Factor by any method. See Examples 1–7. (x+y)^3-(x-y)^3

Verified step by step guidance

1. Recognize the expression as a difference of cubes: \((x+y)^3 - (x-y)^3\).
2. Apply the difference of cubes formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\), where \(a = x+y\) and \(b = x-y\).
3. Substitute \(a = x+y\) and \(b = x-y\) into the formula: \((x+y) - (x-y)\) and \((x+y)^2 + (x+y)(x-y) + (x-y)^2\).
4. Simplify the first factor: \((x+y) - (x-y) = 2y\).
5. Expand and simplify the second factor: \((x+y)^2 + (x+y)(x-y) + (x-y)^2\).

Key Concepts

Difference of Cubes

The expression given involves the difference of two cubes, which can be factored using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, (x+y)^3 and (x-y)^3 are the cubes, where a = (x+y) and b = (x-y). Recognizing this pattern is essential for simplifying the expression.

Binomial Expansion

The binomial expansion theorem allows us to expand expressions of the form (a + b)^n. For n = 3, the expansion is given by (x+y)^3 = x^3 + 3x^2y + 3xy^2 + y^3. Understanding how to apply this theorem is crucial for rewriting the cubes in the expression before factoring.

Factoring Techniques

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Techniques such as grouping, using special products, and recognizing patterns like the difference of squares or cubes are vital for effectively simplifying algebraic expressions.