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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!

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( 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.

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( 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.'

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( 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.'

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( 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.'

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( 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

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( 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

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( 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

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Practice set a

Find the following products.

( x + 5 ) 2

x 2 + 10 x + 25

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( x + 7 ) 2

x 2 + 14 x + 49

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( y 6 ) 2

y 2 12 y + 36

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( 3 a + b ) 2

9 a 2 + 6 a b + b 2

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( 9 m n ) 2

81 m 2 18 m n + n 2

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( 10 x 2 y ) 2

100 x 2 40 x y + 4 y 2

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( 12 a 7 b ) 2

144 a 2 168 a b + 49 b 2

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( 5 h 15 k ) 2

25 h 2 150 h k + 225 k 2

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Exercises

For the following problems, find the products.

( x + 3 ) 2

x 2 + 6 x + 9

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( x + 8 ) 2

x 2 + 16 x + 64

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( y + 9 ) 2

y 2 + 18 y + 81

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( a 4 ) 2

a 2 8 a + 16

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( a 7 ) 2

a 2 14 a + 49

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( b + 15 ) 2

b 2 + 30 b + 225

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( x 12 ) 2

x 2 24 x + 144

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( y 20 ) 2

y 2 40 y + 400

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( 4 x + 2 ) 2

16 x 2 + 16 x + 4

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( 7 x 2 ) 2

49 x 2 28 x + 4

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( 3 a 9 ) 2

9 a 2 54 a + 81

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( 5 a 3 b ) 2

25 a 2 30 a b + 9 b 2

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( 2 h 8 k ) 2

4 h 2 32 h k + 64 k 2

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( a + 1 3 ) 2

a 2 + 2 3 a + 1 9

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( x + 2 5 ) 2

x 2 + 4 5 x + 4 25

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( y 5 6 ) 2

y 2 5 3 y + 25 36

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( x + 1.3 ) 2

x 2 + 2.6 x + 1.69

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( a + 0.5 ) 2

a 2 + a + 0.25

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( x 3.1 ) 2

x 2 6.2 x + 9.61

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( b 0.04 ) 2

b 2 0.08 b + 0.0016

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( x + 5 ) ( x 5 )

x 2 25

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( x + 1 ) ( x 1 )

x 2 1

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( f + 9 ) ( f 9 )

f 2 81

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( 2 y + 3 ) ( 2 y 3 )

4 y 2 9

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( 5 x + 6 ) ( 5 x 6 )

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

4 a 2 49 b 2

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

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

25 h 2 4 k 2

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( x + 1 3 ) ( x 1 3 )

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( a + 2 9 ) ( a 2 9 )

a 2 4 81

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( x + 7 3 ) ( x 7 3 )

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

4 b 2 36 49

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

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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

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

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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

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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.

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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.

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Exercises for review

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

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

1 80

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

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

10 m 2 + 9 m 9

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

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Questions & Answers

What is scarcity.
Npoanlarb Reply
when there is adequate resources
Fiona
why our wants are limited
Npoanlarb Reply
nooo want is unlimited but resources are limited
Ruchi
and do to that there occurs scarcity and we have to make choice in order to have what we need if need be I will explain more
Madara
our wants are not limited but rather the resources
Moses
as we know that there are two principle of microeconomics scarcity of resources and they have alternative uses...
Ruchi
yes .....
Mathias
because our resources are limited./we have a limited resources.
Ijeoma
what is demand
Thank Reply
demand is something wt we called in economic theory of demand it simply means if price of product is increase then demand of product will decrease
Ruchi
inverse relationship between demand and price
Ruchi
in microeconomic
Ruchi
demand is what and how much you want and what's your need...
Shikhar
how can one be so with economics even while you have less knowledge in mathematics.
OKORO Reply
why is it that some products increases everyday by day
Chiamaka Reply
because demand is increase
Ruchi
because demand is increase
Patience
but how demand increases?
Aziz
Because of the Marketing and purchasing power of people.
AmarbirSingh
but how could we know that people's demands have increased everyday by day and how could we know that this is time to produced the products in the market. Is any connection among them
yaqoob
for normal good people demand remain the same if price of product will increase or not
Ruchi
see that some product which increases day by day is comes under normal good which is used by consumer
Ruchi
Seems hot discussing going here
Shamamet
If there are less products demand starts to increase for those products
Shamamet
Economics is really interesting to learn ....
Shamamet
see there is Inferior goods ands normal goods inferior good demand is rarely increase whereas as we talk about normal good demand will absolutely Increase whether price is increase or not
Ruchi
and demand for normal goods increase cause people's income as a while increases time to time
Abhisek
and it might also be that the cost of raw materials are high.
ATTAH
may be
Ruchi
obviously because demand is increasing.....and price is getting low.....
Shikhar
hmmm there is inverse relationship between demand and price
Ruchi
This is because the supply of those products in relation to raw materials are decreasing and they are also necessities. This crate shortage in the market, so sellers will rise the prices of those products.
Abdul
Importance of economics
Odunayomi Reply
the nature and significance of economics studies
Deborah
What is demand
Shuaib Reply
deman is amount of goods and services a consumer is willing and able to buy or purchase at a given price.
Sainabou
the willingness and ability of a body to purchase goods nd servicesbis called demand ,so if she/has ability but doesn't have willingness it's not a demand same if she or he has willingness but doesn't has ability it's not a demand too
Gul
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time and demand can also be defined as the desire or willingness and backed by the ability to pay.
Fadiga
Yeah
Mathias
What is Choice
Kofi
Choice refers to the ability of a consumer or producer to decide which good, service or resource to purchase or provide from a range of possible options. Being free to chose is regarded as a fundamental indicator of economic well being and development.
Shonal
choice is a act of selecting or choosing from the numerous or plenty wants.
Fadiga
demand is want and it is also what you need and able to afford a particular period of time... because demand changes with time.
Ijeoma
Demand refers to the ability of the consumer to pay for a particular product at a given price
Abdul
how does consumer make profit
Clifford Reply
by buying goods in bulk.
Ijeoma
Compare and contract the function of commercial bank and the central bank of Nigeria
Akwi Reply
what do think is the difference between overhead costs and prime cost
Abdoulkarim
what is economics
Mohamed Reply
economics is a social science that study's how resources can be used to produce goods and services for society
Nathan
Economic is a science which studies human behavior as a relationship between ends and scares means which have alternatives uses or purposes.
Fadiga
what is economics
Mohamed Reply
what is the basic economic problem
John Reply
rules
Buayadarat_Gaming
unlimited wants vs limited resources
Nathan
what economics is all about?
Nomuhle Reply
what is a new paradigm shift
Austen Reply
Paradigm shift it is the reconcilliation of fedural goods in production
Shyline
fedural? what is that?
Aziz
factors that affecting economic system
Bemen Reply
crux
Shyline
what is meant by broadening the tax base?
Fiona Reply
Difference between extinct and extici spicies
Amanpreet 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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