/*! 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 41 In the following exercises, fact... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 41

In the following exercises, factor by grouping. $$ u v-9 u+2 v-18 $$

Short Answer

Expert verified
\((u v - 9 u + 2 v - 18) = (v - 9)(u + 2)\).

Step by step solution

01

- Group the terms

Group the terms in pairs that can be factored separately. Write the expression as \((u v - 9 u) + (2 v - 18)\).
02

- Factor out the greatest common factor (GCF) in each group

Identify and factor out the GCF in each grouped pair.In the first group \((u v - 9 u)\), the GCF is \(u\), so factor out \(u\): \(u(v - 9)\). In the second group \((2 v - 18)\), the GCF is \(2\), so factor out \(2\): \(2(v - 18/2) = 2(v - 9)\).The expression now is: \(u(v - 9) + 2(v - 9)\).
03

- Factor out the common binomial factor

Now, both terms have a common binomial factor \((v - 9)\). Factor out \((v - 9)\): \((v - 9)(u + 2)\).

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 the largest factor that two or more terms share.
It's essential to understand GCF because it helps simplify expressions and make calculations easier.
To find the GCF of a pair of terms, identify the common factors of each term and then choose the highest one.
For example, in the expression \(u v - 9 u\), both terms share a common factor of \(u\).
Here鈥檚 another tip to determine the GCF:
  • List the factors of each term.
  • Identify the common factors.
  • Select the greatest one.

By factoring out this GCF, you simplify the expression and prepare it for further steps, like combining terms.
Binomial Factor
A binomial factor is a factor that consists of exactly two terms connected by a plus or minus sign.
Binomials are essential building blocks in algebraic expressions.
For instance, in the expression \(u(v - 9) + 2(v - 9)\), the binomial factor \(v - 9\) is common to both terms.
This means you can factor this binomial out, simplifying the expression further.
How do you recognize binomial factors in an expression? Look for two-term expressions that repeat within the larger expression.
Here are a few key points about binomial factors:
  • They consist of two terms.
  • Binomials can be added or subtracted.
  • Common binomial factors simplify your work.
Recognizing and factoring out binomial factors is a central part of simplifying algebraic expressions.
Factorization
Factorization is the process of breaking down an expression into simpler terms or factors.
It鈥檚 like reversing multiplication to find what multiplied terms give an original expression.
In the example \(u v - 9 u + 2 v - 18\), you factor by grouping, identifying pairs of terms that can be factored separately.
The steps in factorization might include:
  • Grouping the terms.
  • Finding and factoring the GCF from each group.
  • Factoring out a common binomial from the resulting expression.
Factorization helps solve equations, simplify expressions, and understand polynomial structures.
By mastering factorization, you can handle more complex algebraic problems effectively.

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

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ 8-6 x+x^{2} $$

In the following exercises, factor completely using trial and error. $$ 4 q^{2}-7 q-2 $$

In the following exercises, factor using the 'ac' method. $$ 48 z^{3}-102 z^{2}-45 z $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ x^{2}-8 x+7 $$

In the following exercises, factor by grouping. $$ x^{2}-x+4 x-4 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.