The expression \( A^3 - 1 \) can be factored using the difference of cubes formula. The formula for the difference of cubes is:

\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]

In your case, \( A^3 - 1 \) can be written as:

\[
A^3 - 1^3
\]

Using the difference of cubes formula:

\[
A^3 - 1 = (A - 1)(A^2 + A \cdot 1 + 1^2)
\]

Simplifying further:

\[
A^3 - 1 = (A - 1)(A^2 + A + 1)
\]

So, the factored form of \( A^3 - 1 \) is:

\[
(A - 1)(A^2 + A + 1)
\]

Would you like more details or have any questions?

Here are some related questions to explore further:
1. How do you factor the sum of cubes, \( A^3 + 1 \)?
2. What are the roots of the equation \( A^3 - 1 = 0 \)?
3. How does the difference of cubes formula compare to the difference of squares?
4. Can \( A^3 - 1 \) be factored further if \( A \) is a complex number?
5. What is the geometric interpretation of factoring \( A^3 - 1 \)?

**Tip:** Remember that the difference of cubes formula can be applied to any terms raised to the third power, not just simple variables.

Explore the process of factoring the polynomial expression A^3 - 1 using the difference of cubes formula. This detailed step-by-step guide is perfect for high school students studying Algebra, covering core concepts like polynomial factoring and the difference of cubes theorem.

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