<< Chapter < Page Chapter >> Page >
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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Quadratic functions. OpenStax CNX. Mar 10, 2011 Download for free at http://cnx.org/content/col11284/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Quadratic functions' conversation and receive update notifications?

Ask