Problem 27 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$3(x+5)$$

Step-by-Step Solution

Answer

The expression can be rewritten without parentheses as \(3x + 15\).

Step 1: Identify the multiplicative constant and the terms inside the parentheses

In the expression \(3(x+5)\), the multiplicative constant is 3 and the terms inside the parentheses are x and 5.

Step 2: Apply the distributive property

Apply the distributive property by multiplying the constant with each term within the parentheses. According to the distributive law, \(3(x+5) = 3*x + 3*5\). Therefore, it can be rewritten as \(3x + 15\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They do not have an equality sign like equations do, making them more flexible for various mathematical manipulations. In the expression \(3(x+5)\), we see that there are numbers (3 and 5) and a variable (\(x\)) included.
Understanding the structure of algebraic expressions is key to simplifying them or performing operations like solving or expanding. When working with them, always look for recognizable patterns or properties, such as the distributive property, to help you simplify or manipulate the expression effectively.
It's important because it forms the basis for solving complex mathematical problems. When you master algebraic expressions, you're well equipped for dealing with things like factoring, solving equations, and working with functions.

Multiplicative Constant

A multiplicative constant is a standalone number used to multiply terms in an expression. In \(3(x+5)\), the number 3 is our multiplicative constant. It is important because it scales the terms within the expression, affecting their total value when expanded or simplified.

• They are essential for distributing values across terms within parentheses.
• Multiplicative constants impact the magnitude but not the structure of an expression.
• Identifying them allows you to correctly apply the distributive property.

In essence, recognizing the multiplicative constant helps you approach the simplification of expressions by knowing which number to distribute to each part of the bracketed expression.

Distributive Law

The distributive law, or property, is a fundamental concept in algebra that describes how to multiply a number by a sum or difference inside parentheses. It involves spreading, or distributing, the multiplicative constant across all terms present inside the parentheses.
In the expression \(3(x+5)\), the distributive law directs us to multiply 3 by \(x\) and 3 by 5. This results in the expanded expression \(3x + 15\).

• Key for expanding expressions into simpler forms.
• Useful in mental math for simplifying calculations.
• Helps in finding equivalent expressions which can be vital in solving equations.

This law is crucial for breaking down expressions into manageable pieces, making complex calculations more straightforward and approachable.