https://file.aiquickdraw.com/mathbot/file/1727056267218bavhvrl2.jpg

The problem presented in the image is to simplify the expression:

\[
\frac{x^3 - 22x^2 + 121x}{x^3 - 11x^2}
\]

### Steps to simplify:

1. **Factor the numerator and the denominator**:

   - **Numerator**: \( x^3 - 22x^2 + 121x \)
     - Notice that \(x\) is a common factor. So, factor out \(x\):
     \[
     x(x^2 - 22x + 121)
     \]
     - The quadratic \(x^2 - 22x + 121\) can be factored further. It is a perfect square:
     \[
     x^2 - 22x + 121 = (x - 11)^2
     \]
     - So, the numerator becomes:
     \[
     x(x - 11)^2
     \]

   - **Denominator**: \( x^3 - 11x^2 \)
     - Here, \(x^2\) is a common factor. Factor out \(x^2\):
     \[
     x^2(x - 11)
     \]

2. **Simplify the fraction**:
   \[
   \frac{x(x - 11)^2}{x^2(x - 11)}
   \]
   - Cancel out the common factors \(x\) and \(x - 11\):
     \[
     \frac{(x - 11)}{x}
     \]

### Final simplified expression:
\[
\frac{x - 11}{x}
\]

Let me know if you'd like further clarification!

---

Would you like more details or have any questions?

Here are 5 related questions you can explore:
1. How do you factor complex trinomials quickly?
2. What are the common strategies for simplifying rational expressions?
3. How does canceling common factors work in fractions?
4. Can every quadratic expression be factored, or are some irreducible?
5. Why is factoring important in algebraic simplification?

**Tip:** Always check for common factors before simplifying, as this can save you time in long algebraic problems.

Simplify the expression: (x^3 - 22x^2 + 121x) / (x^3 - 11x^2)

Simplify the rational expression (x^3 - 22x^2 + 121x) / (x^3 - 11x^2) using factoring and canceling common terms. This step-by-step solution focuses on factoring techniques and simplifying algebraic expressions. Ideal for high school algebra students.

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