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?

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.