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Chapter 6: Problem 78

In Exercises 77-80, factor the polynomial by grouping. $$ (x+y)(x-y)-x(x-y) $$

Short Answer

Expert verified
The factorized form of the given polynomial is \((y)(x-y)\).

Step by step solution

01

Identify Common Factors

The common factor in this polynomial is \(x-y\). The expression can be rewritten as \((x+y)(x-y) - 1*(x)(x-y)\).
02

Employ the Distributive Property

Using distributive property, the expression can be rewritten by factoring out the common term. Hence, \((x+y-1x)(x-y)\).
03

Simplify the Expression

On simplifying, the final expression is \((y)(x-y)\).

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Key Concepts

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

Understanding Polynomials
Polynomials are fundamental in algebra and consist of terms made up of variables and coefficients. Each term can have variables raised to whole number powers. For example, in the given expression \((x+y)(x-y) - x(x-y)\), we have variables \(x\) and \(y\).
Understanding polynomials is essential because they help us understand the behavior of algebraic equations.
  • Terms: Individual components like \(x\), \(y\), or combinations like \((x+y)\).
  • Coefficient: The number in front of a variable, such as the 1 in \(1 \, \cdot \, x\).
  • Factors: Parts of an expression that multiply together to form the polynomial.
To factor polynomials, we simplify them into the product of simpler expressions, crucial for solving equations.
Exploring Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operations. Unlike equations, they don't have an equals sign. In the original problem, expressions like \((x+y)(x-y)\) and \(x(x-y)\) show how different terms interact.
Let's break down some key ideas:
  • Variables: Symbols like \(x\) and \(y\)
  • Operations: Addition, subtraction, multiplication, and division guide the construction of expressions.
  • Grouping: A technique used to factor expressions by identifying and factoring out common segments.
Building a strong foundation in algebraic expressions allows us to manipulate and solve various mathematical problems effectively.
Mastering the Distributive Property
The distributive property is a key tool in algebra that allows us to simplify and factor expressions. It states that \(a(b+c) = ab + ac\). In our exercise, we used the distributive property to factor by grouping.
During the solution:
  • Identified common factor \((x-y)\) to rewrite the expression.
  • Applied property to factor out comon elements, leading to the simplified form.
  • Simplified the equation further by combining like terms.
Understanding and using the distributive property makes it easier to break down complex expressions, turning them into manageable parts. This technique is vital for factoring polynomials and solving algebraic equations.

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