/*! 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 131 Factor each polynomial completel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 131

Factor each polynomial completely. \(9 a^{2}-48 a b+64 b^{2}-4 c^{2}\)

Short Answer

Expert verified
The complete factorization is \((3a-8b-2c)(3a-8b+2c)\).

Step by step solution

01

- Identify Quadratic Form

The given polynomial is in the quadratic form: \[ 9a^{2} - 48ab + 64b^{2} - 4c^{2} \]
02

- Group Terms

Group the first three terms and the last term separately: \[ 9a^{2} - 48ab + 64b^{2} - 4c^{2} \]
03

- Identify a Common Pattern

Notice that the first three terms \(9a^2 - 48ab + 64b^2\) can be written as a perfect square. Rewrite it: \[ (3a - 8b)^2 - 4c^2 \]
04

- Apply the Difference of Squares Formula

Recognize the expression as a difference of squares: \[ (3a - 8b)^2 - (2c)^2 \]. Use the formula for difference of squares \(x^2 - y^2 = (x-y)(x+y)\): \[ (3a - 8b - 2c)(3a - 8b + 2c) \]

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.

Difference of Squares
The difference of squares is a way to factor certain types of polynomials. Essentially, it applies to expressions that look like this: a square number minus another square number. Notice that this fits the pattern: \[ x^2 - y^2 \] You can factor this easily using: \[ (x - y)(x + y) \] For example, consider \[ (3a - 8b)^2 - (2c)^2 \]. This fits our pattern, where \( x = 3a - 8b \) and \( y = 2c \). So, it factors to: \[ (3a - 8b - 2c)(3a - 8b + 2c) \].
Perfect Square Trinomial
A perfect square trinomial is another type of polynomial that is easy to factor. Look for an expression in the form \[ a^2 + 2ab + b^2 \] or \[ a^2 - 2ab + b^2 \]. Both of these patterns will factor into a binomial squared. Take \[ 9a^2 - 48ab + 64b^2 \], for instance. It matches the form of \( a^2 - 2ab + b^2 \) where \( a = 3a \) and \( b = 8b \). So, it can be written as: \[ (3a - 8b)^2 \]
Quadratic Form
Quadratic forms appear frequently in polynomial factoring, typically in the form \[ Ax^2 + Bx + C \]. Recognizing this form helps simplify complex expressions intricately. In the exercise \[ 9a^2 - 48ab + 64b^2 - 4c^2 \], the first three terms \[ 9a^2 - 48ab + 64b^2 \] represent a quadratic form. Identifying and factoring this part produces \[ (3a - 8b)^2 \], leading into the next conceptual steps. Recognizing these standard patterns is essential for efficient polynomial factoring.
Grouping Terms
Grouping terms involves organizing a polynomial to simplify factoring. In the given polynomial, \[ 9a^2 - 48ab + 64b^2 - 4c^2 \], you start by identifying parts you can manage easier. Here, you can group the first three terms separately from the last term: \[ 9a^2 - 48ab + 64b^2 \] and \[ -4c^2 \]. Within grouped terms, it's easier to see how to apply patterns, such as perfect square trinomials or the difference of squares. Grouping breaks down complex expressions into manageable chunks.

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. \(m^{2}+\frac{2}{3} m+\frac{1}{9}\)

Solve each problem. Two cars left an intersection at the same time. One traveled north. The other traveled 14 mi farther, but to the east. How far apart were they at that time if the distance between them was 4 mi more than the distance traveled east?

Factor the integer 35 as the product of two prime numbers.

Factor each binomial completely. \(x^{3}-343\)

Solve each problem. Find two consecutive odd integers such that their product is 15 more than three times their sum.
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

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.