Problem 44 - Algebra 1: Concepts and Skills

Question

Use the distributive property to rewrite the expression without parentheses. $$ (2+t)(-2) $$

Step-by-Step Solution

Answer

-4 - 2t

Step 1: Applying the Distributive Property

The expression (2 + t)(-2) requires multiplying each term inside the parentheses by -2. Therefore, -2 is distributed to each term inside the parentheses.

Step 2: Multiply the terms

Multiply -2 with the first term inside parentheses: -2 * 2 = -4.\nMultiply -2 with the second term inside parentheses: -2 * t = -2t.

Step 3: Combining the results

Combine -4 and -2t to get -4 - 2t, which is the expression without parentheses.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They are the building blocks of algebra and allow us to express calculations and relationships in a mathematical format. In the given exercise,

• the expression is \( (2 + t)(-2) \)
• which includes a numeric term (2) and a variable term (\( t \)) inside parentheses, being multiplied by \( -2 \).

Understanding how to manipulate algebraic expressions is crucial in algebra, particularly when simplifying or calculating using properties like the distributive property.

Multiplication

Multiplication is one of the fundamental arithmetic operations. When dealing with algebraic expressions, multiplication often involves numbers, variables, or both. In the problem \( (2 + t)(-2) \), multiplication occurs between the term outside the parentheses, -2, and each term inside the parentheses, which are 2 and \( t \). Here,

• first, \( -2 imes 2 = -4 \)
• second, \( -2 imes t = -2t \).

It's crucial to understand how to perform each of these operations correctly. Performing these multiplications correctly allows us to rewrite the expression without parentheses.

Parentheses

Parentheses in algebraic expressions are used to group terms and indicate that the operations enclosed should be completed first, according to the order of operations (PEMDAS/BODMAS). In the expression \( (2 + t)(-2) \), the parentheses indicate that the sum \( 2 + t \) should be considered as a single unit to be multiplied by \(-2\). Hence, you start by using the distributive property to "collectively" distribute \(-2\) to each term inside the parentheses, resulting in \( -4 \text{ and } -2t \). It essentially helps to simplify the expression in a structured manner.

Negative Numbers

Negative numbers are numbers less than zero, marked by a minus sign. When working with algebra, it's vital to know how they affect operations like multiplication. Such knowledge is essential in problems like \( (2 + t)(-2) \), where

• each time \(-2\) is multiplied by another term, it affects the sign of the result:
  • \(-2 imes 2 = -4\)
  • \(-2 imes t = -2t\).

Multiplying a positive number or variable by a negative number flips the sign. This understanding helps ensure the accuracy when distributing negative multipliers across terms in parentheses.