91Ó°ÊÓ

In the realm of algebra, an \textbf{algebraic expression} represents a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. For example, \( (x+1)(x-2)(x+3) \) is an algebraic expression. Understanding these expressions is vital because they provide a way to describe patterns and relationships among quantities symbolically.

When dealing with algebraic expressions, we often perform operations such as expansion, which involves using distributive property to multiply individual terms, and combining like terms to simplify the expression. The steps demonstrated in our given exercise show how to expand a complex algebraic expression by multiplying binomials in a step-by-step fashion.

The \textbf{distributive property} is a cornerstone of simplifying algebraic expressions. It dictates that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products, which is represented as \( a(b+c) = ab + ac \).

When we apply this to polynomials, we distribute each term of one polynomial across each term of the other. In our exercise, we first distribute the term \(x\) across \(x-2\), and then distribute the constant \(1\) across \(x-2\). By adhering to the distributive property, we efficiently multiply binomials and trinomials, and gradually expand more complex expressions like the one given in our exercise.

Simplifying an expression is all about making it as concise and straightforward as possible. This involves combining like terms—those that have the same variable raised to the same power. After using the distributive property to expand an expression, simplifying is an essential step to find the most reduced form of the expression.

In our exercise, simplifying the expression \(x^3 + 3x^2 - x^2 - 3x - 2x - 6\) involves combining the \(x^2\) terms and then the \(x\) terms. The result, \(x^3 + 2x^2 - 5x - 6\), is the simplified form of the expanded expression. Simplifying helps in various contexts, like finding the value for a particular variable or solving equations efficiently.