/*! 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 105 Factor by grouping. \(2 z^{2}+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 105

Factor by grouping. \(2 z^{2}+6 w-4 z-3 w z\)

Short Answer

Expert verified
(z - 2)(2z - 3w)

Step by step solution

01

Group the Terms

Group the polynomial into two pairs so it's easier to factor. The given polynomial is 2z^2 + 6w - 4z - 3wz. Group them as follows: (2z^2 - 4z) + (6w - 3wz).
02

Factor Out the Greatest Common Factor (GCF)

Factor out the GCF from each pair. For the first group (2z^2 - 4z), factor out 2z, and for the second group (6w - 3wz), factor out 3w: 2z(z - 2) + 3w(2 - z).
03

Reorder Terms if Necessary

Observe that (2 - z) can be written as -(z - 2). Therefore, reorder terms to have a common binomial factor: 2z(z - 2) - 3w(z - 2).
04

Factor Out the Common Binomial

Now factor out the common binomial factor (z - 2): (z - 2)(2z - 3w).

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.

factoring polynomials
Factoring polynomials is a method used to break down a polynomial into simpler components called factors. This process is similar to splitting numbers into prime factors. To factor polynomials, you need to look for common patterns or terms that can be grouped together.

For example, consider the polynomial given in the exercise: 2z^2 + 6w - 4z - 3wz. By using the factoring by grouping method, you group terms in pairs which makes it easier to identify common factors.

Grouped as: (2z^2 - 4z) + (6w - 3wz), the polynomial becomes easier to manage. In this way, breaking it down into smaller, more straightforward parts makes factoring polynomials a straightforward process.
greatest common factor (GCF)
The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or expressions. Finding the GCF is crucial when factoring polynomials since it allows you to simplify expressions by extracting common factors.

Let's look at our example again. For the group (2z^2 - 4z), the GCF is 2z as both terms include 2 and z as factors. Similarly, for (6w - 3wz), the GCF is 3w.

Extracting these GCFs, you rewrite the polynomial like this:2z(z - 2) + 3w(2 - z).
By identifying and factoring out the GCF, simplifying a polynomial makes the subsequent steps easier.
binomial factors
Binomial factors are expressions containing two terms that, when multiplied together with other factors, form the original polynomial. Recognizing common binomial factors helps further simplify the polynomial.

In the example provided, after factoring out the GCFs, you are left with: 2z(z - 2) + 3w(2 - z).

Notice that (2 - z) can be rewritten as -(z - 2). Therefore, you can reorder the terms to have a common binomial factor:2z(z - 2) - 3w(z - 2).

It's then easy to factor out the common binomial factor (z - 2), giving you the simplified expression: (z - 2)(2z - 3w).These binomial factors are the simplest components that, when multiplied, will give you the original polynomial.

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

Apply the special factoring rules of this section to factor each polynomial. \(p^{2}-\frac{1}{9}\)

Factor each binomial completely. \(m^{3}+8\)

Apply the special factoring rules of this section to factor each polynomial. \(x^{3}+\frac{1}{64}\)

Factor each trinomial. \(24 x^{4}+17 x^{2}-20\)

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

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

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.