Skip to content Skip to navigation

Connexions

You are here: Home » Content » Algebraic Expressions and Equations: Combining Polynomials Using Multiplication

Navigation

Lenses

What is a lens?

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions 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.
  • CCOT Project display tagshide tags

    This module is included in aLens by: CC Open Textbook ProjectAs a part of collection:"Elementary Algebra"

    Click the "CCOT Project" 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 module is included inLens: Florida Orange Grove Textbooks
    By: Florida Orange GroveAs a part of collection:"Elementary Algebra"

    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 module is included inLens: Connexions Featured Content
    By: ConnexionsAs a part of collection:"Elementary Algebra"

    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.

Also in these lenses

  • SHN CNX Workshop display tagshide tags

    This module is included inLens: Stategic Horizon Network Workshop on Alternative Couseware -- Connexions Session
    By: ConnexionsAs a part of collection:"Elementary Algebra"

    Comments:

    "This textbook by traditionally published authors, Wade and Burzynski, was aquired by the Community College Open Textbook project and put into Connexions for the benefit of the community."

    Click the "SHN CNX Workshop" link to see all content selected in this lens.

    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.

Algebraic Expressions and Equations: Combining Polynomials Using Multiplication

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

User rating (How does the rating system work?)
Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

:
(0 ratings)

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.

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

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)

Solution

3x+24 3x+24

Exercise 2

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

Solution

4a+8 4a+8

Exercise 3

( a 2 −2b+6)2a ( a 2 −2b+6)2a

Solution

2 a 3 −4ab+12a 2 a 3 −4ab+12a

Exercise 4

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

Solution

16 a 3 b 3 +56 a 2 b 4 +24 a 2 b 3 16 a 3 b 3 +56 a 2 b 4 +24 a 2 b 3

Exercise 5

4x(2 x 5 +6 x 4 −8 x 3 − x 2 +9x−11) 4x(2 x 5 +6 x 4 −8 x 3 − x 2 +9x−11)

Solution

8 x 6 +24 x 5 −32 x 4 −4 x 3 +36 x 2 −44x 8 x 6 +24 x 5 −32 x 4 −4 x 3 +36 x 2 −44x

Exercise 6

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

Solution

6 a 3 b 3 +12 a 2 b 4 6 a 3 b 3 +12 a 2 b 4

Exercise 7

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

Solution

5 m 3 n 3 +5 m 2 n+5mn 5 m 3 n 3 +5 m 2 n+5mn

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)

Solution

12.7233 a 3 +48.24 a 2 b 12.7233 a 3 +48.24 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

(6x−1) (6x−1) .

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

(6x−1)=6x−1 (6x−1)=6x−1

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 +7a−18) −(21 a 2 +7a−18) .

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

Example 11

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

Practice Set B

Simplify by removing the parentheses.

Exercise 9

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

Solution

2a+3b 2a+3b

Exercise 10

( a 2 −6a+10) ( a 2 −6a+10)

Solution

a 2 −6a+10 a 2 −6a+10

Exercise 11

−(x+2y) −(x+2y)

Solution

−x−2y −x−2y

Exercise 12

−(5m−2n) −(5m−2n)

Solution

−5m+2n −5m+2n

Exercise 13

−(−3 s 2 −7s+9) −(−3 s 2 −7s+9)

Solution

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

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

(m−3) 2 = (m−3)(m−3) = m⋅m+m(−3)−3⋅m−3(−3) = m 2 −3m−3m+9 = m 2 −6m+9 (m−3) 2 = (m−3)(m−3) = m⋅m+m(−3)−3⋅m−3(−3) = m 2 −3m−3m+9 = m 2 −6m+9

Example 17

(x+5) 3 = (x+5)(x+5)(x+5) Associate the first two factors. = [ (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+10x⋅x+10x⋅5+25⋅x+25⋅5 = 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) Associate the first two factors. = [ (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+10x⋅x+10x⋅5+25⋅x+25⋅5 = 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)

Solution

a 2 +5a+4 a 2 +5a+4

Exercise 15

(m−9)(m−2) (m−9)(m−2)

Solution

m 2 −11m+18 m 2 −11m+18

Exercise 16

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

Solution

2 x 2 +14x+20 2 x 2 +14x+20

Exercise 17

(x+y)(2x−3y) (x+y)(2x−3y)

Solution

2 x 2 −xy−3 y 2 2 x 2 −xy−3 y 2

Exercise 18

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

Solution

15 a 4 +3 a 3 −5 a 2 −a 15 a 4 +3 a 3 −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)

Solution

10 x 5 y + 5 7 x 4 y 4 + x 3 y 3 10 x 5 y + 5 7 x 4 y 4 + x 3 y 3

Exercise 20

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

Solution

a 3 +6 a 2 +15a+18 a 3 +6 a 2 +15a+18

Exercise 21

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

Solution

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

Exercise 22

(r−7)(r−7) (r−7)(r−7)

Solution

r 2 −14r+49 r 2 −14r+49

Exercise 23

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

Solution

x 2 +12x+36 x 2 +12x+36

Exercise 24

(y−8) 2 (y−8) 2

Solution

y 2 −16y+64 y 2 −16y+64

Sample Set D

Perform the following additions and subtractions.

