/*! 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 84 Factor completely. $$ 7 y^{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 84

Factor completely. $$ 7 y^{2}-28 \quad[5.5] $$

Short Answer

Expert verified
7(y + 2)(y - 2)

Step by step solution

01

- Identify Common Factor

First, identify the greatest common factor (GCF) of the terms in the expression. The given expression is 7y^2 - 28. Observe that both terms have a common factor of 7.
02

- Factor Out the GCF

Next, factor out the GCF from each term in the expression. When you factor out 7 from each term, you get: 7(y^2 - 4).
03

- Recognize the Difference of Squares

The expression inside the parentheses is a difference of squares, which can be factored further: y^2 - 4 = (y + 2)(y - 2).
04

- Write the Complete Factored Form

Now, combine the factored GCF with the factored difference of squares: 7(y + 2)(y - 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
Let's dive into understanding the 'Greatest Common Factor' (GCF). The GCF of a set of terms is the largest factor that divides each term in the set.
To find the GCF, follow these steps:
  • List the factors of each term.
  • Identify the largest common factor among them.
In the given expression, 7y^2 - 28, we identify the common factor for both terms:
  • For 7y^2, the factors are: 1, 7, y, y^2.
  • For -28, the factors are: 1, 2, 4, 7, 14, 28.
Here, the greatest common factor is 7. By factoring out this GCF, we simplify the expression to 7(y^2 - 4).
Difference of Squares
The 'Difference of Squares' is a special factoring formula used when an expression is in the form a^2 - b^2. This can be factored into (a + b)(a - b).
In our expression, after factoring out the GCF, we have 7(y^2 - 4). Notice that y^2 - 4 fits the difference of squares pattern.
Here:
  • y^2 is a square term (y * y).
  • 4 is also a square term (2 * 2).
So, the difference of squares formula can be applied: y^2 - 4 = (y + 2)(y - 2). Now, we factor it further into the product of two binomials.
This simplifies our expression to 7(y + 2)(y - 2).
Factoring Process
The 'Factoring Process' involves breaking down an expression into simpler factors or components. This can make solving equations or simplifying expressions easier.
Here's a general method for factoring polynomials, illustrated with our example:
  • Step 1: Identify the Greatest Common Factor (GCF)
    We found that the GCF of 7y^2 - 28 is 7, giving us the expression 7(y^2 - 4).
  • Step 2: Recognize Special Patterns
    We identified y^2 - 4 as a difference of squares that follows the form a^2 - b^2 = (a + b)(a - b).
  • Step 3: Factor Completely
    Finally, applying the difference of squares formula, we factor it to get 7(y + 2)(y - 2).
By following these steps, we can transform complex expressions into simpler, more manageable forms. This makes understanding and further solving them a lot easier!

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

Use a graphing calculator to check Example \(5 .\) Perform the check using $$ \begin{aligned} &y_{1}=\left(9 x^{2}+x^{3}-5\right) /\left(x^{2}-1\right)\\\ &y_{2}=x+9+(x+4) /\left(x^{2}-1\right), \text { and } y_{3}=y_{2}-y_{1} \end{aligned} $$

Find an equation of variation in which: \(y\) varies inversely as the square of \(x,\) and \(y=0.15\) when \(x=0.1\).

A sound's reverberation time \(T\) is the time that it takes for the sound level to decrease by \(60 \mathrm{dB}\) (decibels) after the sound has been turned off. Reverberation time varies directly as the volume \(V\) of a room and inversely as the sound absorption \(A\) of the room. A given sound has a reverberation time of 1.5 sec in a room with a volume of \(90 \mathrm{m}^{3}\) and a sound absorption of \(9.6 .\) What is the reverberation time of the same sound in a room with a volume of \(84 \mathrm{m}^{3}\) and a sound absorption of \(10.5 ?\)

Find an equation of variation in which: \(y\) varies directly as the square of \(x,\) and \(y=0.15\) when \(x=0.1\)

Ultraviolet Index. At an ultraviolet, or UV, rating of \(4,\) those people who are less sensitive to the sun will burn in 75 min. Given that the number of minutes it takes to burn, \(t,\) varies inversely with the UV rating, \(u,\) how long will it take less sensitive people to burn when the UV rating is \(14 ?\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.