To determine the end behavior of the function $f(x) = (x-5)^3 (x+2)^6$, we focus on the terms with the highest degree as $x \to \infty$ and $x \to -\infty$.

Step 1: Find the degree of the polynomial

• The degree of $(x-5)^3$ is 3.
• The degree of $(x+2)^6$ is 6.

The total degree of the polynomial is $3 + 6 = 9$, making this a degree-9 polynomial.

Step 2: Determine the leading term

The leading term will dominate the end behavior. For large values of $|x|$, the behavior of $f(x)$ resembles the behavior of $x^9$, because the coefficients and lower degree terms become insignificant.

Step 3: Identify the sign of the leading coefficient

Since the leading term is positive ($x^9$) and there is no negative coefficient involved in the expansion, the leading coefficient is positive.

Step 4: Analyze the end behavior

• As $x \to \infty$, $f(x) \to \infty$ (because a degree-9 polynomial with a positive leading coefficient increases without bound).
• As $x \to -\infty$, $f(x) \to -\infty$ (because $x^9$ will be negative for negative $x$).

Conclusion

The end behavior of the function $f(x)$ is:

• $f(x) \to \infty$ as $x \to \infty$.
• $f(x) \to -\infty$ as $x \to -\infty$.

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How would the end behavior change if $f(x)$ had an even degree?
2. What happens to the end behavior if the leading coefficient is negative?
3. How does the multiplicity of each factor affect the shape of the graph near its zeros?
4. What is the behavior of $f(x)$ near the zero $x = 5$?
5. How would you find the x-intercepts of $f(x)$?

Tip: The degree of a polynomial determines its end behavior: odd degrees result in opposite end behaviors at $\infty$ and $-\infty$, while even degrees have the same end behavior at both ends.