How do you factor completely x³ - 1?

To factor the expression \(x^3 - 1\) completely, we can use the difference of cubes formula. This formula states that \(a^3 - b^3\) can be factored as \((a - b)(a^2 + ab + b^2)\).

Identifying the Components

In our case, we can set:

• a = x
• b = 1

Applying the Formula

Using the difference of cubes formula, we can rewrite \(x^3 - 1\) as:

\(x^3 - 1 = (x - 1)(x^2 + x \cdot 1 + 1^2)\)

Final Factored Form

This simplifies to:

\(x^3 - 1 = (x - 1)(x^2 + x + 1)\)

Summary

The complete factorization of \(x^3 - 1\) is:

(x - 1)(x² + x + 1)