Problem 78 Factor by grouping. Do not combi... [FREE SOLUTION]

91影视

Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 78

Factor by grouping. Do not combine like terms before factoring. $$ x^{2}+x y-x y-y^{2} $$

Short Answer

Expert verified

The factored form is $(x + y)(x - y)$.

Step by step solution

Identify two pairs of terms

Separate the given expression into two groups for easier factorization. The expression is: $ x^2 + xy - xy - y^2 $. Now, group the terms as: \[(x^2 + xy) + (-xy - y^2)\].

Factor out the greatest common factor (GCF) from each group

Factor the GCF from each of the separated groups. For the first group $(x^2 + xy)$, the GCF is $x$. For the second group $(-xy - y^2)$, the GCF is $-y$. Perform the factorization: \[x(x + y) - y(x + y)\]

Factor by grouping

Notice that $(x + y)$ is a common factor in both terms. Factor $(x + y)$ out of the expression: \[(x + y)(x - y)\]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

greatest common factor

When factoring algebraic expressions, the first step often involves finding the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in a group without leaving a remainder. For instance, in the pair of terms $x^2 + xy$, the common factor is $x$. Similarly, for the pair $-xy - y^2$, the GCF is \-y\. Identifying and factoring out the GCF simplifies the expression and is essential for further operations.

polynomial factorization

Polynomial factorization aims to rewrite a polynomial as the product of simpler polynomials. This helps in simplifying expressions or solving equations. Factoring by grouping is a smart method for polynomials that do not have an obvious GCF across all terms, but may have a GCF within grouped terms. In the provided example, we separate $x^2 + xy - xy - y^2$ into groups $(x^2 + xy) + (-xy - y^2)$ and then factorize each group. This reduces the polynomial to its factors, making it easier to handle.

algebraic expressions

Algebraic expressions consist of variables, constants, and arithmetic operations. Understanding how to manipulate these expressions is crucial in algebra. Consider the expression $x^2+xy-xy-y^2$. Rather than simplifying prematurely, which can obscure factorable parts, proceed with techniques like grouping to help reveal useful patterns. For this expression, separating terms into $(x^2+xy) + (-xy-y^2)$ allows recognizing the common factors and further breaking them down.

grouping method

The grouping method is a powerful technique in factoring complex polynomials. It involves rearranging terms into groups that share common factors. Taking the expression $x^2+xy-xy-y^2$, we first rearrange it as $(x^2+xy) + (-xy-y^2)$. Next, factor the identified GCF from each group, resulting in $x(x+y) - y(x+y)$. Finally, since $x+y$ is common in both groups, factor it out to get the final expression $(x+y)(x-y)$. This method simplifies seemingly complicated expressions effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Factor by grouping. Do not combine like terms before factoring. $$ x^{2}+x y-x y-y^{2} $$

Short Answer

Step by step solution

Identify two pairs of terms

Factor out the greatest common factor (GCF) from each group

Factor by grouping

Key Concepts

greatest common factor

polynomial factorization

algebraic expressions

grouping method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.