https://file.aiquickdraw.com/mathbot/file/1730695177430wkmwiu8u.jpg

To write an equation for the polynomial graph, let's analyze the key characteristics visible in the graph:

### Observations:
1. **Roots**: The polynomial crosses the x-axis at \( x = -3 \), \( x = 0 \), and \( x = 3 \). This suggests factors of \( (x + 3) \), \( x \), and \( (x - 3) \).
2. **Behavior at Roots**: The graph touches the x-axis at \( x = -3 \) and \( x = 3 \) without crossing, which indicates these roots have an even multiplicity (likely 2). The root at \( x = 0 \) crosses the x-axis, suggesting an odd multiplicity (likely 1).
3. **Degree of the Polynomial**: Given the observed behaviors, the smallest degree that could satisfy these conditions is 5:
   - Factors: \( (x + 3)^2 \), \( x \), and \( (x - 3)^2 \).
   - This results in a degree: \( 2 + 1 + 2 = 5 \).

4. **Leading Coefficient**: The graph opens downward for large values of \( x \) (both positive and negative), indicating a negative leading coefficient.

### Polynomial Equation
Based on these observations, we can propose the polynomial equation in the form:
\[
y(x) = -a (x + 3)^2 \cdot x \cdot (x - 3)^2
\]
where \( a \) is a positive constant that controls the vertical stretch.

### Final Equation
Assuming \( a = 1 \) for simplicity (if the scale is not specified), the equation of the polynomial can be:
\[
y(x) = -(x + 3)^2 \cdot x \cdot (x - 3)^2
\]

This equation captures the roots and behavior observed in the graph.

Would you like further details on how to expand this polynomial, or do you have other questions?

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#### Additional Questions to Explore:
1. How does changing the value of \( a \) affect the graph of the polynomial?
2. What would the graph look like if the leading coefficient were positive instead of negative?
3. Why does an even multiplicity at a root cause the graph to "touch" rather than "cross" the x-axis?
4. What are the implications of having a polynomial of degree 5?
5. Can this polynomial be written in standard form, and how would that look?

#### Tip:
When identifying polynomial functions from graphs, note where the graph touches or crosses the x-axis—this indicates the multiplicity of each root.

Write an equation for the polynomial graphed below.

Learn how to find the equation of a polynomial with roots at x = -3, 0, and 3. This polynomial exhibits even and odd root multiplicities, influencing its behavior at the x-axis. Suitable for high school students studying polynomial functions.

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