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The worksheet you've uploaded is titled **"Multiplicity and Graphing Polynomials"** and seems to focus on analyzing the behavior of polynomial functions, specifically examining the **degree of the polynomial** and the number of **turning points** on the graph.

### Here’s a breakdown of what I observe in the worksheet:

1. **Warm-up (Top part):**
   - You are asked to describe the **right-hand and left-hand behavior** of two polynomial functions:
     - \( f(x) = (2x^2 - 5x^3 + 3x^5) \)
     - \( f(x) = x^4 - 29x^3 + 100 \)
   
2. **Lesson Exploration (Middle part):**
   - This part shows four graphs of different polynomial functions, along with their factored forms:
     1. \( y = -(x - 1)(x + 5) \)
     2. \( y = -(x - 1)(x - 3)(x + 2) \)
     3. \( y = (x + 1)(x - 2)^2(x - 4) \)
     4. \( y = (x + 2)^2(x - 3) \)
   - There are tables below the graphs that prompt you to determine:
     - **Degree of the polynomial** for each function.
     - **Number of turning points** of each graph.

3. **Question (Bottom part):**
   - There’s a key question for you to answer:
     - *"Do you see a relationship between the degree of the polynomial and the number of turning points that the graph can have?"*

### Explanation and Key Concepts:

1. **Right-hand and Left-hand behavior** of a polynomial function is determined by the degree and leading coefficient:
   - For \( f(x) = (2x^2 - 5x^3 + 3x^5) \), since the degree is 5 (odd), the graph will fall to the left and rise to the right.
   - For \( f(x) = x^4 - 29x^3 + 100 \), since the degree is 4 (even), both ends of the graph will rise or fall based on the leading coefficient.

2. **Degree of a Polynomial and Turning Points**:
   - The **degree of a polynomial** (the highest power of \(x\)) tells you how the graph behaves.
   - The **number of turning points** is at most \(n - 1\), where \(n\) is the degree of the polynomial.

### Would you like more detailed help with any specific part, such as how to determine turning points, or explanations of behavior based on multiplicity?

Here are 5 related questions:
1. What is the general pattern for the behavior of odd vs. even degree polynomials?
2. How can multiplicity affect the shape of the graph at certain roots?
3. What role does the leading coefficient play in the end behavior of a polynomial?
4. How do you find the degree of a polynomial from its factored form?
5. How do you calculate the number of turning points given the degree?

**Tip:** When analyzing polynomial graphs, always check if the degree is odd or even, as this can quickly tell you about the end behavior of the graph.

What is the relationship between the degree of the polynomial and the number of turning points?

Explore the relationship between the degree of a polynomial and the number of turning points on its graph. Learn key concepts of graphing polynomials, understanding their multiplicity of roots, and determining the maximum number of turning points based on the polynomial degree. Suitable for high school students.

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