Factoring Complex Expressions: Simplifying (x^2 2xy y^2 - a^2)

In the realm of algebra, factorization plays a crucial role in simplifying expressions and solving equations. Today, we explore the process of factoring the complex expression (x^2 2xy y^2 - a^2). This expression can be broken down into simpler components, making it easier to understand and solve. Let's dive into the steps required to achieve this simplification.

Step 1: Recognize the Pattern

The first step in factoring this expression is to recognize the pattern within it. Take a look at the first three terms: (x^2 2xy y^2). This is a perfect square trinomial, which can be factored using the (a b)^2 formula. The steps to recognize this pattern are as follows:

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

In this case, (a x) and (b y), so we have:

(x y)^2

Now, the expression looks like this:

(x y)^2 - a^2

Step 2: Factor Using the Difference of Squares

The next step is to use the difference of squares to further factor the expression. The difference of squares formula is:

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

In our case, (A (x y)) and (B a). Therefore, the expression can be factored as:

(x y a)(x y - a)

Thus, the fully factored form of the original expression (x^2 2xy y^2 - a^2) is (x y a)(x y - a).

Conclusion

In conclusion, through the steps of recognizing the perfect square trinomial and then applying the difference of squares formula, we have successfully factored the complex expression (x^2 2xy y^2 - a^2). This process not only simplifies the expression but also demonstrates the power of algebraic identities in simplifying complex mathematical problems.

Keywords: factorization, algebraic expressions, simplification