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.