Problem 130 - College Algebra with Modeling and Visualization

Question

Factor the expression completely. \(a x-b x-a y+b y\)

Step-by-Step Solution

Answer

The expression factors to \((a-b)(x-y)\).

Step 1: Group the Terms

Start by rearranging and grouping the terms in pairs: \( (a x - b x) + (-a y + b y) \). This setup allows us to factor by grouping.

Step 2: Factor Out the Common Factor in Each Group

For the first group \( (a x - b x) \), factor out \( x \), giving \( x(a-b) \). For the second group \( (-a y + b y) \), factor out \( y \), resulting in \( y(-a + b) \), which is also \( y(b-a) \). Notice that \((a-b) = -(b-a)\) because these terms are negatives of each other. We rewrite this as: \( x(a-b) - y(a-b) \).

Step 3: Factor Out the Common Binomial Factor

Now that both terms have a common binomial factor \((a-b)\), you can factor it out: \((a-b)(x-y)\).

Step 4: Final Expression

The completely factored form is \((a-b)(x-y)\). This is the simplest form, where no further factoring is possible.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. In algebra, letters such as \(a\), \(b\), and \(x\) represent numbers, which allows us to formulate equations that express general relationships.

The purpose of using algebra is to solve problems that can't be easily tackled with arithmetic alone. By converting words into mathematical equations, algebra helps us find solutions that apply to many situations, not just one specific instance.

Moreover, algebra involves operations such as addition, subtraction, multiplication, division, and exponentiation on these letters and numbers to achieve an intended outcome. For example, in the expression \( a x - b x - a y + b y \), the variables \(x\) and \(y\) might represent different quantities whose relationships are defined by the constants \(a\) and \(b\). Understanding these interactions is crucial for factoring and simplifying the expression.

Binomial Factor

The notion of a binomial factor is pivotal in simplifying expressions in algebra. A binomial is simply a polynomial with two terms, such as \(a-b\) or \(b-a\). When factoring expressions, identifying a common binomial factor allows us to simplify the expression significantly.

In the exercise given, the expression \( a x - b x - a y + b y \) can be reorganized and simplified by recognizing a common binomial factor, \((a-b)\). This factor is essential because it groups the expression into a more manageable form, allowing further simplification.

To successfully use binomial factoring, it's important to first group terms in a way that makes the common factors evident. For instance, by reorganizing the expression into \((a x - b x) + (-a y + b y)\), we can directly spot and extract the binomial factor \((a-b)\), ultimately leading us to an expression like \((a-b)(x-y)\). Recognizing and factoring out common binomial factors are key strategies in algebra for simplifying expressions and solving equations.

Expression Simplification

Expression simplification is a fundamental skill in algebra that involves reducing expressions to their simplest form while retaining the same value. The goal is to make expressions as clear and straightforward as possible, which helps in problem-solving and understanding mathematical relationships.

In our example, the expression \( a x - b x - a y + b y \) undergoes a series of transformations to reach a simplified form. By grouping and factoring out common elements, the expression eventually becomes \( (a-b)(x-y) \). This process eliminates redundancy and reduces complexity, making it easier to work with in further calculations or applications.

Simplifying expressions often involves several techniques, such as combining like terms, using the distributive property, and factoring. It is essential to ensure that throughout this process, the integrity of the original expression's value is maintained. Remember, a simplified expression efficiently communicates the same information as the original, merely in a more accessible way.