https://file.aiquickdraw.com/mathbot/file/1729039947879vw1x7u6r.jpg

Let's analyze the given polynomial function \( P(x) = -x^5 - 5x^3 - 5x + 2 \) to determine its end behavior and select the correct end behavior diagram.

### Key information:
1. **Degree of the polynomial**: The highest power of \(x\) in the given polynomial is 5, meaning the polynomial has degree 5.
   
2. **Leading coefficient**: The coefficient of the highest degree term, \( -x^5 \), is negative.

### End behavior rules for polynomials:
- **Odd-degree polynomials** (like \(x^5\)) have opposite behaviors at \(x \to +\infty\) and \(x \to -\infty\).
- **Negative leading coefficient**: If the leading coefficient is negative, the polynomial will fall to \( -\infty \) as \( x \to +\infty \) and rise to \( +\infty \) as \( x \to -\infty \).

### End behavior of \( P(x) \):
- As \( x \to +\infty \), \( P(x) \to -\infty \).
- As \( x \to -\infty \), \( P(x) \to +\infty \).

### Conclusion:
The end behavior diagram should show the function rising on the left (as \(x\) goes to negative infinity) and falling on the right (as \(x\) goes to positive infinity).

From the image, **Option C** corresponds to this behavior, with the left arrow pointing upward and the right arrow pointing downward.

Would you like more details, or do you have any other questions?

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Here are 5 related questions for further exploration:
1. What are the general rules for determining end behavior based on degree and leading coefficient?
2. How does the sign of the leading coefficient affect the shape of a polynomial function?
3. Can a higher-degree term dominate the end behavior if it has a smaller coefficient?
4. What would the end behavior be for a polynomial of degree 6 with a positive leading coefficient?
5. How does graphing a polynomial help in visualizing its end behavior?

**Tip**: Always check the degree and leading coefficient first to quickly determine end behavior of polynomials.

Find the correct end behavior diagram for the given polynomial function: P(x) = -x^5 - 5x^3 - 5x + 2

Discover how to analyze the end behavior of the polynomial function P(x) = -x^5 - 5x^3 - 5x + 2. Learn the rules for odd-degree polynomials and negative leading coefficients with this detailed explanation, suitable for Grades 10-12.

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