Example 18

3x+7+(x−3). We must first remove the parentheses. They are preceded by a "+" sign, so we remove them and leave the sign of ​each term the same. 3x+7+x−3 Combine like terms. 4x+4   3x+7+(x−3). We must first remove the parentheses. They are preceded by a "+" sign, so we remove them and leave the sign of ​each term the same. 3x+7+x−3 Combine like terms. 4x+4  

Example 19

5 y 3 +11−(12 y 3 −2). We first remove the parentheses. They are preceded by a "-" sign, so we remove them and change the sign of each term inside them. 5 y 3 +11−12 y 3 +2 Combine like terms. −7 y 3 +13 5 y 3 +11−(12 y 3 −2). We first remove the parentheses. They are preceded by a "-" sign, so we remove them and change the sign of each term inside them. 5 y 3 +11−12 y 3 +2 Combine like terms. −7 y 3 +13

Example 20

Add 4 x 2 +2x−8 4 x 2 +2x−8 to 3 x 2 −7x−10 3 x 2 −7x−10 .

(4 x 2 +2x−8)+(3 x 2 −7x−10) 4 x 2 +2x−8+3 x 2 −7x−10 7 x 2 −5x−18 (4 x 2 +2x−8)+(3 x 2 −7x−10) 4 x 2 +2x−8+3 x 2 −7x−10 7 x 2 −5x−18

Example 21

Subtract 8 x 2 −5x+2 8 x 2 −5x+2 from 3 x 2 +x−12 3 x 2 +x−12 .

(3 x 2 +x−12)−(8 x 2 −5x+2) 3 x 2 +x−12−8 x 2 +5x−2 −5 x 2 +6x−14 (3 x 2 +x−12)−(8 x 2 −5x+2) 3 x 2 +x−12−8 x 2 +5x−2 −5 x 2 +6x−14

Be very careful not to write this problem as

3 x 2 +x−12−8 x 2 −5x+2 3 x 2 +x−12−8 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 +x−12) 8 x 2 −5x+2−(3 x 2 +x−12)

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 +2y−1+(5 y 2 −18) 6 y 2 +2y−1+(5 y 2 −18)

Solution

11 y 2 +2y−19 11 y 2 +2y−19

Exercise 26

(9m−n)−(10m+12n) (9m−n)−(10m+12n)

Solution

−m−13n −m−13n

Exercise 27

Add 2 r 2 +4r−1 2 r 2 +4r−1 to 3 r 2 −r−7 3 r 2 −r−7 .

Solution

5 r 2 +3r−8 5 r 2 +3r−8

Exercise 28

Subtract 4s−3 4s−3 from 7s+8 7s+8 .

Solution

3s+11 3s+11

Exercises

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

Exercise 29

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

Solution

7x+42 7x+42

Exercise 30

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

Exercise 31

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

Solution

6y+24 6y+24

Exercise 32

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

Exercise 33

5(a−6) 5(a−6)

Solution

5a−30 5a−30

Exercise 34

2(x−10) 2(x−10)

Exercise 35

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

Solution

12x+6 12x+6

Exercise 36

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

Exercise 37

9(4y−3) 9(4y−3)

Solution

36y−27 36y−27

Exercise 38

5(8m−6) 5(8m−6)

Exercise 39

−9(a+7) −9(a+7)

Solution

−9a−63 −9a−63

Exercise 40

−3(b+8) −3(b+8)

Exercise 41

−4(x+2) −4(x+2)

Solution

−4x−8 −4x−8

Exercise 42

−6(y+7) −6(y+7)

Exercise 43

−3(a−6) −3(a−6)

Solution

−3a+18 −3a+18

Exercise 44

−9(k−7) −9(k−7)

Exercise 45

−5(2a+1) −5(2a+1)

Solution

−10a−5 −10a−5

Exercise 46

−7(4x+2) −7(4x+2)

Exercise 47

−3(10y−6) −3(10y−6)

Solution

−30y+18 −30y+18

Exercise 48

−8(4y−11) −8(4y−11)

Exercise 49

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

Solution

x 2 +6x x 2 +6x

Exercise 50

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

Exercise 51

m(m−4) m(m−4)

