Problem 63 - Beginning and Intermediate Algebra

Question

Use the distributive property to write each sum as a product. See Example 5 \(4 \cdot 1+4 \cdot y\)

Step-by-Step Solution

Answer

The sum can be written as the product: \(4 \cdot (1 + y)\).

Step 1: Understand the Distributive Property

The distributive property in mathematics states that for any numbers or algebraic expressions \(a\), \(b\), and \(c\), the equality \(a(b + c) = ab + ac\) holds. This means you can multiply the term \(a\) by both \(b\) and \(c\) individually and then add the results.

Step 2: Identify Common Factor

Look at the terms in the sum: \(4 \cdot 1 + 4 \cdot y\). Identify the common factor, which in this case is \(4\).

Step 3: Factor Out the Common Factor

Rewrite the expression by factoring out the common factor \(4\) using the distributive property: \(4 \cdot (1 + y)\). This expression represents the product form of the original sum.

Key Concepts

Factoring

Factoring in algebra involves writing an expression as a product of its factors. It is similar to "splitting" or "breaking down" numbers or variables.
Think of it like finding all the pieces that make up a whole. For example, when you have a sum like \(4 \cdot 1 + 4 \cdot y\), factoring helps us identify what is common in these terms.
This "commonness" can be extracted as a single factor outside a bracket, simplifying the expression.

Factoring is especially useful in algebra because:

• It simplifies algebraic expressions.
• It solves equations easily.
• It helps in graphing functions.

By recognizing common factors in expressions, we can use factoring to make calculations more efficient and straightforward.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. In algebra, expressions are the foundation upon which many principles and concepts are built.
They help in representing real-world problems in a manageable and numerical way. For instance, \(4 \cdot 1 + 4 \cdot y\) is an algebraic expression.

Key features of algebraic expressions include:

• They do not have an equals sign.
• They can include one or several terms, like \(4x + 5\) or \(7a - 3b + 6\).
• They allow for the application of operations such as addition, subtraction, multiplication, and division.

Understanding algebraic expressions allows us to manipulate them using rules like the distributive property, making tasks like factoring possible.

Common Factor

A common factor is a value shared by two or more terms in an expression. It's what allows us to "factor out" and simplify expressions efficiently.
In the expression \(4 \cdot 1 + 4 \cdot y\), the number \(4\) is the common factor because it's multiplied by both terms in the sum.

This process of identifying and using common factors is crucial because:

• It makes expressions simpler and more concise.
• It reveals patterns and structures within expressions.
• It is a fundamental step in solving equations and inequalities.

By recognizing common factors, we can better use the distributive property and bring algebraic expressions into a cleaner product form like \(4(1+y)\), enhancing both simplicity and readability.