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Special Binomial Products

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: 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 ( a+b ) 2 and ( ab ) 2 ( ab ) 2
  • Expanding ( a+b )( ab ) ( a+b )( ab )

Three binomial products occur so frequently in algebra that we designate them as special binomial products. We have seen them before (Sections (Reference) and (Reference)), 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 ( a+b ) 2      and      ( ab ) 2 ( ab ) 2

and as the sum and difference of two terms.

( a+b )( ab ) ( a+b )( ab )

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(a+b)2 and (ab)2(ab)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 +2ab+ b 2 ( ab ) 2 = a 2 2ab+ b 2 ( a+b ) 2 = a 2 +2ab+ b 2 ( ab ) 2 = a 2 2ab+ b 2

Expanding (a+b)(ab)(a+b)(ab)

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

          

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

Sample Set A

Example 1

( x+4 ) 2 Squarethefirstterm:    x 2 . Theproductofbothtermsis4x.Doubleit:   8x. Squarethelastterm:   16. Addthemtogether:    x 2 +8x+16. ( x+4 ) 2 = x 2 +8x+16 ( x+4 ) 2 Squarethefirstterm:    x 2 . Theproductofbothtermsis4x.Doubleit:   8x. Squarethelastterm:   16. Addthemtogether:    x 2 +8x+16. ( x+4 ) 2 = x 2 +8x+16

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

Example 2

( a8 ) 2 Squarethefirstterm:    a 2 . Theproductofbothtermsis8a.Doubleit:   16a. Squarethelastterm:   64. Addthemtogether:    a 2 +( 16a )+64. ( a8 ) 2 = a 2 16a+64 ( a8 ) 2 Squarethefirstterm:    a 2 . Theproductofbothtermsis8a.Doubleit:   16a. Squarethelastterm:   64. Addthemtogether:    a 2 +( 16a )+64. ( a8 ) 2 = a 2 16a+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   ( )( )=+ ( + )( + )=+   and   ( )( )=+

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

Example 3

( y1 ) 2 Squarethefirstterm:    y 2 . Theproductofbothtermsisy.Doubleit:   2y. Squarethelastterm:   +1. Addthemtogether:    y 2 +( 2y )+1. ( y1 ) 2 Squarethefirstterm:    y 2 . Theproductofbothtermsisy.Doubleit:   2y. Squarethelastterm:   +1. Addthemtogether:    y 2 +( 2y )+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.'

Example 4

( 5x+3 ) 2 Squarethefirstterm:   25 x 2 . Theproductofbothtermsis15x.Doubleit:   30x. Squarethelastterm:   9. Addthemtogether:   25 x 2 +30x+9. ( 5x+3 ) 2 Squarethefirstterm:   25 x 2 . Theproductofbothtermsis15x.Doubleit:   30x. Squarethelastterm:   9. Addthemtogether:   25 x 2 +30x+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.'

Example 5

( 7b2 ) 2 Squarethefirstterm:   49 b 2 . Theproductofbothtermsis14b.Doubleit:   28b. Squarethelastterm:   4. Addthemtogether:   49 b 2 +( 28b )+4. ( 7b2 ) 2 Squarethefirstterm:   49 b 2 . Theproductofbothtermsis14b.Doubleit:   28b. Squarethelastterm:   4. Addthemtogether:   49 b 2 +( 28b )+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.'

Example 6

(x+6)(x6) Squarethefirstterm: x 2 . Subtractthesquareofthesecondterm(36)from thesquareofthefirstterm: x 2 36. (x+6)(x6)= x 2 36 (x+6)(x6) Squarethefirstterm: x 2 . Subtractthesquareofthesecondterm(36)from thesquareofthefirstterm: x 2 36. (x+6)(x6)= x 2 36

Example 7

(4a12)(4a+12) Squarethefirstterm:16 a 2 . Subtractthesquareofthesecondterm(144)from thesquareofthefirstterm:16 a 2 144. (4a12)(4a+12)=16 a 2 144 (4a12)(4a+12) Squarethefirstterm:16 a 2 . Subtractthesquareofthesecondterm(144)from thesquareofthefirstterm:16 a 2 144. (4a12)(4a+12)=16 a 2 144

Example 8

(6x+8y)(6x8y) Squarethefirstterm:36 x 2 . Subtractthesquareofthesecondterm(64 y 2 )from thesquareofthefirstterm: 36 x 2 64 y 2 . (6x+8y)(6x8y)=36 x 2 64 y 2 (6x+8y)(6x8y) Squarethefirstterm:36 x 2 . Subtractthesquareofthesecondterm(64 y 2 )from thesquareofthefirstterm: 36 x 2 64 y 2 . (6x+8y)(6x8y)=36 x 2 64 y 2

Practice Set A

Find the following products.

