Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Elementary Algebra » Combining Polynomials Using Multiplication

Navigation

Table of Contents

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

This content is ...

Endorsed by Endorsed (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
  • College Open Textbooks display tagshide tags

    This collection is included inLens: Community College Open Textbook Collaborative
    By: CC Open Textbook Collaborative

    Comments:

    "Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"

    Click the "College Open Textbooks" link to see all content they endorse.

    Click the tag icon tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
  • OrangeGrove display tagshide tags

    This collection is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange Grove

    Click the "OrangeGrove" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

  • Featured Content display tagshide tags

    This collection is included inLens: Connexions Featured Content
    By: Connexions

    Comments:

    "Elementary Algebra covers traditional topics studied in a modern elementary algebra course. Written by Denny Burzynski and Wade Ellis, it is intended for both first-time students and those […]"

    Click the "Featured Content" link to see all content affiliated with them.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
Download
x

Download collection as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...

Download module as:

  • PDF
  • EPUB (what's this?)

    What is an EPUB file?

    EPUB is an electronic book format that can be read on a variety of mobile devices.

    Downloading to a reading device

    For detailed instructions on how to download this content's EPUB to your specific device, click the "(what's this?)" link.

  • More downloads ...
Reuse / Edit
x

Collection:

Module:

Add to a lens
x

Add collection to:

Add module to:

Add to Favorites
x

Add collection to:

Add module to:

 

Combining Polynomials Using Multiplication

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 multiply a polynomial by a monomial, be able to simplify +(a + b) and -(a - b), be able to multiply a polynomial by a polynomial.

Overview

  • Multiplying a Polynomial by a Monomial
  • Simplifying +(a+b) +(a+b) and (a+b) (a+b)
  • Multiplying a Polynomial by a Polynomial

Multiplying a Polynomial by a Monomial

Multiplying a polynomial by a monomial is a direct application of the distributive property.

Distributive Property

The product of a monomial a and a binomial b plus c is equal to ab plus ac. This is the distributive property. In the expression, there are two arrows originating from the monomial, a, and pointing towards the terms b and c of the binomial.

The distributive property suggests the following rule.

Multiplying a Polynomial by a Monomial

To multiply a polynomial by a monomial, multiply every term of the polynomial by the monomial and then add the resulting products together.

Sample Set A

Example 1

Finding the product of three and the binomial 'x plus nine', using the distributive property. See the longdesc for a full description.

Example 2

Finding the product of six and the binomial 'x cubed minus two x,' using the distributive property. See the longdesc for a full description.

Example 3

Finding the product of the binomial 'x minus seven' and 'x', using the distributive property. See the longdesc for a full description.

Example 4

Finding the product of 'eight a squared' and the trinomial 'three a to the fourth power minus five a cubed plus a,' using the distributive property. See the longdesc for a full description.

Example 5

Finding the product of 'four x squared y to the seventh power z' and the binomial 'x to the fifth power y plus eight y squared z squared,' using the distributive property. See the longdesc for a full description.

Example 6

10a b 2 c(125 a 2 )=1250 a 3 b 2 c 10a b 2 c(125 a 2 )=1250 a 3 b 2 c

Example 7

Finding the product of the binomial 'nine x squared z plus four w' and the product of 'five z and w cubed,' using the distributive property. See the longdesc for a full description.

Practice Set A

Determine the following products.

Exercise 1

3(x+8) 3(x+8)

Exercise 2

(2+a)4 (2+a)4

Exercise 3

( a 2 2b+6)2a ( a 2 2b+6)2a

Exercise 4

8 a 2 b 3 (2a+7b+3) 8 a 2 b 3 (2a+7b+3)

Exercise 5

4x(2 x 5 +6 x 4 8 x 3 x 2 +9x11) 4x(2 x 5 +6 x 4 8 x 3 x 2 +9x11)

Exercise 6

(3 a 2 b)(2a b 2 +4 b 3 ) (3 a 2 b)(2a b 2 +4 b 3 )

