1

. ..

.

1436 / 2015

3

:

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:1 - .

2 -

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4

CHAPTER 1

Operations on The Sets

]1-1 [ ]2-1 [ ]3-1 [

]4-1 [ ]5-1 [

]6-1 [ ]7-1 [

5

:

A,B

AB = {x:x A and x B}

A,B :AB ={x:x A or x B}

, } B = { 2 ,4 , 6 A = { 1 , 2 , 3 , 4 { : AB = {2 , 4{ B , A AB = { 1 , 2 , 3 , 4 , 6 { A , B

.

. Finite Set and Inf inite Set 1-1[ [

: ( }7 , 6 , 5 , 4 , 3 , 2 , 1 }

( 2 20 ( .

( .( 50 .

( . ( 7.

7... . - .

- .

6

) , ( )( )( .

.

. 2 :

C = { 2 , 4 , 6 , 8 , ...{ 5000 = X X

:X = {0 , 1 , 2 , 3 , 4 , ... , 4999 {

]2-1[ : Commutative Property

: 4 5 = 5 4 4 + 5 = 5 + 4 3 4 = 4 3 3 + 4 = 4 + 3

a b = b a , a + b = b + a a b N

1

A = {5 , 4 , 2 { B = { 7 , 5 , 4 {

A B = {5 , 4 , 2 { { 7 , 5 , 4 { : = {5 , 4 {

B A = { 7 , 5 , 4{ { 5 , 4 , 2 { = {5 , 4{

A B = B A

7

2

A = { } B = { {

A B = { } } } } } =

B A = { } } { } } =

A B = B A

: Associative Property

: 6 + )5 + 3( = )6 + 5 ( + 3

6 )5 3 ( = )6 5 ( 3 :

a + ) b + c( = ) a + b( + c

a ) b c( = ) a b( c

a , b , c N

:

3

A = {6 , 5 , 3 , 2 { :B = {4 , 3 , 5 , 2 , 1{ C = {7 , 6 , 4 , 3 , 2 {

A B = { 2 , 3 , 5 { B C = {2 , 3 , 4 {

8

) A B( C = { 5 , 3 , 2{ {2 , 3 , 4 , 6 , 7 {= {3 , 2{

A ) B C( = { 2 , 3 , 5 , 6{ { 4 , 3 , 2 { = {3 , 2 {

)A B ( C = A ) B C( :

4

A = 30 B = 6 C = { 0 , 2 , 3 , 7 {

A B C : A = {2 , 3 , 5{ : B = {1 , 2 , 3 , 6 {

.7

.1

.0

.5

.2

.6

.3

A

B

C

C = {0 , 2 , 3 , 7 { :

A )B C(

A B C = A ) BC( = {2,3,5{ ){1,2,3,6{ {0,2,3,7{( = {2,3,5{ { 3 , 2 { = {3 , 2{

A B C = ) A B(C = ){2,3,5{ {1,2,3,6{ ( {0,2,3,7{ = {3,2{ {0,2,3,7{ = { 3,2{

:

)A B ( C = A ) B C(

:

A B C = )A B ( C = A ) B C (

9

)1 - 1(

1 . )1 - 1( : A = { { B = { { C = { {

A )B C( = { {

A B

C

.8 8

. .6

. .

.4

.5

.

.

7

7

.2

.3

)1 - 1(

2 . )2 - 1(

A = { { ) B = { { )C = { { )

A

B

C

.2 .8

.3 3 . . . .1 .4 4

.5 5

.7 .6

A )B C( = { { ) ( } } = A B( C(

( } } = C B A )1-2(

A = { 2 , 3 , 4 , 5 { 3 .

B = { 3 , 4 , 6 , 7 { C = { 2 , 3 , 4 , 8 {

: A B , B C , A C , A B C

10

4 . A 3 20 B 2 20

A B .

5 . :

A

B

A C

B

)3 - 1( ) 4 - 1(

6 . :( )3(

( ( 100

( 100

11

]3-1[

1

A = {1, 2 , 3 { : B = {2 , 3 , 4 , 5 {

AB = {1 , 2 , 3 {{ 2 , 3 , 4 5 { = {1 , 2 , 3 , 4 , 5 {

BA = {2 , 3 ,4 5 {{1, 2 , 3 { = { 1 , 2 , 3 , 4 , 5 {

AB = BA

2

A = {b , c , d { B = { n , f , d , c {

AB = { b , c , d {{ n , f , d , c { = {n , f , d , c , b{

BA = { n , f , d , c {{ b , c , d { = {n , f , d , c , b {

AB = BA

12

)5-1( :