Exercise 1

( x+5 ) 2 ( x+5 ) 2

Solution

x 2 +10x+25 x 2 +10x+25

Exercise 2

( x+7 ) 2 ( x+7 ) 2

Solution

x 2 +14x+49 x 2 +14x+49

Exercise 3

( y6 ) 2 ( y6 ) 2

Solution

y 2 12y+36 y 2 12y+36

Exercise 4

( 3a+b ) 2 ( 3a+b ) 2

Solution

9 a 2 +6ab+ b 2 9 a 2 +6ab+ b 2

Exercise 5

( 9mn ) 2 ( 9mn ) 2

Solution

81 m 2 18mn+ n 2 81 m 2 18mn+ n 2

Exercise 6

( 10x2y ) 2 ( 10x2y ) 2

Solution

100 x 2 40xy+4 y 2 100 x 2 40xy+4 y 2

Exercise 7

( 12a7b ) 2 ( 12a7b ) 2

Solution

144 a 2 168ab+49 b 2 144 a 2 168ab+49 b 2

Exercise 8

( 5h15k ) 2 ( 5h15k ) 2

Solution

25 h 2 150hk+225 k 2 25 h 2 150hk+225 k 2

Exercises

For the following problems, find the products.

Exercise 9

( x+3 ) 2 ( x+3 ) 2

Solution

x 2 +6x+9 x 2 +6x+9

Exercise 10

( x+5 ) 2 ( x+5 ) 2

Exercise 11

( x+8 ) 2 ( x+8 ) 2

Solution

x 2 +16x+64 x 2 +16x+64

Exercise 12

( x+6 ) 2 ( x+6 ) 2

Exercise 13

( y+9 ) 2 ( y+9 ) 2

Solution

y 2 +18y+81 y 2 +18y+81

Exercise 14

( y+1 ) 2 ( y+1 ) 2

Exercise 15

( a4 ) 2 ( a4 ) 2

Solution

a 2 8a+16 a 2 8a+16

Exercise 16

( a6 ) 2 ( a6 ) 2

Exercise 17

( a7 ) 2 ( a7 ) 2

Solution

a 2 14a+49 a 2 14a+49

Exercise 18

( b+10 ) 2 ( b+10 ) 2

Exercise 19

( b+15 ) 2 ( b+15 ) 2

Solution

b 2 +30b+225 b 2 +30b+225

Exercise 20

( a10 ) 2 ( a10 ) 2

Exercise 21

( x12 ) 2 ( x12 ) 2

Solution

x 2 24x+144 x 2 24x+144

Exercise 22

( x+20 ) 2 ( x+20 ) 2

Exercise 23

( y20 ) 2 ( y20 ) 2

Solution

y 2 40y+400 y 2 40y+400

Exercise 24

( 3x+5 ) 2 ( 3x+5 ) 2

Exercise 25

( 4x+2 ) 2 ( 4x+2 ) 2

Solution

16 x 2 +16x+4 16 x 2 +16x+4

Exercise 26

( 6x2 ) 2 ( 6x2 ) 2

Exercise 27

( 7x2 ) 2 ( 7x2 ) 2

Solution

49 x 2 28x+4 49 x 2 28x+4

Exercise 28

( 5a6 ) 2 ( 5a6 ) 2

Exercise 29

( 3a9 ) 2 ( 3a9 ) 2

Solution

9 a 2 54a+81 9 a 2 54a+81

Exercise 30

( 3w2z ) 2 ( 3w2z ) 2

Exercise 31

( 5a3b ) 2 ( 5a3b ) 2

Solution

25 a 2 30ab+9 b 2 25 a 2 30ab+9 b 2

Exercise 32

( 6t7s ) 2 ( 6t7s ) 2

Exercise 33

(2h8k) 2 (2h8k) 2

Solution

4 h 2 32hk+64 k 2 4 h 2 32hk+64 k 2

Exercise 34

( a+ 1 2 ) 2 ( a+ 1 2 ) 2

Exercise 35

( a+ 1 3 ) 2 ( a+ 1 3 ) 2

Solution

a 2 + 2 3 a+ 1 9 a 2 + 2 3 a+ 1 9

Exercise 36

( x+ 3 4 ) 2 ( x+ 3 4 ) 2

Exercise 37

( x+ 2 5 ) 2 ( x+ 2 5 ) 2

Solution

x 2 + 4 5 x+ 4 25 x 2 + 4 5 x+ 4 25

Exercise 38

( x 2 3 ) 2 ( x 2 3 ) 2

Exercise 39

( y 5 6 ) 2 ( y 5 6 ) 2

Solution

y 2 5 3 y+ 25 36 y 2 5 3 y+ 25 36

Exercise 40

( y+ 2 3 ) 2 ( y+ 2 3 ) 2

Exercise 41

( x+1.3 ) 2 ( x+1.3 ) 2

Solution

