Problem 42 - Elementary and Intermediate Algebra

Question

Factor. $$ (a-b-c) r-(a-b-c) s $$

Step-by-Step Solution

Answer

Factor: \((a-b-c)(r-s)\).

Step 1: Identify Common Factors

Look at the expression \[(a-b-c)r - (a-b-c)s\] and identify the common factor which is \((a-b-c)\). This factor appears in both terms.

Step 2: Factor Out the Common Term

Use the distributive property to factor out the common factor \((a-b-c)\) from both terms of the expression:\[(a-b-c)(r-s)\] This step reduces the expression by taking out the common factor as a single entity.

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions by distributing a factor across terms within parentheses. It states:

• For any numbers or variables, the expression \( a(b + c) = ab + ac \).
• This means that the factor outside the parentheses multiplies each term inside.

In our original exercise, we used the distributive property in reverse to factor out a common factor from two terms. Instead of distributing, we "undistribute" by pulling out the common factor, \( (a-b-c) \), from each term. This process simplifies the expression from\( (a-b-c)r - (a-b-c)s \) to\( (a-b-c)(r-s) \).Recognizing when and how to apply the distributive property in this way is crucial for factoring and simplifying algebraic expressions.

Common Factors

A common factor is a number or expression that divides two or more expressions without remaining values. Finding common factors allows us to simplify expressions by reducing repetition.In the given exercise, the term \( (a-b-c) \) is a common factor since it is present in both terms of the expression\( (a-b-c)r - (a-b-c)s \). Here’s how we identify and utilize common factors:

• Check each term to find repetitive factors.
• Factor them out to streamline expressions.

After identifying \( (a-b-c) \) as a common factor, we can factor it out, as shown by rewriting the expression in a simplified form:\[ (a-b-c)(r-s) \]This process reduces the complexity of expressions and reveals underlying structures that are easier to work with in further algebraic operations.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations (like addition or multiplication) combined together. Understanding how to manipulate these expressions is key in algebra.For example, in the expression \( (a-b-c)r - (a-b-c)s \):

• \( a, b, \) and \( c \) are variables or constants grouped by subtraction.
• \( r \) and \( s \) are multiplied, each by the same expression \( (a-b-c) \).

This tells us:

• How expressions interact in complex ways.
• The importance of recognizing and resolving such patterns.

When we factor expressions, we not only simplify but also transform them into a workable form. This helps in solving equations, performing calculations, and predicting behavior in broader mathematical problems. Understanding algebraic expressions is foundational for higher-level math and real-world problem solving.