The expression \( a^3 + 1 \) can be factored using the sum of cubes formula:

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

This is the factorized form of the expression.

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Here are 8 related questions you might find interesting:

1. How do you factor the difference of cubes, \( a^3 - b^3 \)?
2. What are the roots of the equation \( a^3 + 1 = 0 \)?
3. Can you solve \( a^3 + b^3 = 0 \) using the sum of cubes formula?
4. How can the expression \( a^3 + b^3 + c^3 \) be factored?
5. What is the geometric interpretation of the sum of cubes?
6. How does the factorization of \( a^3 + 1 \) differ when \( a \) is a complex number?
7. What happens when you set \( a = 1 \) in the expression \( a^3 + 1 \)?
8. Can you derive the sum of cubes formula from the expansion of \( (a + b)(a^2 - ab + b^2) \)?

**Tip:** When factoring cubic expressions, always look for patterns like the sum or difference of cubes, which have standard formulas.

Explore how to factorize the expression a³ + 1 using the sum of cubes formula. This detailed solution covers algebraic concepts related to cubic equations and factorization techniques. Suitable for advanced high school students.

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