Problem 39 - Elementary and Intermediate Algebra

Question

Factor. $$ (x+y) u+(x+y) v $$

Step-by-Step Solution

Answer

The expression factors to \((x+y)(u+v)\).

Step 1: Identify Common Factors

Look at the expression \((x+y)u + (x+y)v\). Notice that \((x+y)\) is a common factor in both terms of the expression.

Step 2: Factor Out the Common Factor

Since \((x+y)\) is common in both terms, we can factor it out. The expression becomes: \[(x+y)(u+v)\] This is the factored form of the original expression.

Key Concepts

Common Factors

In algebra, the term **common factors** refers to components that appear in multiple parts of an expression or equation. When we talk about factoring expressions, identifying common factors is one of the most crucial steps. Let's break it down.

A common factor in a mathematical expression is a term that each part of the expression shares. It can be a number, a variable, or even a more complex expression. For instance, in the expression \((x+y)u + (x+y)v\), the term \((x+y)\) is a common factor because it appears in both parts of the expression.

To effectively find common factors, look for repeated components in the expression. Sometimes it requires multiplying and rearranging terms until the commonality becomes evident. By recognizing these shared pieces, one can simplify expressions or solve equations more efficiently.

Algebraic Expressions

**Algebraic expressions** are mathematical statements that include numbers, variables, and operations, like addition or multiplication, to represent a value or relationship. Let's explore what makes these a key component of algebra.

When we talk about algebraic expressions, they can be simple or complex:

• Simple expressions might be something like \(3a + 5b\), where numbers (coefficients) and letters (variables) are used.
• More complex expressions, such as \((x+y)u + (x+y)v\), include multiple terms and operations.

In our specific example, the algebraic expression is comprised of two terms: \((x+y)u\) and \((x+y)v\). Each component has its own variables and potentially coefficients.

Working with these expressions requires understanding the roles of each element:

• Co-efficients: These are the numbers that multiply the variables (e.g., in \(3x\), 3 is the coefficient).
• Variables: Often letters like \(x\) or \(y\), variables act as placeholders and can represent different quantities.

By manipulating these parts, various operations such as simplifying, factoring, and solving are performed.

Factoring Process

The **factoring process** is a method used to simplify algebraic expressions by extracting common elements. This process makes expressions easier to work with and solve. Let’s see how it operates in detail.

Starting with an expression like \((x+y)u + (x+y)v\), the factoring process consists of a few clear steps:

• **Identify Common Factors:** First, as we did in the original problem, find components common to all terms of the expression. In \((x+y)u + (x+y)v\), this is \((x+y)\).
• **Factor Out the Common Factor:** Once identified, factor out the common factor from each term, essentially dividing every part of the expression by the common component. Here, factoring out \((x+y)\) results in \((x+y)(u+v)\).

The beauty of the factoring process is that it turns complex expressions into more manageable forms. This is incredibly powerful in algebra as it allows for easier solving of equations and simplification.

Remember, successful factoring relies on careful recognition of what terms can be "shared." By consistently applying this process, it becomes a fundamental skill for solving algebra problems efficiently.