Exercise 7

5mn( m 2 n 2 +m+ n 0 ),n0 5mn( m 2 n 2 +m+ n 0 ),n0

Exercise 8

Use a calculator. 6.03(2.11 a 3 +8.00 a 2 b) 6.03(2.11 a 3 +8.00 a 2 b)

Simplifying +(a+b)+(a+b) and -(a+b)-(a+b)

+(a+b)+(a+b) and -(a+b)-(a+b)

Oftentimes, we will encounter multiplications of the form

+1(a+b) +1(a+b) or -1(a+b) -1(a+b)

These terms will actually appear as

+(a+b) +(a+b) and -(a+b) -(a+b)

Using the distributive property, we can remove the parentheses.

Removal of a set of parentheses preceded by a plus sign using the distributive property. See the longdesc for a full description.

The parentheses have been removed and the sign of each term has remained the same.

Removal of a set of parentheses preceded by a minus sign using the distributive property. See the longdesc for a full description.

The parentheses have been removed and the sign of each term has been changed to its opposite.

  1. To remove a set of parentheses preceded by a " + + " sign, simply remove the parentheses and leave the sign of each term the same.
  2. To remove a set of parentheses preceded by a “ ” sign, remove the parentheses and change the sign of each term to its opposite sign.

Sample Set B

Simplify the expressions.

Example 8

(6x1) (6x1) .

This set of parentheses is preceded by a “ + + ’’ sign (implied). We simply drop the parentheses.

(6x1)=6x1 (6x1)=6x1

Example 9

(14 a 2 b 3 6 a 3 b 2 +a b 4 )=14 a 2 b 3 6 a 3 b 2 +a b 4 (14 a 2 b 3 6 a 3 b 2 +a b 4 )=14 a 2 b 3 6 a 3 b 2 +a b 4

Example 10

(21 a 2 +7a18) (21 a 2 +7a18) .

This set of parentheses is preceded by a “ ” sign. We can drop the parentheses as long as we change the sign of every term inside the parentheses to its opposite sign.

(21 a 2 +7a18)=21 a 2 7a+18 (21 a 2 +7a18)=21 a 2 7a+18

Example 11

(7 y 3 2 y 2 +9y+1)=7 y 3 +2 y 2 9y1 (7 y 3 2 y 2 +9y+1)=7 y 3 +2 y 2 9y1

Practice Set B

Simplify by removing the parentheses.

Exercise 9

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

Exercise 10

( a 2 6a+10) ( a 2 6a+10)

Exercise 11

(x+2y) (x+2y)

Exercise 12

(5m2n) (5m2n)

Exercise 13

(3 s 2 7s+9) (3 s 2 7s+9)

Multiplying a Polynomial by a Polynomial

Since we can consider an expression enclosed within parentheses as a single quantity, we have, by the distributive property,

Finding the product of the binomials 'a plus b' and 'c plus d', using the distributive property. See the longdesc for a full description.

For convenience we will use the commutative property of addition to write this expression so that the first two terms contain a a and the second two contain b b .

(a+b)(c+d)=ac+ad+bc+bd (a+b)(c+d)=ac+ad+bc+bd

This method is commonly called the FOIL method.

  • F
       
    First terms
  • O
       
    Outer terms
  • I
       
    Inner terms
  • L
       
    Last terms

(a+b)(2+3)= (a+b)+(a+b) 2terms + (a+b)+(a+b)+(a+b) 3terms (a+b)(2+3)= (a+b)+(a+b) 2terms + (a+b)+(a+b)+(a+b) 3terms

Rearranging,

=a+a+b+b+a+a+a+b+b+b =2a+2b+3a+3b =a+a+b+b+a+a+a+b+b+b =2a+2b+3a+3b

Combining like terms,

=5a+5b =5a+5b

This use of the distributive property suggests the following rule.

Multiplying a Polynomial by a Polynomial

To multiply two polynomials together, multiply every term of one polynomial by every term of the other polynomial.

