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We are given the function:

\[
f(x) = \frac{x^2 - 10x + 24}{x^{12} - 6x^{11}} 
\]

### To find the vertical asymptotes:
Vertical asymptotes occur when the denominator is zero and the numerator is non-zero at the same points. Therefore, we need to find where the denominator \( x^{12} - 6x^{11} \) is equal to zero.

#### Step 1: Factor the Denominator
We can factor the denominator \( x^{12} - 6x^{11} \) as follows:

\[
x^{12} - 6x^{11} = x^{11}(x - 6)
\]

Now, set the denominator equal to zero:

\[
x^{11}(x - 6) = 0
\]

This gives us two solutions:

1. \( x^{11} = 0 \)  →  \( x = 0 \)
2. \( x - 6 = 0 \)  →  \( x = 6 \)

#### Step 2: Check the Numerator at These Points
We need to ensure the numerator is non-zero at these values of \( x \).

The numerator is \( x^2 - 10x + 24 \).

- At \( x = 0 \):

\[
0^2 - 10(0) + 24 = 24 \quad (\text{non-zero})
\]

- At \( x = 6 \):

\[
6^2 - 10(6) + 24 = 36 - 60 + 24 = 0 \quad (\text{zero})
\]

### Conclusion:
- There is a vertical asymptote at \( x = 0 \) since the denominator is zero and the numerator is non-zero.
- At \( x = 6 \), there is no vertical asymptote because both the numerator and denominator are zero, indicating a **hole** in the graph rather than an asymptote.

Thus, the function has a **vertical asymptote** at \( x = 0 \), and a **hole** at \( x = 6 \).

Would you like further details or have any questions?

### Relative questions:
1. How do we determine horizontal asymptotes for this function?
2. What is the behavior of the function near the vertical asymptote \( x = 0 \)?
3. How can we find the hole at \( x = 6 \) more precisely?
4. Can you provide more insight into why holes occur in rational functions?
5. What other techniques are available to find vertical asymptotes beyond factoring?

### Tip:
Always check both the numerator and denominator for zeros when determining vertical asymptotes to identify potential holes in the graph.

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following function: f(x) = (x^2 - 10x + 24) / (x^12 - 6x^11).

Identify the vertical asymptotes and holes for the rational function f(x) = (x^2 - 10x + 24) / (x^12 - 6x^11) by factoring and analyzing the numerator and denominator. Suitable for high school students, this problem focuses on key algebraic concepts such as polynomial factorization and rational functions.

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