Solution

m 2 −4m m 2 −4m

Exercise 52

k(k−11) k(k−11)

Exercise 53

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

Solution

3 x 2 +6x 3 x 2 +6x

Exercise 54

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

Exercise 55

6a(a−5) 6a(a−5)

Solution

6 a 2 −30a 6 a 2 −30a

Exercise 56

9x(x−3) 9x(x−3)

Exercise 57

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

Solution

15 x 2 +12x 15 x 2 +12x

Exercise 58

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

Exercise 59

2b(b−1) 2b(b−1)

Solution

2 b 2 −2b 2 b 2 −2b

Exercise 60

7a(a−4) 7a(a−4)

Exercise 61

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

Solution

15 x 4 +12 x 2 15 x 4 +12 x 2

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)

Solution

20 a 7 +12 a 6 +8 a 5 20 a 7 +12 a 6 +8 a 5

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)

Solution

−5 x 3 −10 x 2 −5 x 3 −10 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)

Solution

6 x 4 y 3 −12 x 3 y 6 x 4 y 3 −12 x 3 y

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 (2bx−11) b 5 x 2 (2bx−11)

Solution

2 b 6 x 3 −11 b 5 x 2 2 b 6 x 3 −11 b 5 x 2

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 +y−6) 9 y 3 (2 y 4 −3 y 3 +8 y 2 +y−6)

Solution

18 y 7 −27 y 6 +72 y 5 +9 y 4 −54 y 3 18 y 7 −27 y 6 +72 y 5 +9 y 4 −54 y 3

Exercise 72

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

Exercise 73

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

Solution

a 2 +6a+8 a 2 +6a+8

Exercise 74

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

Exercise 75

(y+6)(y−3) (y+6)(y−3)

Solution

y 2 +3y−18 y 2 +3y−18

Exercise 76

(t+8)(t−2) (t+8)(t−2)

Exercise 77

(i−3)(i+5) (i−3)(i+5)

Solution

i 2 +2i−15 i 2 +2i−15

Exercise 78

(x−y)(2x+y) (x−y)(2x+y)

Exercise 79

(3a−1)(2a−6) (3a−1)(2a−6)

Solution

6 a 2 −20a+6 6 a 2 −20a+6

Exercise 80

(5a−2)(6a−8) (5a−2)(6a−8)

Exercise 81

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

Solution

18 y 2 +93y+110 18 y 2 +93y+110

Exercise 82

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

Exercise 83

(4+x)(3−x) (4+x)(3−x)

Solution

− x 2 −x+12 − x 2 −x+12

Exercise 84

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

Exercise 85

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

Solution

x 3 + x 2 +2x+2 x 3 + x 2 +2x+2

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)

Solution

6 x 4 −7 x 2 −5 6 x 4 −7 x 2 −5

Exercise 88

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

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)

Solution

12 x 5 y 7 +30 x 3 y 5 +12 x 3 y 3 +30xy 12 x 5 y 7 +30 x 3 y 5 +12 x 3 y 3 +30xy

Exercise 90

5(x−7)(x−3) 5(x−7)(x−3)

Exercise 91

4(a+1)(a−8) 4(a+1)(a−8)

Solution

4 a 2 −28a−32 4 a 2 −28a−32

Exercise 92

a(a−3)(a+5) a(a−3)(a+5)

Exercise 93

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

Solution

x 3 +5 x 2 +4x x 3 +5 x 2 +4x

Exercise 94

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

Exercise 95

y 3 (y−3)(y−2) y 3 (y−3)(y−2)

Solution

y 5 −5 y 4 +6 y 3 y 5 −5 y 4 +6 y 3

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)

Solution

5 y 8 +40 y 7 +35 y 6 5 y 8 +40 y 7 +35 y 6

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)(2xy−1) x 3 y 2 (5 x 2 y 2 −3)(2xy−1)

Solution

10 x 6 y 5 −5 x 5 y 4 −6 x 4 y 3 +3 x 3 y 2 10 x 6 y 5 −5 x 5 y 4 −6 x 4 y 3 +3 x 3 y 2

Exercise 100

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

Exercise 101

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

Solution

8 c 3 +40c+88 8 c 3 +40c+88

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)

Solution

24 a 5 b 9 +42 a 4 b 11 +12 a 3 b 13 +18 a 3 b 3 24 a 5 b 9 +42 a 4 b 11 +12 a 3 b 13 +18 a 3 b 3

Exercise 104

(a−4)( a 2 +a−5) (a−4)( a 2 +a−5)

Exercise 105

(x−7)( x 2 +x−3) (x−7)( x 2 +x−3)