Sample Set C

Perform the following multiplications and simplify.

Example 12

Finding the product of 'a plus six' and 'a plus three' using the FOIL method. See the longdesc for a full description.

With some practice, the second and third terms can be combined mentally.

Example 13

Finding the product of two binomials 'x plus y' and 'two x plus four y' using the FOIL method. See the longdesc for a full description.

Example 14

Finding the product of two polynomials 'x squared plus four' and 'x squared plus seven x plus two' using the FOIL method. See the longdesc for a full description.

Example 15

Finding the product of two binomials 'a minus four' and 'a minus three' using the FOIL method. See the longdesc for a full description.

Example 16

(m3) 2 = (m3)(m3) = mm+m(3)3m3(3) = m 2 3m3m+9 = m 2 6m+9 (m3) 2 = (m3)(m3) = mm+m(3)3m3(3) = m 2 3m3m+9 = m 2 6m+9

Example 17

(x+5) 3 = (x+5)(x+5)(x+5) Associatethefirsttwofactors. = [ (x+5)(x+5) ](x+5) = [ x 2 +5x+5x+25 ](x+5) = [ x 2 +10x+25 ](x+5) = x 2 x+ x 2 5+10xx+10x5+25x+255 = x 3 +5 x 2 +10 x 2 +50x+25x+125 = x 3 +15 x 2 +75x+125 (x+5) 3 = (x+5)(x+5)(x+5) Associatethefirsttwofactors. = [ (x+5)(x+5) ](x+5) = [ x 2 +5x+5x+25 ](x+5) = [ x 2 +10x+25 ](x+5) = x 2 x+ x 2 5+10xx+10x5+25x+255 = x 3 +5 x 2 +10 x 2 +50x+25x+125 = x 3 +15 x 2 +75x+125

Practice Set C

Find the following products and simplify.

Exercise 14

(a+1)(a+4) (a+1)(a+4)

Exercise 15

(m9)(m2) (m9)(m2)

Exercise 16

(2x+4)(x+5) (2x+4)(x+5)

Exercise 17

(x+y)(2x3y) (x+y)(2x3y)

Exercise 18

(3 a 2 1)(5 a 2 +a) (3 a 2 1)(5 a 2 +a)

Exercise 19

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

Exercise 20

(a+3)( a 2 +3a+6) (a+3)( a 2 +3a+6)

Exercise 21

(a+4)(a+4) (a+4)(a+4)

Exercise 22

(r7)(r7) (r7)(r7)

Exercise 23

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

Exercise 24

(y8) 2 (y8) 2

Sample Set D

Perform the following additions and subtractions.

Example 18

3x+7+(x3). Wemustfirstremovetheparentheses.Theyareprecededby a"+"sign,soweremovethemandleavethesignofeach termthesame. 3x+7+x3 Combineliketerms. 4x+4 3x+7+(x3). Wemustfirstremovetheparentheses.Theyareprecededby a"+"sign,soweremovethemandleavethesignofeach termthesame. 3x+7+x3 Combineliketerms. 4x+4

Example 19

5 y 3 +11(12 y 3 2). Wefirstremovetheparentheses.Theyareprecededbya "-"sign,soweremovethemandchangethesignofeach terminsidethem. 5 y 3 +1112 y 3 +2 Combineliketerms. 7 y 3 +13 5 y 3 +11(12 y 3 2). Wefirstremovetheparentheses.Theyareprecededbya "-"sign,soweremovethemandchangethesignofeach terminsidethem. 5 y 3 +1112 y 3 +2 Combineliketerms. 7 y 3 +13

Example 20

Add 4 x 2 +2x8 4 x 2 +2x8 to 3 x 2 7x10 3 x 2 7x10 .

