Problem 67 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each sum as a product. See Example 5 \((-1) \cdot 5+(-1) \cdot x\)

Step-by-Step Solution

Answer

\((-1)(5 + x)\)

Step 1: Identify the common factor

In the expression \[(-1) \cdot 5 + (-1) \cdot x\],the term \((-1)\) is a common factor in both parts of the sum. This is a key observation as it allows us to use the distributive property.

Step 2: Apply the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\),\[a(b + c) = ab + ac\].Here, we can factor out the common factor of \((-1)\) from each term:\[ (-1) \cdot 5 + (-1) \cdot x = (-1)(5 + x) \].

Step 3: Write the expression as a product

Having applied the distributive property, the given sum is now written as a single product:\[ (-1)(5 + x) \].This is the expression in its factored form, which is the needed outcome of applying the distributive property.

Key Concepts

Factoring

Factoring is a powerful tool in algebra that simplifies expressions by extracting common factors. Think of it as reverse multiplication, where the goal is to express a sum as the product of simpler terms. This process often begins by identifying common factors that are present in each term of the expression. For example, in the expression \(-1\) \( \cdot \) 5 + \((-1)\) \( \cdot x\), the factor \((-1)\) appears in both terms. Factoring makes it possible to rewrite this expression as \((-1)(5 + x)\), illustrating how multiple terms can be grouped into one expression using the distributive property.

Factoring not only simplifies complex algebraic expressions but also makes it easier to solve equations and understand relationships between variables.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They form the backbone of algebra and serve as a universal language for describing mathematical situations. Consider the algebraic expression \(-1\) \( \cdot \) 5 + \((-1)\) \( \cdot x\). It is composed of constants like 5, variables such as \(x\), and operations like multiplication and addition.

Expressions can be simple or complex, but the key is understanding their structure. Breaking them into manageable pieces helps to spot patterns and simplifications, like finding common factors for factoring.

• Constants: Fixed values, e.g., 5 in the example.
• Variables: Symbols that can represent unknown values, e.g., \(x\).
• Operations: Mathematical processes like addition or multiplication.

The interplay between these elements allows us to manipulate and solve expressions effectively.

Common Factor

A common factor is a number or expression that divides two or more terms without leaving a remainder. Identifying a common factor is an essential step in simplifying algebraic expressions.In the expression \((-1) \cdot 5 + (-1) \cdot x\), \((-1)\) serves as the common factor. This means that \((-1)\) is present in both terms and can be factored out to simplify the expression.

To identify a common factor, follow these steps:

• List all factors of each term.
• Determine which factor(s) are common to all terms.
• Use the common factor in the process of factoring.

Remember, factoring using common factors not only makes expressions neater but also prepares them for solving equations by making subsequent steps easier and clearer.