Problem 44 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ a r-b r+a s-b s $$

Step-by-Step Solution

Answer

The factored form is \((r + s)(a - b)\).

Step 1: Group the Terms

We start by grouping the terms in pairs. The expression is \( ar - br + as - bs \). We will group the first two terms together and the last two terms together: \((ar - br) + (as - bs)\).

Step 2: Factor out the Common Factor from Each Group

Now, within each group, we factor out the greatest common factor. From the first group \((ar - br)\), the common factor is \(r\), so it becomes \(r(a - b)\). From the second group \((as - bs)\), the common factor is \(s\), so it becomes \(s(a - b)\).

Step 3: Combine the Factored Terms

We notice that \((a - b)\) is common in both terms. Thus, the expression can be combined to \((r + s)(a - b)\).

Key Concepts

Understanding the Common Factor

Factoring by grouping is a method used in algebra to simplify a polynomial by identifying and factoring out common factors from different terms. The concept of a "common factor" plays a crucial role here. A common factor is a number or variable that divides exactly into each term of an expression.

When we say that a term has a common factor, we're looking for factors that are shared between terms within a group. In the expression from the problem, \

• In \ \( ar - br \), the common factor is \( r \) because both terms in this group can be divided by \( r \).
• Similarly, in \ \( as - bs \), the common factor is \( s \) as both terms can be divided by \( s \).

After removing the common factor from each group, we simplified the expression, making it easier to handle in further calculations. Recognizing and factoring out a common factor are fundamental skills in simplifying algebraic expressions.

Exploring Polynomials

Polynomials are expressions made up of variables and coefficients that involve operations like addition, subtraction, and multiplication, but no division by a variable. Polynomials can have varying numbers of terms and degrees. In algebra, polynomials with more complex terms are often simplified or rearranged using techniques like factoring by grouping.

Consider the polynomial in the exercise:

• The expression \( ar - br + as - bs \) is a polynomial with four terms.
• Each term is a product of integer coefficients and variables, making it possible to apply grouping.

Factoring by grouping is particularly useful for multi-term polynomials. It involves reordering terms to reveal common factors, simplifying the expression into a more manageable form. Understanding the structure of polynomials and their characteristics is key in identifying when and how to use the grouping method for effective simplification.

Algebraic Expressions Unveiled

Algebraic expressions include polynomials and many other types of expressions formed with variables and constants. When dealing with such expressions, the goal often is to simplify or solve them. Using factoring methods like grouping, we manipulate these expressions to make them simpler and easier to solve or analyze.

The original expression \( ar - br + as - bs \) is an algebraic expression where:

• Terms \( ar, br, as, \) and \( bs \) involve coefficients \( a, b \) with factors \( r \) and \( s \).
• By finding and factoring out the common factors, it allows us to express it in the product of simpler terms as \( (r + s)(a - b) \).

This process of uncovering simpler forms illustrates the flexibility and power of algebraic techniques. Becoming familiar with these methods helps in breaking down complex algebraic expressions effectively, leading to easier computations and insight into mathematical relationships.