/*! 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 132 Factor each expression completel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 132

Factor each expression completely. $$3 a b+9 a-2 b-6$$

Short Answer

Expert verified
The completely factored form of the expression \(3 a b+9 a-2 b-6\) is \((3 a - 2) (b + 3)\).

Step by step solution

01

Group the terms

The first step is to group the terms that have common factors. In this case, \(3 a b\) and \(9 a\) can be grouped together because they share \(a\) as a common factor, as well as \(2 b\) and \(6\) because \(2\) is a common factor. So we can rewrite the expression as \((3 a b + 9 a) - (2 b + 6)\).
02

Apply the distributive property

The next step is to apply the distributive property, also known as factoring out the common factor. For the first group, the common factor is \(3a\), and for the second group, it is \(2\). This gives us an expression of the form: \(3a (b + 3) - 2 (b + 3)\).
03

Factor out the common binomial

Notice that both terms now have common binomial \(b+3\). We factor that out to get \((3 a - 2) (b + 3)\).

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.

Distributive Property
The distributive property is an essential tool in algebra that simplifies expressions and aids in finding common factors. It essentially states that a single term multiplied by terms within a parenthesis can be expanded across each term. This is expressed as \( a(b+c) = ab + ac \). This property not only helps in expansion but also in reverse operations such as factoring, where you "undistribute" or pull out a common factor from terms.

In the problem from the exercise, we see the application of the distributive property in two ways:
  • First, each group of terms is simplified using their common factors. For example, in the group \(3ab + 9a\), the common factor is \(3a\).
  • Next, this common factor is distributed out, resulting in \(3a (b + 3)\).
This leads to a cleaner expression that reveals the underlying structure and relationships between the terms.
Common Factors
Identifying common factors is a crucial step in factoring algebraic expressions. A common factor is a term that appears in multiple parts of an expression. By factoring out common factors, we simplify expressions and make them more manageable.

In our exercise, the common factors allow us to group terms effectively:
  • For the expression \(3ab + 9a\), the common factor is \(3a\), because it is present in both terms.
  • In \(2b + 6\), the common factor is \(2\).
Extracting these factors transforms the expression into \(3a(b+3) - 2(b+3)\). Identifying and using common factors are central in reducing complexity and preparing the expression for further factoring techniques.
Factoring by Grouping
Factoring by grouping is a method used when an expression doesn't yield to simple factorization techniques and contains more terms. This technique involves rearranging and grouping terms so that a common factor can be identified within each group.

The main steps in the exercise using this method are:
  • First, terms are grouped as \((3ab + 9a)\) and \((2b + 6)\).
  • Within each group, a common factor is factored out, resulting in \(3a(b + 3)\) and \(2(b + 3)\).
  • Finally, noticing that \(b+3\) is common to both expressions, \(b+3\) is factored out to achieve \((3a - 2)(b + 3)\).
By using grouping, complex expressions are simplified in a stepwise manner, highlighting common factors systematically and leading to the fully factored form.

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

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$a^{2}+2 a b+b^{2}-4$$

Factor. $$9 x^{2}-6 x+1-d^{2}$$

Factor. If the polynomial is prime, so indicate. $$25+2 u^{2}+3 u$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$10 y+9+y^{2}$$

Factor. $$x^{2}+10 x+25$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.