Factor using the formula for the sum or difference of two cubes. $x^3+64$

Verified step by step guidance

1. Recognize that the expression \(x^3 + 64\) is a sum of two cubes because \(64\) can be written as \$4^3$.
2. Recall the formula for the sum of two cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
3. Identify \(a = x\) and \(b = 4\) in the expression \(x^3 + 4^3\).
4. Apply the sum of cubes formula: \((x + 4)(x^2 - 4x + 16)\).
5. Write the fully factored form as \((x + 4)(x^2 - 4x + 16)\).

Key Concepts

Sum of Cubes Formula

The sum of cubes formula is used to factor expressions of the form a^3 + b^3. It states that a^3 + b^3 = (a + b)(a^2 - ab + b^2). This formula helps break down cubic expressions into simpler polynomial factors.

Identifying Perfect Cubes

To apply the sum or difference of cubes formula, recognize each term as a perfect cube. For example, x^3 is the cube of x, and 64 is the cube of 4 since 4^3 = 64. Correct identification is essential for accurate factoring.

Factoring Polynomials

Factoring polynomials involves rewriting them as products of simpler polynomials. Using special formulas like the sum or difference of cubes simplifies complex expressions, making it easier to solve equations or analyze functions.