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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.

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

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.

Practice set a

If possible, factor the following binomials completely.

m 2 25

( m 5 ) ( m 5 )

36 p 2 81 q 2

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

49 a 4 b 2 c 2

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

x 8 y 4 100 w 12

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

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1
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