A = { { ) B = { { ) C = { { ) AB , BA ) AC , CA )

BC , CB )

A

B

C

.9

.8 8

. . 3 3 . . . .1 . . 2 2

.5

. . 4 4 4 .6

.7

)5 - 1(

3

A = {9, 6 , 3 { B = { 9, 8 , 6 { C = {7, 9 , 3 {

BC = {6 , 8 , 9 {{ 3 , 7 , 9 { = { 3 , 6 , 7, 8 , 9 {

A)BC( = { 3 , 6 , 9 {{ 3 , 6 , 7, 8 , 9 { = {3 , 6 , 7, 8 , 9 {

AB = { 9 , 6 , 3 {{ 9 , 8 , 6 { = {3 , 6 , 8 , 9 {

)AB(C = { 3 , 6 , 8 , 9{{ 7 , 9 , 3 { = {3 , 6 , 8 , 9 , 7 {

: A)BC( = )AB(C

13

4

A = { } B = { { C = { {

BC = { } } { } } =

A) BC ( = { }} {

} } =

AB = { } } { } } =

)A B(C = { }} { } } =

: A)BC( = )A B( C

. ABC )AB( ) BC(

14

)2 - 1(

1 . )6-1( :A = { { )

B = { { ) C = { { )

AB = { { ) AC = { { ) BC = { { )

A

B

C

.9

.3 . . . .1

.5

.2 .7 7

.8

ABC = { { ) )6 - 1(

A = {4 , 3 , 2 , 1 { 2 . B = {7 , 5 , 3 { C = {6 , 4 , 3 {

: ABC )A B( )AC( )A C( )AB(

3 . :

A

B C

A

B C

)CB()CA( )C)AB(

)1-7(

15

Distributive Property 4-1[ [ :

A = {2 , 3 , 4 , 5 { 1

B = { 3 , 5 , 6 { C = { 3 , 4 , 9 {

:BC = { 3 , 5 , 6 , 4 , 9 {

A)BC( = { 2 , 3 , 4 , 5 {{ 3 , 5 , 6 , 4 , 9 { = { 3 , 5 , 4 {.......1

AB = {3 , 5 { AC = { 3 , 4 {

)AB()AC( = { 3 , 4 , 5 {.......2 )1( )2(

A)BC(=)AB()AC (

C = { 7 , 6 , 3{ , B = { 3 , 5 , 6 { , A = { 2 , 3 , 4 { 2

)AB()AC ( = A) BC(

:A) BC ( = {2 , 3 , 4 { { 5 , 6 , 3 , 7 {

= {3{ :

)A B( )AC(AB = { 3 { AC = { 3 {

)AB()AC( = { 3 {{ 3 { = {3{

:A)BC ( = )AB()AC (

16

: :

3

A = {5 , 6 , 7 { :B = { 2 , 5 , 4 { C = { 3 , 4 , 6 {

A)BC(=)AB()AC ( : : :

)A B( )AC( = { 2 , 4 , 5 , 6 ,7 { {3 , 4 , 5 , 6 , 7{= {4 , 5 , 6 , 7 {

:A)BC(

= {5 , 6 , 7{ {4{ = {4 , 5 , 6 , 7{

:A)BC(=)AB()AC (

A = { a , b , c { 4

B = { b , d , e { C = { b , c , e {

A)BC( = )AB()AC( :

: )AB()AC( = {a , b , c , d , e { { a , b , c , e {

= { a , b , c , e {

A)BC( = { a , b , c {{ b , e{ = { a , b , c , e {

A)BC( = )AB()AC (

17

)3 - 1(

1 . A = {2 , 3 ,4 , 5 { B = { 4 , 5 , 8 { C = {3 , 4 , 7 {

2 .

A = {2 ,10 , 12 , 16 { B = { 2 ,4 , 12 , 14 { C = { 6 , 8 , 10 , 12 , 14 {

:)C A( )CB( )

C)AB ( ))C A( )CB( )C)AB ( )

3 . A = { a , b , c , d {

B = { b , c , d { C = {a , c , e {

:)A C( )CB( ))A C( )BC( ))AB( C ))AB( C )

18

Difference Set 5-1[ [ A = { 2 , 5 , 6 , 8 , 7 { B = { 1 , 5 , 6 , 9 {

}8, 7, 2} .B A

.1

.9

.2.5

.6 . . . .7 .8

B

A A - B B , A : } A-B = {2 , 7 ,8 )8 - 1(

}9 , 1} B A B - A A B

B - A = {1 , 9 { :

.B A = A - B .A B = B - A

A-B = {x:x A , x B}B-A ={x:x B , x A}

1

A = 13 , 1. B = 8.

