To determine the end behavior of the polynomial \( f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7 \), we focus on the leading term of the polynomial because it dominates the behavior of the function as \( x \) approaches infinity or negative infinity.

### Leading Term
The leading term of \( f(x) \) is \( x^4 \). This term will dictate the end behavior of the polynomial.

### Characteristics of the Leading Term
1. **Degree**: The degree of the polynomial is 4, which is an even number.
2. **Leading Coefficient**: The leading coefficient is 1, which is positive.

### End Behavior Analysis
For a polynomial \( f(x) = ax^n \) with a positive leading coefficient \( a \) and even degree \( n \):
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

Since \( x^4 \) is the leading term of \( f(x) \), the end behavior of \( f(x) \) will be similar to that of \( x^4 \):

1. **As \( x \to \infty \)**: The function \( f(x) \) grows without bound, i.e., \( f(x) \to \infty \).
2. **As \( x \to -\infty \)**: The function \( f(x) \) also grows without bound, i.e., \( f(x) \to \infty \).

### Conclusion
The end behavior of the polynomial \( f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7 \) is:
- As \( x \to \infty \), \( f(x) \to \infty \).
- As \( x \to -\infty \), \( f(x) \to \infty \).

Would you like more details or have any questions?

Here are some more questions you might want to ask:
1. How do you determine the end behavior of a polynomial?
2. What are the key characteristics of polynomial functions?
3. How do odd-degree polynomials behave as \( x \) approaches infinity or negative infinity?
4. Can you explain the significance of the leading coefficient in polynomial functions?
5. What are some examples of polynomials with different end behaviors?

Tip: Understanding the leading term of a polynomial is crucial for predicting its end behavior. Focus on the degree and leading coefficient to quickly assess how the polynomial behaves as \( x \) approaches large positive or negative values.

Understand how to identify the end behavior of a polynomial function using the leading term analysis. Example provided for the polynomial f(x) = x^4 - 8x^3 - 8x^2 + 8x + 7. Learn about the impact of degree and leading coefficient on the function's behavior as x approaches infinity or negative infinity. Suitable for advanced high school students.

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