/*! 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 49 In the following exercises, fact... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 49

In the following exercises, factor by grouping. $$ 2 x^{2}-14 x-5 x+35 $$

Short Answer

Expert verified
(x - 7)(2x - 5)

Step by step solution

01

- Group terms

Group the terms in pairs to make it easier to factor. Rewrite the expression: o(2x^{2} - 14x) - (5x - 35)
02

- Factor out the GCD

Factor out the greatest common divisor (GCD) from each pair. In this case, we factor out a 2 from the first pair and a -5 from the second pair: o 2x(x - 7) - 5(x - 7)
03

- Factor common binomial

Notice that o(x - 7) is a common factor. Factor it out: o (x - 7)(2x - 5)

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 divisor
The Greatest Common Divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. In factoring algebraic expressions, finding the GCD within a set of terms is crucial as it allows us to simplify the expression.
Let's break down how we use the GCD in the given exercise:

Consider the expression: \[ 2x^{2} - 14x - 5x + 35 \]
Group the terms to make it easier to factor: \[ (2x^{2} - 14x) - (5x - 35) \]
Next, find the GCD for each group:
  • The GCD of \2x^{2} \ and \ -14x \ is \ 2x \
  • The GCD of \ -5x \ and \ 35 \ is \ -5 \
We factor out these GCDs from each group:
  • \[ 2x(x - 7) - 5(x - 7) \]
This step simplifies the expression and sets us up for the next phase of factoring by grouping.
binomial factors
Binomial factors are two-term polynomials that we get after factoring an expression. In our example, once we applied factoring by grouping, we identified a common binomial factor in each group:

We started with: \[ 2x(x - 7) - 5(x - 7) \]
Notice the binomial \ (x - 7) \ is a common factor in both terms. To further simplify, we factor out \ (x - 7) \ from the entire expression. Our expression then becomes:
  • \[ (x - 7)(2x - 5) \]
Breaking down the steps:
1. Finding the common binomial factor (x - 7).
2. Factoring it out to rewrite the original expression as a product of binomials.
This aids in simplifying expressions and solving equations by breaking down complex polynomials into manageable factors.
algebraic expressions
Algebraic expressions are combinations of variables, constants, and operators (such as addition or subtraction). Understanding how to manipulate these expressions through factoring makes solving algebraic problems easier.
In our example, the original algebraic expression was: \[ 2x^{2} - 14x - 5x + 35 \]
We used the technique of factoring by grouping:
1. Grouped terms into pairs:
  • \[ (2x^{2} - 14x) - (5x + 35) \]
2. Factored out the GCD from each pair:
  • \[ 2x(x - 7) - 5(x - 7) \]
3. Identified and factored out the common binomial factor:
  • \[ (x - 7)(2x - 5) \]
This rewrite into a product of simpler polynomials illustrates how recognizing patterns and applying the correct steps can help simplify complex algebraic expressions.

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 using the 'ac' method. $$ 4 k^{2}-16 k+15 $$

In the following exercises, factor each expression using any method. $$ 5 r^{2}+25 r+30 $$

In the following exercises, factor each expression using any method. $$ 7 x^{2}-21 x $$

In the following exercises, factor each trinomial of the form \(x^{2}+b x+c .\) $$ p^{2}+11 p+30 $$

What is the GCF of \(y^{4}, y^{5},\) and \(y^{10} ?\) Write a general rule that tells you how to find the GCF of \(y^{a}, y^{b},\) and \(y^{c}\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.