A - B B - A A = {2 , 4 , 6 , 8 , 10 , 12 { :B = {1 , 2 , 3 , 4 , 5 , 6 , 7 {

A - B = { 8 , 10 , 12 { B - A = {1 , 3 , 5 , 7 {

A - B B - A :

19

2

A = {1 , 2 , 3 { , B = { 2 , 3 , 5 { :

A - B = { 1{ B - A = { 5{ A - A =

:Universal Set 6-1[ [

U :1 A = { 3 , 4 , 5 , 6 , 7 , 9 { B = { 2 , 3 , 7 , 8 { C = { 1 , 6 , 7 , 8 , 9 {

A , B , C )9 - 1( :

U = {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 { , A U

A B

C

.4

.5

.3 ..2

.9 .6

.7 .8

.1

, B U C U )9 - 1(

U A B C U N

A N , B N , C N U , N

: :

.U :

20

2

A = { }

B = { { C = { {

U = { }

A , B , C U : )10 - 1(

A

B C

.

.

. .

.

.. . . . . .

. .

U

)10 - 1(

20

21

:Complement Set 7-1[ [A = { 2 4 6 { U = {2 , 4 , 6 , 8 , 10 , 12 , 14 {

A U U - A :

U - A = { 8 , 10 , 12 , 14 { :

A U : A A A = U - A : :

A = { 8 , 10 , 12 , 14 { U

.10 .14

A

.12 .8

.2.4 .6

: A( = A ( A A . )1-11(

U = { 16,0 x : x { A = { 2 , 6 , 8 , 10{ , B = {2 4 6 {

:

A , B , A B , ) A B( , A B U = { 2 , 4 , 6 , 8 , 10 , 12 , 14 { : A = { 4 , 12 , 14 { B = { 8 , 10 , 12 , 14 { A B = { 2 , 6 {

)A B ( = { 4 , 8 , 10 , 12 , 14 { A B = { 4 , 8 , 10 , 12 , 14 { )A B( = A B :

22

)4 - 1(

1 . } A = { 2 , 4 , 6 { , B = { 1 , 2 , 3 { , C = { 3 , 4 , 6 : B - C ) C - A ) C - B ) A - C ) B - A ) A - B )

B = { 2 , 5 , 7 { , A = { 2 , 5 , 8 , 9 { 2 . A B :

B - A ) A - B ) A B ) A B )

3 . :

A B A B

A B A B

A = { 4 , 5 , 7, 8{ , B ={ 5 ,7 , 3{ , C = {9 2 { 4 . .

5 . :U = 30 .

:( 5.

( 3.

23

( 6. ( 8.

6 - U = 10 A = { 1 , 2 , 3 , 5{ B = { 2 , 3 , 5 , 7 {

: A - B , B - A )

B , A ) A B , A B ) )A B( , A B ) )A B( , A B ) ) A( )

24

CHAPTER 2

Relations

]1-2 [ ]2-2 [

]1-2-2 [ ]2-2-2 [ ]3-2-2 [

]4-2-2 [

25

Relations 1-2[ [

Ordered Pair Y X R . Y X

X Y XY .Y X

R = { )a , b ( : a X , b Y {

1

X = { 1 , 2 , 3 {Y = { 1 , 4 , 5 , 6 {R = { )1 , 1( , )1 , 4( , )1 , 5( , )1 , 6( , )2 , 5( , )2 , 6( , )3 , 6( {

RXY : Y X R R )1-2(

RX Y

1.

2.

3.

.1

.4

.5

6

.

.

.

.

.

...

YX

)2-1(

26

}Y = { 2 , 3 , 5 , X = { 2 , 3 , 4 , 6 , 8 { 2 :

X X ( X X ( Y Y (

X X ( Y X ( X (

X Y ( 1 :

( a < b a,b X :R1 = { )2 , 3( , )2 , 4( , )2 , 6( , )2 , 8( , )3 , 4( , )3 , 6( , )3 , 8( , )4 , 6( , )4 , 8( , )6 , 8( {

)2-2( : 4 > 3 8 > 2 6 > 2 4 > 2 3 > 2

3 b :

