91Ó°ÊÓ

The distributive property is a fundamental concept in mathematics, especially when working with algebraic expressions. It allows us to simplify complex expressions by distributing one term over others within parentheses.

In our problem, we begin by applying the distributive property to the expression \( (x-y)(x^2 + xy + y^2) \). This involves multiplying the entire \((x-y)\) term by each component of the second factor inside the parenthesis.

Using this property, your expression can look something like:

• \( (x-y) \cdot x^2 \)
• \( (x-y) \cdot xy \)
• \( (x-y) \cdot y^2 \)

After applying the distributive property, we break down the initial complex expression into simpler terms, allowing further steps for simplification. It's like unpacking a complicated idea into simpler parts.

Simplifying expressions is a vital step in algebra that makes working with them more straightforward. Once we've applied the distributive property, we need to manage each resulting term separately.

In our exercise, we expanded the following expressions:

• \( (x-y) \cdot x^2 = x^3 - yx^2 \)
• \( (x-y) \cdot xy = x^2y - y^2x \)
• \( (x-y) \cdot y^2 = xy^2 - y^3 \)

These operations help in transforming the expression into smaller, easier-to-manage pieces.

By simplifying, you also reduce the risk of errors in subsequent calculations, making this step indispensable in algebra.

After simplifying the expressions, the final task is to check for like terms. Like terms are terms that contain the same variables raised to the same power. They can be combined to further simplify the expression.

In our example, the expression after simplification is \( x^3 - yx^2 + x^2y - y^2x + xy^2 - y^3 \).
While inspecting this expression, you will note that it already contains no like terms together to combine further.

This is an essential step, as merging like terms reduces the expression to its simplest form. Recognizing and combining like terms highlight patterns within algebraic expressions, making calculations more approachable.