(4 x 2 +2x8)+(3 x 2 7x10) 4 x 2 +2x8+3 x 2 7x10 7 x 2 5x18 (4 x 2 +2x8)+(3 x 2 7x10) 4 x 2 +2x8+3 x 2 7x10 7 x 2 5x18

Example 21

Subtract 8 x 2 5x+2 8 x 2 5x+2 from 3 x 2 +x12 3 x 2 +x12 .

(3 x 2 +x12)(8 x 2 5x+2) 3 x 2 +x128 x 2 +5x2 5 x 2 +6x14 (3 x 2 +x12)(8 x 2 5x+2) 3 x 2 +x128 x 2 +5x2 5 x 2 +6x14

Be very careful not to write this problem as

3 x 2 +x128 x 2 5x+2 3 x 2 +x128 x 2 5x+2

This form has us subtracting only the very first term, 8 x 2 8 x 2 , rather than the entire expression. Use parentheses.
Another incorrect form is

8 x 2 5x+2(3 x 2 +x12) 8 x 2 5x+2(3 x 2 +x12)

This form has us performing the subtraction in the wrong order.

Practice Set D

Perform the following additions and subtractions.

Exercise 25

6 y 2 +2y1+(5 y 2 18) 6 y 2 +2y1+(5 y 2 18)

Exercise 26

(9mn)(10m+12n) (9mn)(10m+12n)

Exercise 27

Add 2 r 2 +4r1 2 r 2 +4r1 to 3 r 2 r7 3 r 2 r7 .

Exercise 28

Subtract 4s3 4s3 from 7s+8 7s+8 .

Exercises

For the following problems, perform the multiplications and combine any like terms.

Exercise 29

7(x+6) 7(x+6)

Exercise 30

4(y+3) 4(y+3)

Exercise 31

6(y+4) 6(y+4)

Exercise 32

8(m+7) 8(m+7)

Exercise 33

5(a6) 5(a6)

Exercise 34

2(x10) 2(x10)

Exercise 35

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

Exercise 36

6(3x+4) 6(3x+4)

Exercise 37

9(4y3) 9(4y3)

Exercise 38

5(8m6) 5(8m6)

Exercise 39

9(a+7) 9(a+7)

Exercise 40

3(b+8) 3(b+8)

Exercise 41

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

Exercise 42

6(y+7) 6(y+7)

Exercise 43

3(a6) 3(a6)

Exercise 44

9(k7) 9(k7)

Exercise 45

5(2a+1) 5(2a+1)

Exercise 46

7(4x+2) 7(4x+2)

Exercise 47

3(10y6) 3(10y6)

Exercise 48

8(4y11) 8(4y11)

Exercise 49

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

Exercise 50

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

Exercise 51

m(m4) m(m4)

Exercise 52

k(k11) k(k11)

Exercise 53

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

Exercise 54

4y(y+7) 4y(y+7)

Exercise 55

6a(a5) 6a(a5)

Exercise 56

9x(x3) 9x(x3)

Exercise 57

3x(5x+4) 3x(5x+4)

Exercise 58

4m(2m+7) 4m(2m+7)

Exercise 59

2b(b1) 2b(b1)

Exercise 60

7a(a4) 7a(a4)

Exercise 61

3 x 2 (5 x 2 +4) 3 x 2 (5 x 2 +4)

Exercise 62

9 y 3 (3 y 2 +2) 9 y 3 (3 y 2 +2)

Exercise 63

4 a 4 (5 a 3 +3 a 2 +2a) 4 a 4 (5 a 3 +3 a 2 +2a)

Exercise 64

2 x 4 (6 x 3 5 x 2 2x+3) 2 x 4 (6 x 3 5 x 2 2x+3)

Exercise 65

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

Exercise 66

6 y 3 (y+5) 6 y 3 (y+5)

Exercise 67

2 x 2 y(3 x 2 y 2 6x) 2 x 2 y(3 x 2 y 2 6x)

Exercise 68

8 a 3 b 2 c(2a b 3 +3b) 8 a 3 b 2 c(2a b 3 +3b)

Exercise 69

b 5 x 2 (2bx11) b 5 x 2 (2bx11)