R2 = { )3 , 2( , )4 , 2( , )4 , 3( , )6 , 2( , )6 , 3( , )6 , 4( , )8 , 2( , )8 , 3( , )8 , 4( , )8 , 6( {

)3-2(

27

2 8

3 6

4

R2

)2-3(

2 < 8 2 < 6 3 < 4 2 < 4 2 < 3 6 < 8 3 < 8 6 > 3 6 > 4 8 > 4

R2 X X X X R2

( = : R3 = { )2 , 2( , )3 , 3( , )5 , 5( {

:2 = 2 3 = 3 5 = 5

Y Y )4-2(

R3 532

)2-4(

( a b | :R4 = { )2 , 2( , )2 , 4( , )2 , 6( , )2 , 8( , )3 , 3( , )3 , 6( , )4 , 4( , )4 , 8( , )8 , 8( {

28

4 4 4 8 8 8 2 2 2 4 6 2 2 8 3 3 3 6

X X )5-2( .

2

8 3

6 4 R4

)2-5(

Y X (

R5 = { )4 , 2( , )6 , 3( { 4 2 6 3

)2-6( X Y .

2.3.4.6.8.

.2

.3

.5

R5X Y

)2-6(

X ( } )R6 = { )6 , 2 6 2

X X

29

X Y ( 1 :

R7 = { )3 , 2( , )3 , 4( , )5 , 6( , )2 , 3( , )5 , 4( { )7-2( :

2.3.5.

.2

.3

.4

.6

.8

R7Y X

)2-7(

:

.

Y = { 2 , 3 , 6 { X = { 2 , 4 , 6 , 8 {

a , b X , Y a , b ( R ( a R b :

X X ( Y Y (

Y X ( Y X ( 2

X Y ( 2 X ( 4

X ( X Y (

)( )( )( )( .

30

)1 - 2(

X = { 1 , 2 , 3 , 4 , 6 , 8 { 1. Y = { 2 , 4 , 6 {

: ( X Y .( X Y .

X ( Y (

X = { 1 , 3 , 5 { 2 . Y = { 1 , 2 , 4 , 6 {

:( X X

X ( Y (

( X = { 0 , 2 , 4 { 3 . Y = { 0 1 , 2 {

R R :( X

( Y ( X , Y

4 . R a 3 b X X = { 1 2 , 3 , 4 , 5 { a , b X

)4 , 2( R R )2 , 1( ( (

)1 , 4( R ( )1 , 3( R ( )4 , 1( R ( )5 , 2( R (

31

]2-2[ Reflexive Property 1-2-2[ [

a R X X )a,a( R X . a X

1

X= { 3 , 4 , 5 {

X R1 = { )3 , 3( , )4 , 4( , )4 , 3( , )5 , 5( {R2 = { )3 , 3( , )4 , 4( , )3 , 4( , )4,3( , )5 , 5( { R1 , R2 X

)2-9( )2-8 (

5 3

4 R1

)2-8(

5 3

4

R2

4

)2-9(

R1 , R2 X

32

2

X= { 1 , 2 , 3 { ( R = { )2 , 1( , )2 , 3( , )2 , 2( , )3 , 3( {

:

( )2-10(

2 3

1 R

)2-10(

R ) 1 , 1( X 1 1

X= { 3 , 4 , 5 { ( R = { )5 , 5( , )4 , 4( , )3 , 3 ({

:

.X R ( )11-2(

5 3

4

R

4

)2-11(

33

3 } X= { 2 , 3 , 5 , 6 :

( XX . X (

. X ( a,b X ( a b

:(

XX = { )2 , 2( , )2 , 3( , )2 , 5( , )2 , 6( , )3 , 2( , )3 , 3( , )3 , 5( , )3 , 6( , )5 , 2( , )5 , 3( , )5 , 5( , )5 , 6( , )6 , 2( , )6 , 3( , )6 , 5( , )6 , 6( { XX 16 = 44

X ( R1 = { )2 , 2( , )3 , 3( , )5 , 5( , )6 , 6( {

6 5 3 2 R1

R1 X : )2 , 2( , )3 , 3( , )5 , 5( , )6 , 6( R1X (

R2 = { )2 , 2( , )3 , 2( , )5 , 2( , )6 , 2( , )3 , 3( , )5 , 3( , )6 , 3( , )5 , 5( , )6 , 5( , )6 , 6( {

:)2 , 2( , )3 , 3( , )5 , 5( , )6 , 6( R2

aX .R2

a,b X ( a b R3 = { )2 , 2( , )3 , 3( , )3 , 6( , )2 , 6( , )5 , 5( , )6 , 6( {

34

2 2 3 3 5 5 6 6)a , a( R3 a X a a

R3 a X )a , a(

4 } X= { 1 , 2 , 3 4 5 R

R = { )1 , 1( , )2 , 2( , )4 , 4( , )5 , 5( , )1 , 2( , )2 , 3( , )3 , 5( {

R : )12-2(

1 2 3 3 4 5 5 R

)2-12(

R 3 . :

