/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 83 Factor by grouping. \(y^{2}+3 ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 83

Factor by grouping. \(y^{2}+3 x+3 y+x y\)

Short Answer

Expert verified
Factor: (y + 3)(y + x)

Step by step solution

01

Group Terms

Group the terms in pairs to facilitate factoring by grouping:y^2 + 3y + 3x + xy
02

Factor out the Greatest Common Factor (GCF) from Each Pair

Factor out the greatest common factor from each pair of grouped terms:y(y + 3) + x(3 + y)
03

Recognize and Factor Out the Common Binomial Factor

Notice that y + 3 and 3 + y are the same binomial (since addition is commutative). Factor out the common binomial factor:y(y + 3) + x(y + 3) = (y + 3)(y + x)

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
The Greatest Common Factor (GCF) is an important concept in algebra. It's the largest number or algebraic expression that divides each term perfectly without any remainder. Finding the GCF simplifies expressions and equations.
In the exercise provided, we identify the GCF in each group of terms:
  • For the first group, we have \(y^2 + 3y\). The GCF is \(y\), since both terms contain \(y\).
  • For the second group, \(3x + xy\), the GCF is \(x\), as both terms include \(x\).
Factoring out the GCF allows us to see the common binomial factors more clearly.
Binomial Factor
A binomial factor is simply a factor that contains two terms. When factoring expressions, recognizing common binomial factors can simplify the process.
In our example, when we grouped and factored terms, we got \(y(y + 3)\) and \(x(y + 3)\). Both expressions share the common binomial factor \(y + 3\).
  • By factoring out \(y + 3\), we simplify the expression to \((y + 3)(y + x)\).
  • This shows that both parts of the equation have a shared factor, making further simplification easier.
Recognizing common binomial factors can often make complex algebraic expressions more manageable.
Algebraic Expressions
Understanding algebraic expressions is fundamental to solving many math problems. An algebraic expression contains variables, numbers, and operations.
In our exercise, \(y^2 + 3y + 3x + xy\) is an algebraic expression. We can simplify it by:
  • Grouping terms: \(y^2 + 3y\) and \(3x + xy\).
  • Factoring out the GCF: \(y(y + 3)\) and \(x(y + 3)\).
  • Combining like terms: \((y + 3)(y + x)\).
This approach, called factoring by grouping, makes it easier to handle and solve equations. Recognizing how parts of an expression work together helps in manipulating and understanding algebra.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Solve each equation, and check your solutions. $$ 6 x^{2}(2 x+3)=4(2 x+3)+5 x(2 x+3) $$

Solve each equation. $$ 3 t+2=0 $$

Factor each trinomial completely. $$ 18 x^{2}-48 x y+32 y^{2} $$

Find each product. \((x+2)(x+7)\)

Brain Busters Factor each polynomial. ( Hint: As the first step, factor out the greatest common factor.) $$ 25 q^{2}(m+1)^{3}-5 q(m+1)^{3}-2(m+1)^{3} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.