The Difference of Squares: (xy)(x+y)
In mathematics, the expression (xy)(x+y) is a common and important pattern known as the difference of squares. Understanding this pattern can significantly simplify algebraic expressions and calculations.
Expanding the Expression
To see why (xy)(x+y) is called the difference of squares, let's expand the expression:
 Step 1: Apply the distributive property.
 (xy)(x+y) = x(x+y)  y(x+y)
 Step 2: Distribute further.
 x(x+y)  y(x+y) = x² + xy  xy  y²
 Step 3: Simplify by combining like terms.
 x² + xy  xy  y² = x²  y²
The Result: x²  y²
As you can see, expanding (xy)(x+y) results in x²  y², which is the difference of two squares: x² and y². This is why the expression is called the difference of squares.
Applications and Significance
Understanding the difference of squares pattern is crucial in algebra for several reasons:
 Factoring: It allows you to easily factor expressions in the form x²  y² into (xy)(x+y). This is helpful for solving equations and simplifying expressions.
 Simplifying expressions: The pattern can be used to quickly simplify more complex expressions involving the difference of squares.
 Solving equations: The difference of squares pattern can be used to solve quadratic equations of the form ax²  c = 0.
Examples
Here are some examples of how the difference of squares pattern can be applied:

Factoring: Factor the expression x²  9.
 Notice that 9 is a perfect square (3²).
 Therefore, x²  9 can be factored as (x  3)(x + 3).

Simplifying expressions: Simplify the expression (2x  3y)(2x + 3y).
 Using the difference of squares pattern, we get: (2x)²  (3y)² = 4x²  9y²

Solving equations: Solve the equation x²  16 = 0.
 Factor the equation: (x  4)(x + 4) = 0
 Set each factor to zero and solve:
 x  4 = 0 => x = 4
 x + 4 = 0 => x = 4
Conclusion
The difference of squares pattern is a fundamental concept in algebra with numerous applications in simplifying expressions, factoring, and solving equations. By understanding and applying this pattern, you can significantly improve your understanding and proficiency in algebra.