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A common factor in algebraic expressions refers to an element that is shared among all terms of the expression. Understanding how to identify and use common factors is essential in simplifying algebraic equations.

When tasked with simplifying an expression, the first step is to look at each term and spot any elements that are identical across them. These elements can be constants, variables, or even a combination of both. In our original problem, the expression was:

• \(3r(a - b) + s(a - b)\)

Here, \((a-b)\) is present in both terms of the expression, making it a common factor. This is the key to simplifying the expression efficiently.

Recognizing common factors not only helps in factoring expressions but also in solving equations and reducing expressions to its simplest form.

Factorization is the process of breaking down an algebraic expression into products of simpler expressions. It is a crucial skill in algebra, enabling you to simplify and solve equations with ease.

In the problem at hand, after identifying the common factor \((a-b)\), factorization involves rewriting the expression by taking \((a-b)\) out of the terms. The algebraic expression \(3r(a - b) + s(a - b)\) reduces to:

• \((a-b)(3r+s)\)

This means that the original expression can be expressed as the product of two simpler terms: \((a-b)\) and \((3r+s)\).

Factorization is vital as it can often reveal the solution to algebraic equations by providing a simpler form that is more manageable to work with or easier to solve.

Algebraic expressions are combinations of variables, numbers, and operators that represent a particular quantity. These expressions form the building blocks of algebra and are used to represent real-world problems in mathematical form.

An example of an algebraic expression is \(3r(a - b) + s(a - b)\). The expression can include:

• Variables like \(r\), \(s\), and \((a-b)\)
• Coefficients such as the number 3
• Operators like addition and multiplication

By working with algebraic expressions, one can solve equations, compare quantities, and analyze relationships between variables. Understanding how to manipulate these expressions, through operations such as factoring, is essential for solving complex problems in algebra.