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The distributive property is a fundamental principle in algebra that helps simplify expressions. It involves distributing or spreading out one term across other terms inside parentheses. For example, if we have \( a(b + c) \), we can use the distributive property to rewrite it as \( ab + ac \).

In the given exercise, we start with: \( \( x^{3} - x^{2}y \) - \( 4xy^{2} - y^{3} \) + \( 3x^{2}y - xy^{2} \) \).

We need to distribute the negative sign from the second parentheses first: \( x^{3} - x^{2} y - 4xy^{2} + y^{3} + 3x^{2} y - x y^{2} \).

By distributing negative signs properly, we make the expression easier to work with in the following steps.

Like terms are terms in an algebraic expression that have identical variables raised to the same power. This similarity allows them to be combined to simplify the expression. For instance, \( 2x^{2} \) and \( 3x^{2} \) are like terms because both involve \( x^{2} \), whereas \( x^{2} \) and \( xy \) are not because the variable parts are different.

After distributing, our expression is: \( x^{3} - x^{2} y - 4xy^{2} + y^{3} + 3x^{2} y - x y^{2} \).

We then group like terms together:

• \( x^{3} \)
• \( -x^{2} y \) and \( 3x^{2} y \)
• \( -4xy^{2} \) and \( -xy^{2} \)
• \( y^{3} \)

Grouping like terms helps organize the expression and ultimately combine terms to simplify further.

An algebraic expression includes variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions represent real-world situations and are simplified or solved using algebraic techniques.

For this exercise, we combined like terms as follows: \( x^{3} \) is alone, so it remains \( x^{3} \).
\( -x^{2} y \) and \( 3x^{2} y \) simplify to \( 2x^{2} y \).
\( -4xy^{2} \) and \( -xy^{2} \) simplify to \( -5xy^{2} \). Finally, \( y^{3} \) also remains the same.

Thus, the simplified expression is: \( x^{3} + 2x^{2} y - 5xy^{2} + y^{3} \).

Understanding algebraic expressions and how to manipulate them is crucial for solving more complex mathematical problems.