Problem 41 - Elementary and Intermediate Algebra

Question

Factor. $$ 5(a-b+c)-t(a-b+c) $$

Step-by-Step Solution

Answer

The expression factors to \((5-t)(a-b+c)\).

Step 1: Recognize Common Terms

First, observe that the expression is of the form \(5(a-b+c) - t(a-b+c)\). Notice that \(a-b+c\) is a common term in both parts of the expression.

Step 2: Apply Factorization by Grouping

Since \(a-b+c\) is a common factor, we can apply the distributive property (factorization by grouping) to factor it out from the expression. This involves writing the expression as \((5-t)(a-b+c)\).

Step 3: Simplify the Expression

The simplified expression is \((5-t)(a-b+c)\). This is the factorized form of the given expression.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions and solve equations. It allows you to multiply a single term across a sum or difference inside parentheses. For example, if you have an expression of the form \( a(b + c) \), using the distributive property, it becomes \( ab + ac \). When factoring, it works in reverse—extracting a common factor from terms.

In the original exercise, the distributive property is reversed to factor the expression. Recognizing the common term \((a-b+c)\) in \(5(a-b+c) - t(a-b+c)\), you can factor it out to write the expression in a simpler form: \((5-t)(a-b+c)\). This step shows how the distributive property helps you combine terms efficiently.

Common Factor

Finding a common factor is an essential part of algebra, especially when simplifying expressions. A common factor is a term or number that divides two or more terms evenly. Factoring is a method used to identify and extract these common factors to simplify algebraic expressions and solve equations.

In our given problem, \((a-b+c)\) is identified as a common factor within both terms of the expression \(5(a-b+c) - t(a-b+c)\). By factoring \((a-b+c)\) out, you're left with the expression \((5-t)(a-b+c)\). Finding the common factor not only simplifies the expression but also makes it easier to understand and manipulate further in problem-solving contexts. It is a fundamental skill to master in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division) that represent a specific value. These expressions are the foundation of algebra and allow us to generalize mathematical relationships.

Understanding and manipulating algebraic expressions is crucial for problem-solving. For the exercise given, the expression \(5(a-b+c) - t(a-b+c)\) is broken down using algebraic techniques like finding common factors and applying the distributive property. These processes transform a complex expression into a simpler one, making it easier to work with.

• Enable generalized calculations for different values
• Serve as a framework for writing equations
• Simplify complex mathematical problems

Learning to handle algebraic expressions is vital in developing deeper mathematical knowledge and an ability to tackle various algebraic challenges.