Problem 73 - Elementary and Intermediate Algebra

Question

Factor. $$ y(x+2)+3(x+2) $$

Step-by-Step Solution

Answer

The factored form is \((x+2)(y+3)\).

Step 1: Identify the Common Factor

The expression is \( y(x+2) + 3(x+2) \). To factor it, first identify the common factor in both terms, which is \((x+2)\).

Step 2: Factor out the Common Expression

Since \((x+2)\) is common to both terms, we can factor it out:\[y(x+2) + 3(x+2) = (x+2)(y+3)\]

Step 3: Write the Factored Form

The factored form of the expression is \((x+2)(y+3)\). This shows the expression in its simplest factored state.

Key Concepts

Understanding Common Factor

In algebra, finding a common factor is an essential skill when simplifying expressions. A common factor is a term or number that is present in each part of an algebraic expression. Identifying it allows you to combine these parts into a simpler form. Think of a common factor like a shared ingredient in a recipe.

For example, in the expression \(y(x+2) + 3(x+2)\), the parentheses \((x+2)\) act as a common factor. This expression can be likened to two baskets, both containing \((x+2)\). By recognizing this, you can simplify the expression more easily. Once spotted, this common factor can be factored out from each term, much like taking the common ingredient out of two dishes and reusing it.

• Helps simplify calculations.
• Makes solving equations easier and more efficient.
• A foundational concept in algebra.

Revealing the Factored Form

Once you've identified and factored out the common factor, the expression can be rewritten in the factored form. This transformation not only simplifies the expression but also highlights its structure. In the expression \(y(x+2) + 3(x+2)\), once we've determined that \((x+2)\) is common, we can factor it out.

The factored form here becomes \((x+2)(y+3)\). This tells us that there are two parts in the simplified expression, represented by multiplying two binomials. It’s similar to converting a complex sentence into a much simpler one without losing its original meaning.

Achieving the factored form provides a clear view of the roots and value solutions of an equation, especially useful when solving quadratic equations.

Decoding Algebraic Expressions

Algebraic expressions are the building blocks of algebra, composed of numbers, variables, and operation symbols. Understanding these components is key to grasping more complex algebraic concepts. Think of algebraic expressions like a puzzle. Each piece or term fits together according to the rules of math.

Their structure varies: they may consist of a simple term like \(3x\), or a combination of multiple terms such as \(y(x+2) + 3(x+2)\). This ability to break down expressions into simpler parts like factoring is crucial. It allows for easier computation and understanding.

When working with algebraic expressions, always look for opportunities to factor or simplify.

• Combines constants and variables.
• Comprised of one or more terms.
• Can represent real-world relationships mathematically.