https://file.aiquickdraw.com/mathbot/file/172625746117047j7le69.jpg

The graph provided appears to represent a cubic polynomial function (since it changes direction twice and crosses the x-axis at multiple points). Let's analyze the key points and behavior:

### Key observations:
1. **X-intercepts**: These are the points where the graph crosses the x-axis. Based on the graph:
   - One intercept occurs at \( x = -5 \).
   - Another intercept occurs at \( x = 0 \).
   - A third intercept occurs at \( x = 3 \).

2. **End behavior**: The graph seems to fall to the right, which suggests the leading term has a negative coefficient.

3. **Multiplicity of roots**:
   - The root at \( x = 0 \) seems to have an odd multiplicity, likely 1, since the graph passes directly through this point.
   - The other roots at \( x = -5 \) and \( x = 3 \) appear to have multiplicity 1 as well.

### Equation construction:
Since we are looking for the lowest-degree polynomial and there are three distinct roots, the polynomial will be of degree 3.

The general form of the polynomial can be written as:
\[
f(x) = a(x + 5)(x)(x - 3)
\]
Where:
- \( x + 5 \) corresponds to the root \( x = -5 \),
- \( x \) corresponds to the root at \( x = 0 \),
- \( x - 3 \) corresponds to the root at \( x = 3 \).

### Choosing the leading coefficient:
Since the graph falls to the right, the leading coefficient \( a \) must be negative. We can choose \( a = -1 \) (as per the instructions to use \( 1 \) or \( -1 \)).

Thus, the equation is:
\[
f(x) = -(x + 5)(x)(x - 3)
\]

Would you like to expand this equation or ask further questions about this problem?

### Related Questions:
1. How do we determine the multiplicity of a root from a graph?
2. What happens if the leading coefficient is positive instead of negative?
3. Can a cubic polynomial have a turning point at one of the roots?
4. How does the shape of the graph change with different root multiplicities?
5. How would you find the y-intercept of this polynomial?

### Tip:
To verify your polynomial function, you can substitute values of \( x \) from the graph into your equation and see if the corresponding \( y \)-values match the graph points.

Write an equation for the lowest-degree polynomial function with the graph and intercepts shown in the figure. For this exercise, make the leading coefficient be 1 or -1.

Learn how to derive the equation of a cubic polynomial based on its graph and x-intercepts. This step-by-step guide explores polynomial roots, their multiplicities, and how to identify the leading coefficient. Suitable for high school students in Grades 10-12.

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