In Exercises 75–94, factor using the formula for the sum or difference of two cubes. (x − y)³ − y³

Verified step by step guidance

1. Identify the expression as a difference of cubes: \((x - y)^3 - y^3\).
2. Recognize the formula for the difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
3. Assign \(a = x - y\) and \(b = y\) to match the formula.
4. Apply the formula: \((x - y) - y)((x - y)^2 + (x - y)y + y^2)\).
5. Simplify the expression: \((x - 2y)((x - y)^2 + (x - y)y + y^2)\).

Key Concepts

Difference of Cubes

The difference of cubes is a specific algebraic identity that states a³ - b³ = (a - b)(a² + ab + b²). This formula allows us to factor expressions where one term is the cube of a variable and the other is the cube of another variable, facilitating simplification and solving of equations.

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, including the difference of cubes, is essential for simplifying polynomials and solving algebraic equations effectively.

Cubic Expressions

Cubic expressions are polynomial expressions of degree three, typically in the form ax³ + bx² + cx + d. Recognizing the structure of cubic expressions is crucial for applying appropriate factoring methods, such as the difference of cubes, to simplify or solve them.