Problem 59 - Algebra for College Students

Question

Factor by grouping. $$a x-b y+b x-a y$$

Step-by-Step Solution

Answer

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

Step 1: Group the Terms

Start by organizing the given expression into two groups. The expression is given as \( ax - by + bx - ay \). We can group it as follows: \((ax - ay) + (bx - by)\). This is advantageous because we will then factor each group separately.

Step 2: Factor Each Group

Now, look at each group separately:\((ax - ay)\) and \((bx - by)\). - For the first group \((ax - ay)\), factor out the common factor \(a\), which gives \(a(x - y)\).- For the second group \((bx - by)\), factor out the common factor \(b\), which gives \(b(x - y)\).

Step 3: Factor Out the Common Binomial

Notice that both terms \(a(x - y)\) and \(b(x - y)\) share a common binomial factor \((x - y)\). Factor \((x - y)\) out of the entire expression to get:\( (x - y)(a + b) \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are like the building blocks of algebra, forming equations and functions. In an expression, we see numbers (known as constants), letters (known as variables), and operations like addition, subtraction, multiplication, and division.
For instance, the expression from our original exercise, \( ax - by + bx - ay \), includes variables \(a\), \(b\), \(x\), and \(y\), as well as operations like addition and subtraction. It's essential to understand how these elements interact in an expression to simplify or factor it correctly.

• Variables: Symbols that represent unknown values, like \(x\) and \(y\).
• Constants: Fixed numerical values that do not change within an expression.
• Operations: Mathematical processes such as addition and multiplication that combine numbers and variables.

Recognizing these components allows you to manipulate expressions through various algebraic techniques, like factoring or expanding. This understanding is key when working to simplify or solve equations.

The Art of Binomial Factoring

Binomial factoring involves breaking down an expression into simpler components, specifically focusing on pairs of terms. A binomial expression has exactly two terms, like \(x + y\) or \(ax - ay\). The goal is to factor these terms so that they become a single product of smaller expressions.
In our exercise, the expression when grouped is \((ax - ay) + (bx - by)\). Each group is a perfect candidate for binomial factoring.

• Identify the Groups: Notice pairs or sets of terms that can factor together.
• Factor the Common Terms: Look for common factors in each group. For example, in \(ax - ay\), you can factor out \(a\), yielding \(a(x - y)\).
• Combine Similar Structures: Use the common factors to rewrite the expression into a single product.

Mastering binomial factoring simplifies expressions significantly and is often a stepping stone to solving more complex algebraic problems.

Exploring the Grouping Method

The grouping method is a useful strategy for factoring polynomials, especially when an expression doesn't immediately appear factorable. This technique involves rearranging and combining terms to reveal common factors.
Let’s dissect the original task of factoring \(ax - by + bx - ay\) using grouping:

• Group by Similarity: Create separate terms based on commonalities. Here, grouping terms into \((ax - ay)\) and \((bx - by)\) exposes common variables.
• Factor Each Group: Extract the greatest common factor from each subgroup. From \((ax - ay)\), remove \(a\), resulting in \(a(x-y)\); similarly, remove \(b\) from \((bx - by)\), giving \(b(x-y)\).
• Observe Common Binomials: Both resulting expressions contain \((x - y)\) as a factor, allowing further simplification into a single expression: \((x - y)(a + b)\).

The grouping method is a powerful technique for restructuring polynomials to reveal underlying patterns and simplifications. Practice and awareness of this strategy enhance problem-solving efficiency in algebra.