# 6.5 Factoring two special products

 Page 1 / 2
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses.The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. Objectives of this module: know the fundamental rules of factoring, be able to factor the difference of two squares and perfect square trinomials.

## Overview

• The Difference of Two Squares
• Fundamental Rules of Factoring
• Perfect Square Trinomials

## The difference of two squares

Recall that when we multiplied together the two binomials $\left(a+b\right)$ and $\left(a-b\right)$ , we obtained the product ${a}^{2}-{b}^{2}$ .

$\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$

## Perfect square

Notice that the terms ${a}^{2}$ and ${b}^{2}$ in the product can be produced by squaring $a$ and $b$ , respectively. A term that is the square of another term is called a perfect square . Thus, both ${a}^{2}$ and ${b}^{2}$ are perfect squares. The minus sign between ${a}^{2}$ and ${b}^{2}$ means that we are taking the difference of the two squares.
Since we know that $\left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}$ , we need only turn the equation around to find the factorization form.

${a}^{2}-{b}^{2}=\left(a+b\right)\left(a-b\right)$

The factorization form says that we can factor ${a}^{2}-{b}^{2}$ , the difference of two squares, by finding the terms that produce the perfect squares and substituting these quantities into the factorization form.
When using real numbers (as we are), there is no factored form for the sum of two squares. That is, using real numbers,

${a}^{2}+{b}^{2}$ cannot be factored

## Sample set a

Factor ${x}^{2}-16$ . Both ${x}^{2}$ and 16 are perfect squares. The terms that, when squared, produce ${x}^{2}$ and 16 are $x$ and 4, respectively. Thus,

${x}^{2}-16=\left(x+4\right)\left(x-4\right)$

We can check our factorization simply by multiplying.

$\begin{array}{lll}\left(x+4\right)\left(x-4\right)\hfill & =\hfill & {x}^{2}-4x+4x-16\hfill \\ \hfill & =\hfill & {x}^{2}-16.\hfill \end{array}$

$49{a}^{2}{b}^{4}-121$ . Both $49{a}^{2}{b}^{4}$ and 121 are perfect squares. The terms that, when squared, produce $49{a}^{2}{b}^{4}$ and 121 are $7a{b}^{2}$ and 11, respectively. Substituting these terms into the factorization form we get

$49{a}^{2}{b}^{4}-121=\mathrm{\left(7}a{b}^{2}+11\right)\left(7a{b}^{2}-11\right)$

We can check our factorization by multiplying.

$\begin{array}{lll}\left(7a{b}^{2}+11\right)\left(7a{b}^{2}-11\right)\hfill & =\hfill & 49{a}^{2}{b}^{4}-11a{b}^{2}+11a{b}^{2}-121\hfill \\ \hfill & =\hfill & 49{a}^{2}{b}^{4}-121\hfill \end{array}$

$3{x}^{2}-27$ . This doesn’t look like the difference of two squares since we don’t readily know the terms that produce $3{x}^{2}$ and 27. However, notice that 3 is common to both the terms. Factor out 3.

$3\left({x}^{2}-9\right)$

Now we see that ${x}^{2}-9$ is the difference of two squares. Factoring the ${x}^{2}-9$ we get

$\begin{array}{lll}3{x}^{2}-27\hfill & =\hfill & 3\left({x}^{2}-9\right)\hfill \\ \hfill & =\hfill & 3\left(x+3\right)\left(x-3\right)\hfill \end{array}$

Be careful not to drop the factor 3.

## Practice set a

If possible, factor the following binomials completely.

${m}^{2}-25$

$\left(m+5\right)\left(m-5\right)$

$36{p}^{2}-81{q}^{2}$

$9\left(2p-3q\right)\left(2p+3q\right)$

$49{a}^{4}-{b}^{2}{c}^{2}$

$\left(7{a}^{2}+bc\right)\left(7{a}^{2}-bc\right)$

${x}^{8}{y}^{4}-100{w}^{12}$

$\left({x}^{4}{y}^{2}+10{w}^{6}\right)\left({x}^{4}{y}^{2}-10{w}^{6}\right)$

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Eleven fifteenths of two more than a number is eight.
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.