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Chapter 5: Problem 138

Explain how you would factor \(x(a-b)+y(b-a)\).

Short Answer

Expert verified
The factored form of the expression \(x(a-b)+y(b-a)\) is \((a - b) * (x - y)\).

Step by step solution

01

Identify common factors

The algebraic expression is \(x(a-b)+y(b-a)\). We can see that \((a - b)\) and \((b - a)\) are essentially the same if we think of \((b - a)\) as \(-(a - b)\). Hence, \((a - b)\) is the common factor here.
02

Factor out the common factor

Since \(b - a = -(a - b)\), we can rewrite the expression as \(x(a - b) - y(a - b)\). Now, we can factor out \((a - b)\) from both terms, giving us \((a - b) * (x - y)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Factors
When we talk about common factors in algebra, we refer to terms that are shared by different parts of an expression. In simpler terms, a common factor is a piece of a mathematical expression that appears in each part of the equation. For example, in the expression \(x(a-b)+y(b-a)\), the common factor is \((a-b)\).
A common factor allows us to simplify expressions by factoring out this shared component, making it easier to see the individual elements of the expression. By identifying common factors, you can transform a complex expression into something more straightforward, thus making it easier to solve or understand.
  • Look for terms that appear in multiple parts of the expression.
  • The expression \((b-a)\) can be rewritten as \(-(a-b)\), highlighting the commonality in the equation.
  • Once identified, the common factor can be used to simplify the expression by factoring it out, providing clearer insight into the remaining structure.
Difference of Terms
In algebra, understanding the difference of terms is crucial. The difference means subtraction, and it is a basic operation in algebraic expressions. The difference can sometimes play a pivotal role when factoring or simplifying expressions.
Consider the expression \((a-b)\) and \((b-a)\). These expressions appear different at first glance due to the reversed order of terms. However, they are negative reflections of each other. Mathematically, \((b-a) = -(a-b)\), meaning the subtraction order impacts the sign but not the fundamental relationship between the numbers.
  • Highlight the interaction and relevance of the sign when terms are switched.
  • Recognizing differences in terms helps in understanding how to transform and factor complex expressions.
  • Once you identify that reversing the terms leads to a sign change, it becomes easier to find common factors or further simplify equations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent mathematical relationships and properties without providing numerical answers. Understanding these expressions is like understanding the language of algebra.
In the expression \(x(a-b)+y(b-a)\), each component carries its own role:
  • Variables \(x\) and \(y\) can represent numbers that may change, showing flexibility in expressions.
  • Parenthetical terms like \((a-b)\) denote a specific operation to perform, ensuring clarity in complex algebraic instructions.
    • li>These expressions can often be transformed by spotting patterns and common factors, which makes them easier to solve or rearrange.
Grasping the nuances of algebraic expressions lets you effectively decipher and manipulate the relationships they depict, paving the way for better problem-solving skills in mathematics.

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