/*! 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 75 Factor by grouping. \(18 r^{2}... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 75

Factor by grouping. \(18 r^{2}+12 r y-3 x r-2 x y\)

Short Answer

Expert verified
(3r + 2y)(6r - x)

Step by step solution

01

Group the terms

Group the quadratic expression into two pairs: group_1: (18r^{2} + 12ry) group_2: (-3xr - 2xy)
02

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

Identify and factor out the GCF from each pair: group_1: 6r(3r + 2y) group_2: -x(3r + 2y)
03

Factor out the common binomial factor

Notice that both groups contain the common binomial factor (3r + 2y). So factor out (3r + 2y) from both groups: (3r + 2y)(6r - 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.

quadratic expressions
Quadratic expressions are a central concept in algebra. They are polynomials of degree two, meaning the highest power of the variable is squared. A typical quadratic expression looks like ax^2 + bx + c, where a, b, and c are constants, and x represents the variable.

In the given exercise, the expression is 18r^2 + 12ry - 3xr - 2xy. Here, the terms involve both r and x variables, making it slightly more complex. To simplify and solve these kinds of expressions, factoring by grouping is a useful technique. This method involves rearranging the expression into groups and looking for common factors within those groups.
greatest common factor (GCF)
The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring expressions, finding the GCF is a useful first step to simplify the terms.

For example, consider the pairs from the grouped expression: group_1: (18r^2 + 12ry) and group_2: (-3xr - 2xy).
  • In group_1, the GCF is 6r since both 18r^2 and 12ry are divisible by 6r. This gives us 6r(3r + 2y).
  • In group_2, the GCF is -x as both -3xr and -2xy are divisible by -x. This leads to -x(3r + 2y).

Identifying the GCF simplifies the expression, making it easier to proceed to the next step.
binomial factorization
Binomial factorization involves rewriting a quadratic or polynomial expression as a product of two binomials. After grouping and factoring out the GCF, you often find a common binomial factor in each group.

In the solution to our exercise, after factoring the GCF from each group, we get 6r(3r + 2y) and -x(3r + 2y). Notice that both groups now contain the binomial factor (3r + 2y).
We can factor out this binomial to consolidate the expression:

By grouping terms and factoring, we achieve the final factored form: (3r + 2y)(6r - x).

  • This technique can be generally applied to various polynomial expressions to simplify and solve for variables more efficiently.

Whether you're new to algebra or refreshing your skills, understanding these steps is essential for mastering factorization.

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 each trinomial completely. $$ 64 x^{2}+48 x y+9 y^{2} $$

Factor completely. $$ y^{3} z+3 y^{2} z^{2}-54 y z^{3} $$

Factor each polynomial. $$ (2 p+q) r^{2}-12(2 p+q) r+27(2 p+q) $$

Factor by grouping. \(5 m-6 p-2 m p+15\)

Factor each trinomial completely. $$ x^{2}-8 x+16 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.