Problem 67 - Algebra A Combined Function

Question

Use the distributive property to write each sum as a product. See Examples 13 and 14. $$ (-1) \cdot 5+(-1) \cdot x $$

Step-by-Step Solution

Answer

\(-1 (5 + x)\)

Step 1: Identify the Common Factor

In the expression \((-1) \cdot 5 + (-1) \cdot x\), the common factor is \(-1\), as it is multiplied to both terms, 5 and \(x\).

Step 2: Apply the Distributive Property

Using the distributive property, factor out \(-1\) from both terms. This gives us: \(-1 (5 + x)\).

Step 3: Simplify the Expression

The simplified expression that represents the product, after factoring out \(-1\), is \(-1 \cdot (5 + x)\).

Key Concepts

Factoring

When dealing with algebraic expressions, factoring is an essential skill that helps simplify problems and reveal their underlying structure. Factoring involves finding common factors in an expression and rewriting it as a product of these factors and another expression.

In the expression \(-1 \cdot 5 + (-1) \cdot x\), the process of factoring involves identifying the common factor, which in this case is \(-1\). This means both terms are multiplied by \(-1\), making it our common factor.

By factoring out \(-1\), you are essentially using the distributive property in reverse. It transforms the expression into a simpler product form: \(-1(5 + x)\). This step of factoring simplifies calculations and clarifies relationships between terms, making it easier to solve complex problems or perform other mathematical operations later.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations, representing mathematical quantities. They can vary in complexity, from simple expressions like \(x + 2\) to more involved ones like \(-1 \cdot 5 + (-1) \cdot x\).

Understanding algebraic expressions is key in algebra, as they are used to demonstrate relationships and solve equations. When looking at an expression, it's crucial to identify the different parts:

• Constants, like 5 in the example. They are fixed values.
• Variables, such as \(x\), which represent unknown values.
• Coefficients, like \(-1\), are numbers that multiply the variables.

Notice how the rules of algebra allow you to manipulate expressions—factoring is one aspect of this.

By rewriting expressions, such as in our example \(-1 \cdot (5 + x)\), we make them easier to handle in equations and understand the mathematical operations involved.

Mathematical Operations

Mathematical operations are foundational actions performed on numbers and algebraic expressions. Involving actions like addition, subtraction, multiplication, and division, these operations dictate how we calculate expressions.

In our example \(-1 \cdot 5 + (-1) \cdot x\), the operations involved are multiplication and addition. First, each term is multiplied by \(-1\); then, the products are summed to form the expression.

Every operation follows specific rules and properties that dictate how they interact with numbers and variables:

• Addition combines values or expressions.
• Multiplication amplifies them based on a factor, like \(-1\) here, which also affects the sign.

Using the distributive property, a specific rule for distributing multiplication over addition, allows us to rewrite the original expression as a product: \(-1(5 + x)\).

This manipulation highlights the operations performed within the original expression and shows how factoring simplifies solving problems by reducing the expression to a clearer structure.