In Exercises 69–80, factor completely.(x − y)⁴ − 4(x − y)²

Verified step by step guidance

1. Recognize that the expression \((x - y)^4 - 4(x - y)^2\) is a difference of squares.
2. Let \(u = (x - y)^2\). Then the expression becomes \(u^2 - 4u\).
3. Factor out the greatest common factor from \(u^2 - 4u\), which is \(u\), resulting in \(u(u - 4)\).
4. Substitute back \(u = (x - y)^2\) into the factored expression to get \((x - y)^2((x - y)^2 - 4)\).
5. Recognize that \((x - y)^2 - 4\) is a difference of squares and factor it further as \((x - y + 2)(x - y - 2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or expressions. This process is essential for simplifying expressions and solving equations. In this case, recognizing common factors and applying techniques such as the difference of squares will aid in factoring the given expression.

Difference of Squares

The difference of squares is a specific factoring technique that applies to expressions of the form a² - b², which can be factored into (a - b)(a + b). In the given problem, the expression can be viewed as a difference of squares, allowing for further simplification and factorization of the polynomial.

Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the factoring process. In this case, letting u = (x - y)² transforms the original expression into a more manageable quadratic form, making it easier to factor completely and then revert back to the original variables.