Let's solve the expression \( x^5 - x^3 + 27x^2 - 27 \).

To analyze this expression, we can look for possible factorization or simplify it directly. Here's the expression:

\[ x^5 - x^3 + 27x^2 - 27 \]

### Factorization and Simplification:
1. **Grouping Terms**: Grouping might help in factoring:
   \[
   (x^5 - x^3) + (27x^2 - 27)
   \]
   - The first group can be factored by taking out \( x^3 \):
     \[
     x^3(x^2 - 1)
     \]
   - The second group can be factored by taking out \( 27 \):
     \[
     27(x^2 - 1)
     \]
   - So, the expression becomes:
     \[
     x^3(x^2 - 1) + 27(x^2 - 1)
     \]
2. **Factoring Out Common Factor**: Notice \( (x^2 - 1) \) is common in both terms:
   \[
   (x^2 - 1)(x^3 + 27)
   \]

3. **Further Factorization**:
   - \( x^2 - 1 \) can be further factored as:
     \[
     (x - 1)(x + 1)
     \]
   - \( x^3 + 27 \) is a sum of cubes, which can be factored as:
     \[
     (x + 3)(x^2 - 3x + 9)
     \]

### Final Factored Form:
So, the complete factorization of the given expression is:
\[
(x - 1)(x + 1)(x + 3)(x^2 - 3x + 9)
\]

Would you like more details on the factorization process or have any other questions?

Here are 5 related questions that could expand your understanding:
1. How do you factor expressions that involve higher powers like \(x^5\)?
2. What is the sum of cubes formula, and how is it applied?
3. Can you demonstrate the difference of squares using \(x^2 - 1\)?
4. How can you verify the factorization of a polynomial by expanding the factors?
5. What are the possible real roots of the factored expression?

**Tip**: When factoring polynomials, always check for common factors first before moving on to more complex techniques like grouping or special identities.

Explore the factorization process of the polynomial x^5 - x^3 + 27x^2 - 27 with a detailed step-by-step solution. Understand polynomial factorization techniques including grouping, factoring common terms, and the sum of cubes formula. Suitable for advanced high school students.

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