Exercise 70

4x(3 x 2 6x+10) 4x(3 x 2 6x+10)

Exercise 71

9 y 3 (2 y 4 3 y 3 +8 y 2 +y6) 9 y 3 (2 y 4 3 y 3 +8 y 2 +y6)

Exercise 72

a 2 b 3 (6a b 4 +5a b 3 8 b 2 +7b2) a 2 b 3 (6a b 4 +5a b 3 8 b 2 +7b2)

Exercise 73

(a+4)(a+2) (a+4)(a+2)

Exercise 74

(x+1)(x+7) (x+1)(x+7)

Exercise 75

(y+6)(y3) (y+6)(y3)

Exercise 76

(t+8)(t2) (t+8)(t2)

Exercise 77

(i3)(i+5) (i3)(i+5)

Exercise 78

(xy)(2x+y) (xy)(2x+y)

Exercise 79

(3a1)(2a6) (3a1)(2a6)

Exercise 80

(5a2)(6a8) (5a2)(6a8)

Exercise 81

(6y+11)(3y+10) (6y+11)(3y+10)

Exercise 82

(2t+6)(3t+4) (2t+6)(3t+4)

Exercise 83

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

Exercise 84

(6+a)(4+a) (6+a)(4+a)

Exercise 85

( x 2 +2)(x+1) ( x 2 +2)(x+1)

Exercise 86

( x 2 +5)(x+4) ( x 2 +5)(x+4)

Exercise 87

(3 x 2 5)(2 x 2 +1) (3 x 2 5)(2 x 2 +1)

Exercise 88

(4 a 2 b 3 2a)(5 a 2 b3b) (4 a 2 b 3 2a)(5 a 2 b3b)

Exercise 89

(6 x 3 y 4 +6x)(2 x 2 y 3 +5y) (6 x 3 y 4 +6x)(2 x 2 y 3 +5y)

Exercise 90

5(x7)(x3) 5(x7)(x3)

Exercise 91

4(a+1)(a8) 4(a+1)(a8)

Exercise 92

a(a3)(a+5) a(a3)(a+5)

Exercise 93

x(x+1)(x+4) x(x+1)(x+4)

Exercise 94

x 2 (x+5)(x+7) x 2 (x+5)(x+7)

Exercise 95

y 3 (y3)(y2) y 3 (y3)(y2)

Exercise 96

2 a 2 (a+4)(a+3) 2 a 2 (a+4)(a+3)

Exercise 97

5 y 6 (y+7)(y+1) 5 y 6 (y+7)(y+1)

Exercise 98

a b 2 ( a 2 2b)(a+ b 4 ) a b 2 ( a 2 2b)(a+ b 4 )

Exercise 99

x 3 y 2 (5 x 2 y 2 3)(2xy1) x 3 y 2 (5 x 2 y 2 3)(2xy1)

Exercise 100

6( a 2 +5a+3) 6( a 2 +5a+3)

Exercise 101

8( c 3 +5c+11) 8( c 3 +5c+11)

Exercise 102

3 a 2 (2 a 3 10 a 2 4a+9) 3 a 2 (2 a 3 10 a 2 4a+9)

Exercise 103

6 a 3 b 3 (4 a 2 b 6 +7a b 8 +2 b 10 +14) 6 a 3 b 3 (4 a 2 b 6 +7a b 8 +2 b 10 +14)

Exercise 104

(a4)( a 2 +a5) (a4)( a 2 +a5)

Exercise 105

(x7)( x 2 +x3) (x7)( x 2 +x3)

Exercise 106

(2x+1)(5 x 3 +6 x 2 +8) (2x+1)(5 x 3 +6 x 2 +8)

Exercise 107

(7 a 2 +2)(3 a 5 4 a 3 a1) (7 a 2 +2)(3 a 5 4 a 3 a1)

Exercise 108

(x+y)(2 x 2 +3xy+5 y 2 ) (x+y)(2 x 2 +3xy+5 y 2 )

Exercise 109

(2a+b)(5 a 2 +4 a 2 bb4) (2a+b)(5 a 2 +4 a 2 bb4)

Exercise 110

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

Exercise 111

(x+1) 2 (x+1) 2

Exercise 112

(x5) 2 (x5) 2

Exercise 113

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

Exercise 114

(a9) 2 (a9) 2

Exercise 115

(3x5) 2 (3x5) 2

Exercise 116

(8t+7) 2 (8t+7) 2

For the following problems, perform the indicated operations and combine like terms.

