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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 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 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

Practice Set A

Determine the following products.

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

3x+24 3x+24

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

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

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

Solution

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

Exercise 8

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.

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

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.

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

2a+3b 2a+3b

Exercise 10

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

Solution

a 2 6a+10 a 2 6a+10

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

x2y x2y

(5m2n) (5m2n)

5m+2n 5m+2n

Exercise 13

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

Solution

3 s 2 +7s9 3 s 2 +7s9

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,

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 .

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

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

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)

Solution

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

Exercise 15

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

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

Solution

2 x 2 xy3 y 2 2 x 2 xy3 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

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

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

(y8) 2 (y8) 2

Solution

y 2 16y+64 y 2 16y+64

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)

Solution

11 y 2 +2y19 11 y 2 +2y19

Exercise 26

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

m13n m13n

Exercise 27

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

Solution

5 r 2 +3r8 5 r 2 +3r8

Exercise 28

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

3s+11 3s+11

Exercises

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

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

7x+42 7x+42

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

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

6y+24 6y+24

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

5(a6) 5(a6)

5a30 5a30

2(x10) 2(x10)

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

12x+6 12x+6

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

9(4y3) 9(4y3)

36y27 36y27

5(8m6) 5(8m6)

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

9a63 9a63

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

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

4x8 4x8

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

3(a6) 3(a6)

3a+18 3a+18

9(k7) 9(k7)

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

10a5 10a5

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

3(10y6) 3(10y6)

30y+18 30y+18

8(4y11) 8(4y11)

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

x 2 +6x x 2 +6x

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

m(m4) m(m4)

m 2 4m m 2 4m

k(k11) k(k11)

Exercise 53

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

Solution

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

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

Exercise 55

6a(a5) 6a(a5)

Solution

6 a 2 30a 6 a 2 30a

9x(x3) 9x(x3)

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(b1) 2b(b1)

Solution

2 b 2 2b 2 b 2 2b

7a(a4) 7a(a4)

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

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

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 +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)

Solution

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

Exercise 74

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

Exercise 75

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

Solution

y 2 +3y18 y 2 +3y18

Exercise 76

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

Exercise 77

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

Solution

i 2 +2i15 i 2 +2i15

Exercise 78

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

Exercise 79

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

Solution

6 a 2 20a+6 6 a 2 20a+6

Exercise 80

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

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

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 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)

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

Exercise 91

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

Solution

4 a 2 28a32 4 a 2 28a32

Exercise 92

a(a3)(a+5) a(a3)(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 (y3)(y2) y 3 (y3)(y2)

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

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

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

Exercise 105

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

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

Solution

21 a 7 22 a 5 15 a 3 7 a 2 2a2 21 a 7 22 a 5 15 a 3 7 a 2 2a2

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)

Solution

10 a 3 +8 a 3 b+4 a 2 b 2 +5 a 2 b b 2 8a4b2ab 10 a 3 +8 a 3 b+4 a 2 b 2 +5 a 2 b b 2 8a4b2ab

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

Exercise 111

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

Solution

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

(x5) 2 (x5) 2

Exercise 113

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

Solution

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

(a9) 2 (a9) 2

Exercise 115

(3x5) 2 (3x5) 2

Solution

9 x 2 +30x25 9 x 2 +30x25

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)

Solution

7 x 2 5x7 7 x 2 5x7

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 )

Solution

2 y 2 5xy12x 2 y 2 5xy12x

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

Solution

6 x 2 33x1 6 x 2 33x1

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

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 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

Solution

17 a 3 11 a 2 7a6 17 a 3 11 a 2 7a6

Exercise 126

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

Exercise 127

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

12x9 12x9

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) .

7x10 7x10

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 .

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 8x3 8x3 .

20x18 20x18

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 .

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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 y5 x 3 y 3 4x y 3 +3 x 5 y5 x 3 y 3 , and write the numerical coefficient of each term.

Solution

degreeis6;4,3,5 degreeis6;4,3,5

Exercise 140

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

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