Factor using the formula for the sum or difference of two cubes. $x^3-27$

Verified step by step guidance

1. Recognize that the expression \(x^3 - 27\) is a difference of cubes because \(x^3\) is a cube and \(27\) can be written as \$3^3$.
2. Recall the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Identify \(a = x\) and \(b = 3\) in the expression \(x^3 - 27\).
4. Apply the difference of cubes formula by substituting \(a\) and \(b\): \((x - 3)(x^2 + 3x + 9)\).
5. Write the fully factored form as the product of the binomial and trinomial: \((x - 3)(x^2 + 3x + 9)\).

Key Concepts

Difference of Cubes Formula

The difference of cubes formula is used to factor expressions of the form a³ - b³. It states that a³ - b³ = (a - b)(a² + ab + b²). Recognizing this pattern allows you to break down cubic expressions into simpler polynomial factors.

Identifying Perfect Cubes

To apply the sum or difference of cubes formula, you must identify terms that are perfect cubes. For example, x³ is a cube of x, and 27 is a cube of 3 since 3³ = 27. Recognizing these helps in correctly setting a and b in the formula.

Factoring Polynomials

Factoring polynomials involves rewriting them as products of simpler polynomials. This process simplifies solving equations and analyzing functions. Using special formulas like the difference of cubes is a key technique in polynomial factoring.