<< Chapter < Page Chapter >> Page >
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to expand (a + b)^2, (a - b)^2, and (a + b)(a - b).

Overview

  • Expanding ( a + b ) 2 and ( a b ) 2
  • Expanding ( a + b ) ( a b )

Three binomial products occur so frequently in algebra that we designate them as special binomial products . We have seen them before (Sections [link] and [link] ), but we will study them again because of their importance as time saving devices and in solving equations (which we will study in a later chapter).

These special products can be shown as the squares of a binomial

( a + b ) 2      and      ( a b ) 2

and as the sum and difference of two terms .

( a + b ) ( a b )

There are two simple rules that allow us to easily expand (multiply out) these binomials. They are well worth memorizing, as they will save a lot of time in the future.

Expanding ( a + b ) 2 And ( a b ) 2

Squaring a binomial

To square a binomial: *

  1. Square the first term.
  2. Take the product of the two terms and double it.
  3. Square the last term.
  4. Add the three results together.

( a + b ) 2 = a 2 + 2 a b + b 2 ( a b ) 2 = a 2 2 a b + b 2

Expanding ( a + b ) ( a b )

Sum and difference of two terms

To expand the sum and difference of two terms:

  1. Square the first term and square the second term.
  2. Subtract the square of the second term from the square of the first term.

( a + b ) ( a b ) = a 2 b 2


* See problems 56 and 57 at the end of this section.
See problem 58.

Sample set a

( x + 4 ) 2 Square the first term:    x 2 . The product of both terms is 4 x . Double it:    8 x . Square the last term:   16 . Add them together:    x 2 + 8 x + 16. ( x + 4 ) 2 = x 2 + 8 x + 16

Note that ( x + 4 ) 2 x 2 + 4 2 . The 8 x term is missing!

Got questions? Get instant answers now!

( a 8 ) 2 Square the first term:    a 2 . The product of both terms is 8 a . Double it:    16 a . Square the last term:    64. Add them together:    a 2 + ( 16 a ) + 64. ( a 8 ) 2 = a 2 16 a + 64

Notice that the sign of the last term in this expression is “ + .” This will always happen since the last term results from a number being squared . Any nonzero number times itself is always positive.

( + ) ( + ) = +    and    ( ) ( ) = +

The sign of the second term in the trinomial will always be the sign that occurs inside the parentheses.

Got questions? Get instant answers now!

( y 1 ) 2 Square the first term:    y 2 . The product of both terms is y . Double it:    2 y . Square the last term:    + 1. Add them together:    y 2 + ( 2 y ) + 1.

The square of the binomial 'y minus one' is equal to y squared minus two y plus one. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'minus.' The sign of the last term of the trinomial is labeled as 'plus.'

Got questions? Get instant answers now!

( 5 x + 3 ) 2 Square the first term:    25 x 2 . The product of both terms is 15 x . Double it:    30 x . Square the last term:   9 . Add them together:    25 x 2 + 30 x + 9.

The square of the binomial 'five x plus three' is equal to twenty five x squared plus thirty x plus nine. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'plus.' The sign of the last term of the trinomial is also labeled as 'plus.'

Got questions? Get instant answers now!

( 7 b 2 ) 2 Square the first term:    49 b 2 . The product of both terms is 14 b . Double it:    28 b . Square the last term:   4 . Add them together:    49 b 2 + ( 28 b ) + 4.

The square of the binomial 'seven b minus two' is equal to forty-nine b squared minus twenty-eight b plus four. The sign inside the parentheses and the sign of the middle term of the trinomial are the same, and are labeled as 'minus.' The sign of the last term of the trinomial is labeled as 'plus.'

Got questions? Get instant answers now!

( x + 6 ) ( x 6 ) Square the first term: x 2 . Subtract the square of the second term ( 36 ) from the square of the first term: x 2 36. ( x + 6 ) ( x 6 ) = x 2 36

Got questions? Get instant answers now!

( 4 a 12 ) ( 4 a + 12 ) Square the first term: 16 a 2 . Subtract the square of the second term ( 144 ) from the square of the first term: 16 a 2 144. ( 4 a 12 ) ( 4 a + 12 ) = 16 a 2 144

Got questions? Get instant answers now!

( 6 x + 8 y ) ( 6 x 8 y ) Square the first term: 36 x 2 . Subtract the square of the second term ( 64 y 2 ) from the square of the first term: 36 x 2 64 y 2 . ( 6 x + 8 y ) ( 6 x 8 y ) = 36 x 2 64 y 2

Got questions? Get instant answers now!

Practice set a

Find the following products.

( x + 5 ) 2

x 2 + 10 x + 25

Got questions? Get instant answers now!

( x + 7 ) 2

x 2 + 14 x + 49

Got questions? Get instant answers now!

( y 6 ) 2

y 2 12 y + 36

Got questions? Get instant answers now!

( 3 a + b ) 2

9 a 2 + 6 a b + b 2

Got questions? Get instant answers now!

( 9 m n ) 2

81 m 2 18 m n + n 2

Got questions? Get instant answers now!

( 10 x 2 y ) 2

100 x 2 40 x y + 4 y 2

Got questions? Get instant answers now!

( 12 a 7 b ) 2

144 a 2 168 a b + 49 b 2

Got questions? Get instant answers now!

( 5 h 15 k ) 2

25 h 2 150 h k + 225 k 2

Got questions? Get instant answers now!

Exercises

For the following problems, find the products.

( x + 3 ) 2

x 2 + 6 x + 9

Got questions? Get instant answers now!

( x + 8 ) 2

x 2 + 16 x + 64

Got questions? Get instant answers now!

( y + 9 ) 2

y 2 + 18 y + 81

Got questions? Get instant answers now!

