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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
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please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
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50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
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physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
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what is ogarnic chemistry
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determine the slope giving that 3y+ 2x-14=0
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Another formula for Acceleration
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a=v/t. a=f/m a
IHUMA
innocent
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pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
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Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
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No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
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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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