Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Use a form of the distributive property to rewrite each algebraic expression without parentheses. $$3(x-2)$$

Step-by-Step Solution

Answer

The algebraic expression \(3(x - 2)\) simplifies to \(3x - 6\) by using the distributive property.

Step 1: Apply the Distributive property

The distributive property of multiplication over addition is a basic property of numbers which says that the product of the sum of two numbers is the sum of the products. In this case, it entails multiplying '3' by each of the terms inside the parentheses. So you would multiply 3*x and 3*-2.

Step 2: Perform the Multiplication

Multiply 3 and x to get \(3x\). Multiply 3 and -2 to get \(-6\). So the expression \(3(x-2)\) expands to \(3x - 6\).

Step 3: Write the Final Expression

Without parentheses, the algebraic expression is now written as \(3x - 6\). This is the simplest form of the expression.

Key Concepts

Algebraic Expressions

Algebraic expressions form the backbone of algebra, providing a way to represent numbers and variables in a mathematical format. An algebraic expression is a combination of numbers, variables, and operators (such as addition, subtraction, multiplication, and division) organized together without an equal sign. For example, the expression \(3(x - 2)\) combines the number 3, the variable \(x\), and the operation of subtraction within parentheses.

• Constants: These are fixed values such as 3 in the expression \(3(x-2)\).
• Variables: Symbols like \(x\) that can represent unknown quantities or varying values.
• Operators: Symbols that indicate mathematical operations. In our example, subtraction is used within the parentheses, and multiplication is implied between the 3 and the parenthesis.

Understanding how these components interact is essential for manipulating and simplifying expressions. It involves recognizing how operations such as the distributive property work to modify and expand expressions.

Simplifying Expressions

Simplifying expressions entails reducing them to their simplest form so they are easier to understand and use. The process involves performing all possible operations and reducing the expression step-by-step. In our example \(3(x - 2)\), simplifying the expression required using the distributive property:

• Distribute: Multiply the number outside the parentheses by each term inside. Here, 3 is multiplied by both \(x\) and -2.
• Combine: After distributing, combine the resulting terms. This gives us \(3x - 6\).

Simplifying helps in quickly evaluating expressions or solving equations. It streamlines algebraic processes by minimizing complexity. By breaking down expressions systematically, one can avoid mistakes and better understand their structure and meaning.

Multiplication in Algebra

Multiplication plays a crucial role in algebra, especially when it comes to dealing with expressions involving parentheses and variables. The distributive property is key in handling such multiplication tasks. It involves multiplying a single term by each term within parentheses and is expressed as \(a(b + c) = ab + ac\).
In the example \(3(x - 2)\), the term 3 is multiplied by each of the two terms inside the parentheses, yielding two separate terms:

• Multiply 3 by \(x\) to get \(3x\).
• Multiply 3 by \(-2\) to get \(-6\).

The multiplication in algebra not only helps in simplifying expressions but also aids in solving equations and understanding patterns between varying quantities. It provides a foundation upon which many other algebraic concepts are built, making it a fundamental process in mathematics.