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In algebra, identifying the common factor in an expression is a crucial first step in the factoring process. A common factor is a term that appears in each part of an algebraic expression. Think of it as a piece shared by all parts of the expression, like a common toy in different gift boxes. In our exercise, the expression is

• \(3(m+n+p) + x(m+n+p)\),

and the common factor is

• \((m+n+p)\).

Recognizing this common factor helps in simplifying the expression, allowing you to "factor out" this shared piece, making the overall expression more manageable and often revealing more insights into its structure.

An algebraic expression is, at its core, a mathematical phrase that can include numbers, variables, and operation symbols. They are used to represent real-world quantities and relationshipsabstractly. In every algebraic expression, the terms are like puzzle pieces that can be moved around or grouped together. In the context of our original exercise, the algebraic expression is

• \(3(m+n+p) + x(m+n+p)\),

with two main terms divided by the plus sign. Each term includes variables, which are letters representing unknown values (in this case, \(m+n+p\) is considered as one whole entity). Understanding how to manipulate these expressions lays the groundwork for solving equations and making sense of algebraic relationships.

Obtaining the factored form of an algebraic expression means expressing the expression as a product of its factors. This process is akin to peeling away layers to get to the core components of the expression. In our given exercise, we've identified a common factor \((m+n+p)\). By factoring it out, or taking it outside the parentheses, we transform the original expression

• \(3(m+n+p) + x(m+n+p)\)

into its factored form

• \((m+n+p)(3 + x)\).

This factored form underscores the relationship between the terms and simplifies the expression for further mathematical operations, making it an essential skill in algebra. It can also make it easier to solve equations or find the values of the variables involved.