: 
     - \( x = 5 \) (multiplicity 1, crosses the x-axis)
     - \( x = -4 \) (multiplicity 2, touches the x-axis)

7. \( g(x) = 3(x + 5)(x + 2)^2 \)
   - **Zeros**: 
     - \( x = -5 \) (multiplicity 1, crosses the x-axis)
     - \( x = -2 \) (multiplicity 2, touches the x-axis)

8. \( h(x) = 4(x - 3)(x + 6)^3 \)
   - **Zeros**:
     - \( x = 3 \) (multiplicity 1, crosses the x-axis)
     - \( x = -6 \) (multiplicity 3, crosses the x-axis but flattens out)

9. \( F(x) = -3\left(x + \frac{1}{2}\right)(x - 4)^3 \)
   - **Zeros**:
     - \( x = -\frac{1}{2} \) (multiplicity 1, crosses the x-axis)
     - \( x = 4 \) (multiplicity 3, crosses the x-axis but flattens out)

10. \( G(x) = x^3 - 2x^2 + x \)
    - **Factorization**: \( G(x) = x(x - 1)^2 \)
    - **Zeros**:
      - \( x = 0 \) (multiplicity 1, crosses the x-axis)
      - \( x = 1 \) (multiplicity 2, touches the x-axis)

11. \( H(x) = x^3 + 4x^2 + 4x \)
    - **Factorization**: \( H(x) = x(x + 2)^2 \)
    - **Zeros**:
      - \( x = 0 \) (multiplicity 1, crosses the x-axis)
      - \( x = -2 \) (multiplicity 2, touches the x-axis)

12. \( f(x) = x^3 + 7x^2 - 4x - 28 \)
    - **Factorization**: \( f(x) = (x - 2)(x + 7)(x + 2) \)
    - **Zeros**:
      - \( x = 2 \) (multiplicity 1, crosses the x-axis)
      - \( x = -7 \) (multiplicity 1, crosses the x-axis)
      - \( x = -2 \) (multiplicity 1, crosses the x-axis)

13. \( g(x) = x^3 + 5x^2 - 9x - 45 \)
    - **Factorization**: \( g(x) = (x - 3)(x + 5)(x + 3) \)
    - **Zeros**:
      - \( x = 3 \) (multiplicity 1, crosses the x-axis)
      - \( x = -5 \) (multiplicity 1, crosses the x-axis)
      - \( x = -3 \) (multiplicity 1, crosses the x-axis)

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Would you like further details or explanations on any of these steps? Here are 5 related questions:

1. What conditions determine whether a function is a polynomial?
2. How can you quickly identify the degree of a polynomial?
3. Why do negative exponents disqualify a function from being a polynomial?
4. What is the significance of the multiplicity of a zero?
5. How can graph behavior change based on the multiplicity of zeros?

**Tip:** The degree of a polynomial function is always determined by the highest power of the variable with a non-zero coefficient.

Solve polynomial function exercises, including identifying degrees and finding zeros with multiplicity. This solution covers polynomial functions, zeros, and graph behaviors for Grades 9-12. Learn how to determine which functions are polynomial, find their degrees, and analyze graph behavior using zeros and multiplicity.

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