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This module teaches about multiplying binomials. Specifically about common patterns that can be memorized and using the "FOIL" method.

The following three formulae should be memorized.

x + a 2 = x 2 + 2 ax + a 2 size 12{ left (x+a right ) rSup { size 8{2} } =x rSup { size 8{2} } +2 ital "ax"+a rSup { size 8{2} } } {}
x a 2 = x 2 2 ax + a 2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {}
x + a x a = x 2 a 2 size 12{ left (x+a right ) left (x - a right )=x rSup { size 8{2} } - a rSup { size 8{2} } } {}

It is important to have these three formulae on the top of your head. It is also nice to be able to show why these formulae work, for instance by using FOIL. But the most important thing of all is knowing what these three formulae mean, and how to use them .

These three are all “algebraic generalizations,” as discussed in the first unit on functions. That is, they are equations that hold true for any values of x size 12{x} {} and a size 12{a} {} . It may help if you think of the second equation above as standing for:

Anthing Anything Else 2 = Anything 2 2 Anything Else 2 size 12{ left ("Anthing" - "Anything Else" right ) rSup { size 8{2} } ="Anything" rSup { size 8{2} } - 2 left ("Anything Else" right ) rSup { size 8{2} } } {} For instance, suppose the Anything (or x size 12{x} {} ) is 5, and the Anything Else (or a size 12{a} {} ) is 3.

x a 2 = x 2 2 ax + a 2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {} , when x = 5 size 12{x=5} {} , a = 3 size 12{a=3} {} .

  • 5 3 2 = ? 5 2 2 3 5 + 3 2 size 12{5 - 3 rSup { size 8{2} } { {}={}} cSup { size 8{?} } 5 rSup { size 8{2} } - 2 left (3 right ) left (5 right )+3 rSup { size 8{2} } } {}
  • 2 2 = ? 25 30 + 9 size 12{2 rSup { size 8{2} } { {}={}} cSup { size 8{?} } "25" - "30"+9} {}
  • 4 = 4 size 12{4=4} {}

It worked! Now, let’s leave the Anything as x size 12{x} {} , but play with different values of a size 12{a} {} .

More examples of x a 2 = x 2 2 ax + a 2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {}

a = 1 : ( x - 1 ) 2 = x 2 - 2 x + 1 a = 2 : ( x - 2 ) 2 = x 2 - 4 x + 4 a = 3 : ( x - 3 ) 2 = x 2 - 6 x + 9 a = 5 : ( x - 5 ) 2 = x 2 - 10 x + 25 a = 10 : ( x - 10 ) 2 = x 2 - 20 x + 100

Once you’ve seen a few of these, the pattern becomes evident: the number doubles to create the middle term (the coefficient of x size 12{x} {} ), and squares to create the final term (the number).

The hardest thing about this formula is remembering to use it . For instance, suppose you are asked to expand:

2y 6 2 size 12{ left (2y - 6 right ) rSup { size 8{2} } } {}

There are three ways you can approach this.

2y 6 2 , computed three different ways
Square each term FOIL Using the formula above
2y 6 2 2y 2 2 6 2y + 6 2 4y 2 24 y + 36 size 12{ matrix { {} # left (2y - 6 right ) rSup { size 8{2} } {} ##={} {} # left (2y right ) rSup { size 8{2} } - 2 left (6 right ) left (2y right )+6 rSup { size 8{2} } {} ## ={} {} # 4y rSup { size 8{2} } - "24"y+"36"{}} } {} 2y 6 2y 6 2y 2y 2y 6 2y 6 + 36 4y 2 12 y 12 y + 36 4y 2 24 y + 36 size 12{ matrix { {} # left (2y - 6 right ) left (2y - 6 right ) {} ##={} {} # left (2y right ) left (2y right ) - left (2y right )6 - left (2y right )6+"36" {} ## ={} {} # 4y rSup { size 8{2} } - "12"y - "12"y+"36" {} ##={} {} # 4y rSup { size 8{2} } - "24"y+"36"{} } } {} 2y 6 2 2y 2 2 6 2y + 6 2 4y 2 24 y + 36 size 12{ matrix { {} # left (2y - 6 right ) rSup { size 8{2} } {} ##={} {} # left (2y right ) rSup { size 8{2} } - 2 left (6 right ) left (2y right )+6 rSup { size 8{2} } {} ## ={} {} # 4y rSup { size 8{2} } - "24"y+"36"{}} } {}

Did it work? If a formula is true, it should work for any y size 12{y} {} -value; let’s test each one with y = 5 size 12{y=5} {} . (Note that the second two methods got the same answer, so we only need to test that once.)

2y 6 2 = ? 4y 2 36 2 5 6 2 = ? 4y 2 36 10 6 2 = ? 100 36 4 2 = ? 64 size 12{ matrix { left (2y - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "36" {} ##left (2 cdot 5 - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "36" {} ## left ("10" - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "100" - "36" {} ##4 rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "64"{} } } {} 2y 6 2 = ? 4y 2 24 y + 36 2 5 6 2 = ? 4 5 2 24 5 + 36 10 6 2 = ? 100 120 + 36 4 2 = ? 16 size 12{ matrix { left (2y - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4y rSup { size 8{2} } - "24"y+"36" {} ##left (2 cdot 5 - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # 4 left (5 right ) rSup { size 8{2} } - "24" cdot 5+"36" {} ## left ("10" - 6 right ) rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "100" - "120"+"36" {} ##4 rSup { size 8{2} } {} # { {}={}} cSup { size 8{?} } {} # "16"{} } } {}

We conclude that squaring each term individually does not work. The other two methods both give the same answer, which works.

The first method is the easiest, of course. And it looks good. 2y 2 size 12{ left (2y right ) rSup { size 8{2} } } {} is indeed 4y 2 size 12{4y rSup { size 8{2} } } {} . And 6 2 size 12{6 rSup { size 8{2} } } {} is indeed 36. But as you can see, it led us to a false answer —an algebraic generalization that did not hold up.

I just can’t stress this point enough. It sounds like a detail, but it causes errors all through Algebra II and beyond. When you’re adding or subtracting things, and then squaring them, you can’t just square them one at a time. Mathematically, x + a 2 x 2 + a 2 size 12{ left (x+a right ) rSup { size 8{2} }<>x rSup { size 8{2} } +a rSup { size 8{2} } } {} . You can confirm this with numbers all day. 7 + 3 2 = 100 size 12{ left (7+3 right ) rSup { size 8{2} } ="100"} {} , but 7 2 + 3 2 = 58 size 12{7 rSup { size 8{2} } +3 rSup { size 8{2} } ="58"} {} . They’re not the same.

So that leaves the other two methods. FOIL will never lead you astray. But the third approach, the formula, has three distinct advantages.

  1. The formula is faster than FOIL.
  2. Using these formulae is a specific case of the vital mathematical skill of using any formula—learning how to plug numbers and variables into some equation that you’ve been given, and therefore understanding the abstraction that formulae represent.
  3. Before this unit is done, we will be completing the square, which requires running that particular formula backward —which you cannot do with FOIL.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
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Source:  OpenStax, Quadratic functions. OpenStax CNX. Mar 10, 2011 Download for free at http://cnx.org/content/col11284/1.2
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