Solution

x 3 −6 x 2 −10x+21 x 3 −6 x 2 −10x+21

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 −a−1) (7 a 2 +2)(3 a 5 −4 a 3 −a−1)

Solution

21 a 7 −22 a 5 −15 a 3 −7 a 2 −2a−2 21 a 7 −22 a 5 −15 a 3 −7 a 2 −2a−2

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 b−b−4) (2a+b)(5 a 2 +4 a 2 b−b−4)

Solution

10 a 3 +8 a 3 b+4 a 2 b 2 +5 a 2 b− b 2 −8a−4b−2ab 10 a 3 +8 a 3 b+4 a 2 b 2 +5 a 2 b− b 2 −8a−4b−2ab

Exercise 110

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

Exercise 111

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

Solution

x 2 +2x+1 x 2 +2x+1

Exercise 112

(x−5) 2 (x−5) 2

Exercise 113

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

Solution

a 2 +4a+4 a 2 +4a+4

Exercise 114

(a−9) 2 (a−9) 2

Exercise 115

− (3x−5) 2 − (3x−5) 2

Solution

−9 x 2 +30x−25 −9 x 2 +30x−25

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 +5x−2+(4 x 2 −10x−5) 3 x 2 +5x−2+(4 x 2 −10x−5)

Solution

7 x 2 −5x−7 7 x 2 −5x−7

Exercise 118

−2 x 3 +4 x 2 +5x−8+( x 3 −3 x 2 −11x+1) −2 x 3 +4 x 2 +5x−8+( x 3 −3 x 2 −11x+1)

Exercise 119

−5x−12xy+4 y 2 +(−7x+7xy−2 y 2 ) −5x−12xy+4 y 2 +(−7x+7xy−2 y 2 )

Solution

2 y 2 −5xy−12x 2 y 2 −5xy−12x

Exercise 120

(6 a 2 −3a+7)−4 a 2 +2a−8 (6 a 2 −3a+7)−4 a 2 +2a−8

Exercise 121

(5 x 2 −24x−15)+ x 2 −9x+14 (5 x 2 −24x−15)+ x 2 −9x+14

Solution

6 x 2 −33x−1 6 x 2 −33x−1

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 b−3ab+12a b 2 )+a b 2 +2ab (9 a 2 b−3ab+12a b 2 )+a b 2 +2ab

Solution

9 a 2 b+13a b 2 −ab 9 a 2 b+13a b 2 −ab

Exercise 124

6 x 2 −12x+(4 x 2 −3x−1)+4 x 2 −10x−4 6 x 2 −12x+(4 x 2 −3x−1)+4 x 2 −10x−4

Exercise 125

5 a 3 −2a−26+(4 a 3 −11 a 2 +2a)−7a+8 a 3 +20 5 a 3 −2a−26+(4 a 3 −11 a 2 +2a)−7a+8 a 3 +20

Solution

17 a 3 −11 a 2 −7a−6 17 a 3 −11 a 2 −7a−6

Exercise 126

2xy−15−(5xy+4) 2xy−15−(5xy+4)

Exercise 127

Add 4x+6 4x+6 to 8x−15 8x−15 .

Solution

12x−9 12x−9

Exercise 128

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

Exercise 129

Add 3(x+6) 3(x+6) to 4(x−7) 4(x−7) .

Solution

7x−10 7x−10

Exercise 130

Add −2( x 2 −4) −2( x 2 −4) to 5( x 2 +3x−1) 5( x 2 +3x−1) .

Exercise 131

Add four times 5x+2 5x+2 to three times 2x−1 2x−1 .

Solution

26x+5 26x+5

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 −8x−3 −8x−3 .

Solution

−20x−18 −20x−18

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 .

Solution

0

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.

Solution

1.98× 10 5 1.98× 10 5

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 y−5 x 3 y 3 4x y 3 +3 x 5 y−5 x 3 y 3 , and write the numerical coefficient of each term.

Solution

degree is 6;  4,3,−5 degree is 6;  4,3,−5

Exercise 140

((Reference)) Simplify 3(4x−5)+2(5x−2)−(x−3) 3(4x−5)+2(5x−2)−(x−3) .

Content actions

Give Feedback:

E-mail the module authors | Rate module ( How does the rating system work?)

Rating system

Ratings

Ratings allow you to judge the quality of modules. If other users have ranked the module then its average rating is displayed below. Ratings are calculated on a scale from one star (Poor) to five stars (Excellent).

How to rate a module

Hover over the star that corresponds to the rating you wish to assign. Click on the star to add your rating. Your rating should be based on the quality of the content. You must have an account and be logged in to rate content.

(0 ratings)

Download:

Module PDF

Add module to:

My Favorites (?)

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

| A lens (?)

Definition of a lens

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions 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