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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.


  • 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 ( a b ) and ( a b ) , we obtained the product a 2 b 2 .

( a b ) ( a b ) 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 ( a b ) ( a b ) a 2 b 2 , we need only turn the equation around to find the factorization form.

a 2 b 2 ( a b ) ( a b )

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 ( x 4 ) ( x 4 )

We can check our factorization simply by multiplying.

( x + 4 ) ( x - 4 ) = x 2 - 4 x + 4 x - 16 = x 2 - 16.

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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 7 a b 2 and 11, respectively. Substituting these terms into the factorization form we get

49 a 2 b 4 121 (7 a b 2 11) (7 a b 2 11)

We can check our factorization by multiplying.

( 7 a b 2 + 11 ) ( 7 a b 2 - 11 ) = 49 a 2 b 4 - 11 a b 2 + 11 a b 2 - 121 = 49 a 2 b 4 - 121

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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 ( x 2 9 )

Now we see that x 2 9 is the difference of two squares. Factoring the x 2 9 we get

3 x 2 - 27 = 3 ( x 2 - 9 ) = 3 ( x + 3 ) ( x - 3 )

Be careful not to drop the factor 3.

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Practice set a

If possible, factor the following binomials completely.

36 p 2 81 q 2

9 ( 2 p 3 q ) ( 2 p 3 q )

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49 a 4 b 2 c 2

( 7 a 2 + b c ) ( 7 a 2 b c )

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x 8 y 4 100 w 12

( x 4 y 2 + 10 w 6 ) ( x 4 y 2 10 w 6 )

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Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
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where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Eleven fifteenths of two more than a number is eight.
Leny Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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