Mathematical Simplification: Resolving the Expression (x-y)xy

Mathematics often presents us with expressions that require simplification to understand their true form and identify their antecedents. The expression in question, (x-y)xy, is one such example. What is the solution to this expression?

At first glance, the expression (x-y)xy might seem complex. However, with the appropriate mathematical techniques, such as the FOIL method and the Difference of Squares, we can simplify this expression effectively. The FOIL method stands for First, Outer, Inner, Last, which is a process used to multiply two binomials.

Using FOIL Method

Let's begin by breaking down the expression using the FOIL method:

First: Multiply the first terms, x and x, resulting in (x^2) Outer: Multiply the outer terms, x and -y, resulting in -xy Inner: Multiply the inner terms, -y and x, resulting in -xy Last: Multiply the last terms, -y and -y, resulting in (y^2)

Putting these together, we have:

[begin{align*} (x-y)xy x^2 - xy - xy y^2 x^2 - 2xy y^2 x^2 - y^2 - 2xy end{align*}]

However, it seems this approach was overcomplicated. The problem can be simplified further recognizing a more straightforward pattern: the difference of squares.

The Difference of Squares

The more direct approach involves recognizing the expression (x-y)xy as a difference of squares. This is a fundamental algebraic identity:

[begin{align*} (x-y)xy (x-y)(xy) x^2 - y^2 end{align*}]

Here, the term (xy) cancels out as follows:

[begin{align*} (x-y)xy xxy - yxy x^2y - y^2x x^2 - y^2 tag{cancel out (xy)}end{align*}]

Context and Application

Understanding and applying the Difference of Squares is incredibly useful in several contexts. This identity is fundamental in algebraic simplification, calculus, and various other mathematical operations.

For instance, the expression (x^2 - y^2) is the standard form for the difference of squares, which can be factored as ((x-y)(x y))

The identity is applicable for all real numbers x and y. Therefore, the simplified form of (x-y)xy is:

[boxed{x^2 - y^2}]

Understanding these simplification techniques is crucial for tackling more complex algebraic problems and proofs in advanced mathematics.