Understanding the Expansion of (xy)(x^2 + xy + y^2)
The expression (xy)(x^2 + xy + y^2) represents a special case of algebraic manipulation known as the difference of cubes. This pattern arises frequently in algebra and is essential for simplifying expressions and solving equations.
Expanding the Expression
To understand the simplification, we'll use the distributive property:

Multiply the first term of the first factor (x) by each term in the second factor:
 x * x^2 = x^3
 x * xy = x^2y
 x * y^2 = xy^2

Multiply the second term of the first factor (y) by each term in the second factor:
 y * x^2 = x^2y
 y * xy = xy^2
 y * y^2 = y^3

Combine the resulting terms:
 x^3 + x^2y + xy^2  x^2y  xy^2  y^3

Notice that the middle terms cancel out:
 x^3  y^3
The Difference of Cubes Pattern
The result, x^3  y^3, is the difference of cubes. This pattern holds true for any values of x and y. It can be generalized as:
a^3  b^3 = (a  b)(a^2 + ab + b^2)
This pattern is important to recognize as it allows you to quickly factor expressions and simplify complex equations.
Applications
The difference of cubes pattern has applications in various areas of mathematics, including:
 Factoring expressions: You can factor any expression that fits the difference of cubes pattern.
 Solving equations: Knowing the pattern can help you solve equations containing cubic terms.
 Calculus: The difference of cubes pattern can be used to simplify expressions in calculus problems.
By understanding the expansion and pattern of (xy)(x^2 + xy + y^2), you gain a valuable tool for simplifying algebraic expressions and solving problems.