<< 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

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
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