( a 4 ) 2

a 2 8 a + 16

Got questions? Get instant answers now!

( a 7 ) 2

a 2 14 a + 49

Got questions? Get instant answers now!

( b + 15 ) 2

b 2 + 30 b + 225

Got questions? Get instant answers now!

( x 12 ) 2

x 2 24 x + 144

Got questions? Get instant answers now!

( y 20 ) 2

y 2 40 y + 400

Got questions? Get instant answers now!

( 4 x + 2 ) 2

16 x 2 + 16 x + 4

Got questions? Get instant answers now!

( 7 x 2 ) 2

49 x 2 28 x + 4

Got questions? Get instant answers now!

( 3 a 9 ) 2

9 a 2 54 a + 81

Got questions? Get instant answers now!

( 5 a 3 b ) 2

25 a 2 30 a b + 9 b 2

Got questions? Get instant answers now!

( 2 h 8 k ) 2

4 h 2 32 h k + 64 k 2

Got questions? Get instant answers now!

( a + 1 3 ) 2

a 2 + 2 3 a + 1 9

Got questions? Get instant answers now!

( x + 2 5 ) 2

x 2 + 4 5 x + 4 25

Got questions? Get instant answers now!

( y 5 6 ) 2

y 2 5 3 y + 25 36

Got questions? Get instant answers now!

( x + 1.3 ) 2

x 2 + 2.6 x + 1.69

Got questions? Get instant answers now!

( a + 0.5 ) 2

a 2 + a + 0.25

Got questions? Get instant answers now!

( x 3.1 ) 2

x 2 6.2 x + 9.61

Got questions? Get instant answers now!

( b 0.04 ) 2

b 2 0.08 b + 0.0016

Got questions? Get instant answers now!

( x + 5 ) ( x 5 )

x 2 25

Got questions? Get instant answers now!

( x + 1 ) ( x 1 )

x 2 1

Got questions? Get instant answers now!

( f + 9 ) ( f 9 )

f 2 81

Got questions? Get instant answers now!

( 2 y + 3 ) ( 2 y 3 )

4 y 2 9

Got questions? Get instant answers now!

( 5 x + 6 ) ( 5 x 6 )

Got questions? Get instant answers now!

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

4 a 2 49 b 2

Got questions? Get instant answers now!

( 7 x + 3 t ) ( 7 x 3 t )

Got questions? Get instant answers now!

( 5 h 2 k ) ( 5 h + 2 k )

25 h 2 4 k 2

Got questions? Get instant answers now!

( x + 1 3 ) ( x 1 3 )

Got questions? Get instant answers now!

( a + 2 9 ) ( a 2 9 )

a 2 4 81

Got questions? Get instant answers now!

( x + 7 3 ) ( x 7 3 )

Got questions? Get instant answers now!

( 2 b + 6 7 ) ( 2 b 6 7 )

4 b 2 36 49

Got questions? Get instant answers now!

Expand ( a + b ) 2 to prove it is equal to a 2 + 2 a b + b 2 .

Got questions? Get instant answers now!

Expand ( a b ) 2 to prove it is equal to a 2 2 a b + b 2 .

( a b ) ( a b ) = a 2 a b a b + b 2 = a 2 2 a b + b 2

Got questions? Get instant answers now!

Expand ( a + b ) ( a b ) to prove it is equal to a 2 b 2 .

Got questions? Get instant answers now!

Fill in the missing label in the equation below.

The square of the binomial 'a plus b' is equal to a squared plus two ab plus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

first term squared

Got questions? Get instant answers now!

Label the parts of the equation below.

The square of the binomial 'a minus b' is equal to a squared minus two ab plus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

Got questions? Get instant answers now!

Label the parts of the equation below.

The product of the binomial 'a plus b' and the binomial 'a minus b' is equal to a squared minus b squared. Fill in the missing labels for the equation. See the longdesc for a full description.

(a) Square the first term.
(b) Square the second term and subtract it from the first term.

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify ( x 3 y 0 z 4 ) 5 .

Got questions? Get instant answers now!

( [link] ) Find the value of 10 1 2 3 .

1 80

Got questions? Get instant answers now!

( [link] ) Find the product. ( x + 6 ) ( x 7 ) .

Got questions? Get instant answers now!

( [link] ) Find the product. ( 5 m 3 ) ( 2 m + 3 ) .

10 m 2 + 9 m 9

Got questions? Get instant answers now!

( [link] ) Find the product. ( a + 4 ) ( a 2 2 a + 3 ) .

Got questions? Get instant answers now!

Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
Hi
mostafa
hi
LEMLEM
hello
Sammol
hi
Mahesh
bi
Ruqayat
hi
Ruqayat
Hi fellas
Nyawa
hey
Sammol
hi
God
hello
Jahara
Good morning
Jorge
hi
abubakar
hi
Nmesoma
hi
Mahesh
Hi
Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
Hi
Isiaka
Hi
Geli
hy
Man
👋
Bahunda
hy how are you?
Man
ys
Amisha
how are you guys
Sekou
f9 guys
Amisha
how are you guys
Sekou
ys am also fine
Amisha
fine and you guys
Geli
from Nepal
Amisha
nawalparasi district from belatari
Amisha
nd u
Amisha
I am Camara from Guinea west Africa... happy to meet you guys here
Sekou
ma management ho
Amisha
ahile becheclor ho
Amisha
hjr ktm bta ho ani k kaam grnu hunxa tw
Amisha
belatari
Amisha
1st year ho
Amisha
nd u
Amisha
ahh
Amisha
kaha biratnagar
Amisha
ys
Amisha
kina k vo
Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
Mercy is it clear?😋
Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask