Factoring the Expression x2 2xy y2 - a2 - 2ab - b2 into Two Factors

Factoring is a fundamental skill in algebra, enabling us to simplify expressions and solve equations more easily. This article delves into the process of factoring the expression x^2 2xy y^2 - a^2 - 2ab - b^2 into two factors, using algebraic identities and the difference of squares formula.

Step-by-Step Breakdown

Step 1: Analyzing the Expression

First, consider the given expression:

x^2 2xy y^2 - a^2 - 2ab - b^2

We notice that the first three terms, x^2 2xy y^2, can be rewritten as a perfect square trinomial:

x^2 2xy y^2 (x y)^2

Thus, our expression can be rewritten as:

(x y)^2 - a^2 - 2ab - b^2

Step 2: Simplifying the Remaining Terms

Next, we need to simplify the remaining terms. Notice that:

-a^2 - 2ab - b^2 - (a^2 2ab b^2)

The expression inside the parentheses is another perfect square trinomial:

a^2 2ab b^2 (a b)^2

Thus, we can rewrite the expression as:

(x y)^2 - (a b)^2

Step 3: Applying the Difference of Squares Formula

The difference of squares formula states that for any two terms A and B:

A^2 - B^2 (A - B)(A B)

Here, we have:

A x y

B a b

Thus, applying the difference of squares formula, we get:

(x y)^2 - (a b)^2 (x y - (a b))(x y (a b))

Final Answer:

The factorization of the expression is:

boxed{(x y - (a b))(x y (a b))}

Summary of Key Steps

Rewrite the expression as (x y)^2 - (a b)^2. Apply the difference of squares formula: A^2 - B^2 (A - B)(A B). Identify A x y and B a b. Simplify the expression to (x y - (a b))(x y (a b)).

Related Keywords

Factoring Algebraic Expressions Difference of Squares Algebraic Identities

Additional Resources

To further explore and practice factoring algebraic expressions, you can refer to the following resources:

Math is Fun: Polynomials Khan Academy: Factoring Expressions Purplemath: Factoring Special Products and Formulas

These resources will provide additional examples and exercises to help you master factoring techniques.