x 2 +2.6x+1.69 x 2 +2.6x+1.69

Exercise 42

( x+5.2 ) 2 ( x+5.2 ) 2

Exercise 43

( a+0.5 ) 2 ( a+0.5 ) 2

Solution

a 2 +a+0.25 a 2 +a+0.25

Exercise 44

( a+0.08 ) 2 ( a+0.08 ) 2

Exercise 45

( x3.1 ) 2 ( x3.1 ) 2

Solution

x 2 6.2x+9.61 x 2 6.2x+9.61

Exercise 46

( y7.2 ) 2 ( y7.2 ) 2

Exercise 47

( b0.04 ) 2 ( b0.04 ) 2

Solution

b 2 0.08b+0.0016 b 2 0.08b+0.0016

Exercise 48

( f1.006 ) 2 ( f1.006 ) 2

Exercise 49

( x+5 )( x5 ) ( x+5 )( x5 )

Solution

x 2 25 x 2 25

Exercise 50

( x+6 )( x6 ) ( x+6 )( x6 )

Exercise 51

( x+1 )( x1 ) ( x+1 )( x1 )

Solution

x 2 1 x 2 1

Exercise 52

( t1 )( t+1 ) ( t1 )( t+1 )

Exercise 53

( f+9 )( f9 ) ( f+9 )( f9 )

Solution

f 2 81 f 2 81

Exercise 54

( y7 )( y+7 ) ( y7 )( y+7 )

Exercise 55

( 2y+3 )( 2y3 ) ( 2y+3 )( 2y3 )

Solution

4 y 2 9 4 y 2 9

Exercise 56

( 5x+6 )( 5x6 ) ( 5x+6 )( 5x6 )

Exercise 57

( 2a7b )( 2a+7b ) ( 2a7b )( 2a+7b )

Solution

4 a 2 49 b 2 4 a 2 49 b 2

Exercise 58

( 7x+3t )( 7x3t ) ( 7x+3t )( 7x3t )

Exercise 59

( 5h2k )( 5h+2k ) ( 5h2k )( 5h+2k )

Solution

25 h 2 4 k 2 25 h 2 4 k 2

Exercise 60

( x+ 1 3 )( x 1 3 ) ( x+ 1 3 )( x 1 3 )

Exercise 61

( a+ 2 9 )( a 2 9 ) ( a+ 2 9 )( a 2 9 )

Solution

a 2 4 81 a 2 4 81

Exercise 62

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

Exercise 63

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

Solution

4 b 2 36 49 4 b 2 36 49

Exercise 64

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

Exercise 65

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

Solution

( ab )( ab )= a 2 abab+ b 2 = a 2 2ab+ b 2 ( ab )( ab )= a 2 abab+ b 2 = a 2 2ab+ b 2

Exercise 66

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

Exercise 67

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.

Solution

first term squared

Exercise 68

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.

Exercise 69

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.

Solution

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

Exercises for Review

Exercise 70

((Reference)) Simplify ( x 3 y 0 z 4 ) 5 ( x 3 y 0 z 4 ) 5 .

Exercise 71

((Reference)) Find the value of 10 1 2 3 10 1 2 3 .

Solution

1 80 1 80

Exercise 72

((Reference)) Find the product. ( x+6 )( x7 ) ( x+6 )( x7 ) .

Exercise 73

((Reference)) Find the product. ( 5m3 )( 2m+3 ) ( 5m3 )( 2m+3 ) .

Solution

10 m 2 +9m9 10 m 2 +9m9

Exercise 74

((Reference)) Find the product. (a+4)( a 2 2a+3) (a+4)( a 2 2a+3) .

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