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$$  $$(i) Identify the turning points, zeros, and x-intercepts.

(ii) Do you find any point or zero which has a multiplicity in the graph? If so, specify them with multiplicity and explain the reason.

(iii) Identify the degree and the polynomial as well as identify the domain in which the polynomial is increasing and decreasing.

(iv) Do we have local maximum/minimum here? If yes, find them.

(v) Find the remainder when the polynomial is divided by x-4.

Let's analyze the given graph step by step:

### (i) Turning Points, Zeros, and X-intercepts
1. **Turning Points:**
   - $ A = (-1.569, -3.124) $
   - $ B = (0.319, 8.643) $
   
2. **Zeros/X-intercepts:**
   - $ D = (-2, 0) $
   - $ E = (-1, 0) $
   - $ C = (2, 0) $

### (ii) Multiplicity
- **Multiplicity:**
  - At $ D = (-2, 0) $, $ E = (-1, 0) $, and $ C = (2, 0) $, the graph crosses the x-axis, indicating that each zero has an odd multiplicity.
  - Given the polynomial appears to be a cubic function (as observed by the three distinct x-intercepts and two turning points), each zero likely has a multiplicity of 1.

### (iii) Degree and Polynomial Behavior
- **Degree:**
  - The polynomial appears to be cubic (degree 3), due to the presence of two turning points and three x-intercepts.

- **Increasing and Decreasing Intervals:**
  - **Increasing:**
    - From $ x = -\infty $ to $ x = -1.569 $
    - From $ x = -1 $ to $ x = 0.319 $
    - From $ x = 2 $ to $ x = \infty $
  - **Decreasing:**
    - From $ x = -1.569 $ to $ x = -1 $
    - From $ x = 0.319 $ to $ x = 2 $

### (iv) Local Maximum and Minimum
- **Local Minimum:**
  - At $ A = (-1.569, -3.124) $
  
- **Local Maximum:**
  - At $ B = (0.319, 8.643) $

### (v) Remainder when Divided by $ x-4 $
To find the remainder of the polynomial $ f(x) $ when divided by $ x - 4 $, use the Remainder Theorem. The remainder is $ f(4) $.

However, since the polynomial function $ f(x) $ is not explicitly given in the problem, we cannot directly calculate $ f(4) $. If we had the polynomial function, we could substitute $ x = 4 $ into the polynomial to find the remainder.

Would you like a detailed explanation or any further assistance with these concepts? Here are some questions for you to consider:

1. How do you determine the degree of a polynomial from its graph?
2. What is the significance of the turning points in a polynomial graph?
3. How can you use the Remainder Theorem for polynomial division?
4. What information can you gather from the x-intercepts of a polynomial graph?
5. How can the multiplicity of zeros affect the shape of a polynomial graph?

**Tip:** When analyzing polynomial graphs, always look for key features such as turning points, intercepts, and the end behavior to understand the degree and nature of the polynomial.

Learn how to analyze polynomial graphs to identify turning points, zeros, x-intercepts, and more. This detailed explanation covers concepts such as polynomial degrees, local extrema, and the Remainder Theorem for polynomial division. Suitable for high school students studying advanced algebra and graph analysis.

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