In Exercises 57–64, factor using the formula for the sum or difference of two cubes. x^3−27

Key Concepts

Difference of Cubes

The difference of cubes is a specific algebraic identity that states a^3 - b^3 can be factored as (a - b)(a^2 + ab + b^2). In the given expression x^3 - 27, we recognize that 27 is 3^3, allowing us to apply this formula by identifying a as x and b as 3.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Understanding various factoring techniques, such as factoring out the greatest common factor or using special formulas like the difference of cubes, is essential for simplifying algebraic expressions effectively.

Polynomial Expressions

A polynomial expression is a mathematical expression that involves variables raised to whole number exponents, combined using addition, subtraction, and multiplication. In this case, x^3 - 27 is a polynomial of degree three, and recognizing its structure is crucial for applying the appropriate factoring methods.