Exercise 117

3 x 2 +5x2+(4 x 2 10x5) 3 x 2 +5x2+(4 x 2 10x5)

Exercise 118

2 x 3 +4 x 2 +5x8+( x 3 3 x 2 11x+1) 2 x 3 +4 x 2 +5x8+( x 3 3 x 2 11x+1)

Exercise 119

5x12xy+4 y 2 +(7x+7xy2 y 2 ) 5x12xy+4 y 2 +(7x+7xy2 y 2 )

Exercise 120

(6 a 2 3a+7)4 a 2 +2a8 (6 a 2 3a+7)4 a 2 +2a8

Exercise 121

(5 x 2 24x15)+ x 2 9x+14 (5 x 2 24x15)+ x 2 9x+14

Exercise 122

(3 x 3 7 x 2 +2)+( x 3 +6) (3 x 3 7 x 2 +2)+( x 3 +6)

Exercise 123

(9 a 2 b3ab+12a b 2 )+a b 2 +2ab (9 a 2 b3ab+12a b 2 )+a b 2 +2ab

Exercise 124

6 x 2 12x+(4 x 2 3x1)+4 x 2 10x4 6 x 2 12x+(4 x 2 3x1)+4 x 2 10x4

Exercise 125

5 a 3 2a26+(4 a 3 11 a 2 +2a)7a+8 a 3 +20 5 a 3 2a26+(4 a 3 11 a 2 +2a)7a+8 a 3 +20

Exercise 126

2xy15(5xy+4) 2xy15(5xy+4)

Exercise 127

Add 4x+6 4x+6 to 8x15 8x15 .

Exercise 128

Add 5 y 2 5y+1 5 y 2 5y+1 to 9 y 2 +4y2 9 y 2 +4y2 .

Exercise 129

Add 3(x+6) 3(x+6) to 4(x7) 4(x7) .

Exercise 130

Add 2( x 2 4) 2( x 2 4) to 5( x 2 +3x1) 5( x 2 +3x1) .

Exercise 131

Add four times 5x+2 5x+2 to three times 2x1 2x1 .

Exercise 132

Add five times 3x+2 3x+2 to seven times 4x+3 4x+3 .

Exercise 133

Add 4 4 times 9x+6 9x+6 to 2 2 times 8x3 8x3 .

Exercise 134

Subtract 6 x 2 10x+4 6 x 2 10x+4 from 3 x 2 2x+5 3 x 2 2x+5 .

Exercise 135

Substract a 2 16 a 2 16 from a 2 16 a 2 16 .

Exercises for Review

Exercise 136

((Reference)) Simplify ( 15 x 2 y 6 5x y 2 ) 4 ( 15 x 2 y 6 5x y 2 ) 4 .

Exercise 137

((Reference)) Express the number 198,000 using scientific notation.

Exercise 138

((Reference)) How many 4 a 2 x 3 's 4 a 2 x 3 's are there in 16 a 4 x 5 16 a 4 x 5 ?

Exercise 139

((Reference)) State the degree of the polynomial 4x y 3 +3 x 5 y5 x 3 y 3 4x y 3 +3 x 5 y5 x 3 y 3 , and write the numerical coefficient of each term.

Exercise 140

((Reference)) Simplify 3(4x5)+2(5x2)(x3) 3(4x5)+2(5x2)(x3) .

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Reuse / Edit:

Reuse or edit collection (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.

| Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.