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Chapter 1: Problem 28

Simplify the given algebraic expressions. $$-[(A-B)-(B-A)]$$

Short Answer

Expert verified
The simplified expression is \[2(B - A)\].

Step by step solution

01

Distribute the Minus Sign Inside the Bracket

To simplify the expression \[-[(A-B)-(B-A)]\]start by distributing the negative sign inside the bracket. This means you change the signs of everything inside:\[-(A-B) + (B-A)\]
02

Expand the Expression

Next, expand the terms \[-(A-B)\] and \[(B-A)\]. This results in:\[-A + B + B - A\]
03

Combine Like Terms

Combine the like terms in the expression \[-A + B + B - A\]. This results in:\[-2A + 2B\]
04

Factor the Expression

To simplify further, factor the expression \[-2A + 2B\] by taking \[2\] as a common factor:\[2(B - A)\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When working with algebraic expressions, we often encounter terms that can be simplified by combining them. This is known as combining like terms. **What are like terms?**
  • Like terms are terms in an algebraic expression that have the same variable part raised to the same power.
  • For example, in the expression \(-A + B + B - A\), the terms \(-A\) and \(-A\) are like terms because they both include the variable \(A\).
**How to combine them:**
  • Add or subtract their coefficients, while keeping the variable part unchanged.
  • In the expression from our exercise, we combine \(-A\) and \(-A\) to get \(-2A\), and \(B + B\) gives us \(2B\).
Combining like terms is a crucial step in simplifying algebraic expressions because it makes the expression shorter and often more manageable.
Distribution
Distribution is a method used to eliminate parentheses in an expression by multiplying each term inside the parentheses by the factor outside.**Applying distribution in the example:**
  • In the expression \(-[(A-B)-(B-A)]\), the minus sign is distributed across the terms within the parentheses.
  • It involves changing the signs of each term inside to obtain \(-(A-B) + (B-A)\).
**Why distribution is important:**
  • It allows us to simplify expressions that might initially seem complex due to the presence of parentheses.
  • It paves the way for further simplification, such as combining like terms.
Remember, the key to mastering distribution is noticing how each term within the parentheses is affected by the outside factor.
Factoring Expressions
Factoring is a technique in algebra used to simplify expressions by finding common factors among terms.**Factoring in the previous solution:**
  • We had the expression \(-2A + 2B\) towards the end of our simplification.
  • Here, the common factor between the terms \(-2A\) and \(2B\) is \(2\).
**How to factor:**
  • Identify the greatest common factor (GCF) among the terms. This is the largest factor that divides each term equally.
  • Factor out the GCF and rewrite the expression as a product of the factor and the remaining terms. For example, \(-2A + 2B\) becomes \(2(B - A)\).
Factoring helps in presenting the expression in a more elegant and reduced form, making it easier to analyze and work with in future algebraic operations.

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Most popular questions from this chapter

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