)3 , 3( R

: ) a,a ( R aX X R

35

Symmetric Property 2-2-2[ [

a,b X ) b,a( R ) a,b ( R

a b

1

} X= { 1 , 3 , 4 :R = XX ( (

:(

XX = { )1 , 1( , )1 , 3( , )1 , 4( , )3 , 3( , )3 , 1( , )3 , 4( , )4 , 4( , )4 , 1( , )4 , 3( {

1

4 3 3

R

)2-13(

XX X 3 1 1 3 1 4 4 1 3 4 4 3.

R .

36

)14-2( )15-2( )16-2( X= { 1 , 3 , 4 {

)1(R1 = { )1 , 1( , )3 , 3( , )4 , 4( {

1 3 4 R1

)2-14( )2(

R2 = { )4 , 3( , )3 , 4( , )1 , 1( {

3 4 1 R2

)2-15( )3(

R3 = { )1 , 4( , )4 , 1( {

4 1R3

)2-16( )17-2( 1 4

4 1

1 4 3 3 R4

)2-17(

37

2

)18-2(

2 4 6 6 R

)2-18(

:R = { )4 , 4( , )6 , 6( , )6 , 2( {

. ) 2 ,6 ( R 2, 6 ( ( R )18-2( 6 2

2 6

R X :

a ,b X ) b ,a ( R ) a ,b ( R

Transitive Property 3-2-2[ [ a b b c a c

a ,b,c X )a , c( R ) a ,b ( R , ) b ,c( R

b

ca

38

} X= { 1 , 4 , 7 : 1 R1 = { )1 , 4( , )4 , 1( , )1 , 1( , )4 , 4( , )7 , 7( {

)19-2(

1 4 7 7

)2-19( :

)1 ,4( R1 , )4 ,1( R1 ) 1 ,1 ( R1 )4 ,1( R1 , )1 ,4( R1 ) 4 ,4 ( R1

R1

} X= { 2 , 4 , 6 X 2

R1 = { )2 , 6( , )6 , 2( , )2 , 2( , )6 ,6 ( { ( R2 = { )2 , 4( , )4 , 6( { )R3 = { )2 , 4( { (

:

(

)2 , 6( R1 , )6 , 2( R1 )2 , 2( R1

)6 , 2( R1 , )2 , 2( R1 )6 , 2( R1

)6 , 2( R1 , )2 , 6( R1 )6 , 6( R1

)2 , 6( R1 , )6 , 6( R1 )2 , 6( R1

)2 , 2( R1 , )2 , 6( R1 )2 , 6( R1

)6 , 6( R1 , )6 , 2( R1 )6 , 2( R1

R1 X

39

( )2 , 4( R2 , )4 , 6( R2 )2 , 6( R2

)2 , 6( R2 R2

)a,b( R3 )R3( ( :

)1( )a,c( ) b,c ( )2( )b,c( R3

3 )20-2(

3 4 5 5 X

)2-20(

:

R = { )3 , 4( , )4 , 5( , )5 , 5( {

)3 , 5( R R )4 , 3( )4 , 5( R

: R X :

a ,b , c X aRb , bRc aRc )a ,b( R , )b ,c( R ) a ,c ( R

40

4 )21-2( )22-2(

2 4 6 6 2 4 6 6

)2-22( )2-21( : )21-2(

R = { )2 , 2( , )4 , 4( , )6 , 6( {

)2 , 4( R )4 , 6( R )2-22 (

)2 , 6( R

Equivalenc Relation 4-2-2[ [

X X

X= { 1 , 2 , 3 { 1

X XX : XX = { )1 , 1( , )1 , 2( , )1 , 3( , )2 , 1( , )2 , 2( , )2 , 3( , )3 , 1( , )3 , 2( , )3 , 3( {

. )a,a( XX aX XX X

)1 , 2( , )2 , 1()1 , 3( , )3 , 1()2 , 3( , )3 , 2()1 , 1( , )2 , 2( )3 , 3(

41

)2-23(

3

1 2

)2-23(

a b b a X

} X= { 1 , 2 , 3 , 4 R1 2 R1 = { )1 , 2( , )2 , 3( , )1 , 3( , )2 , 4( , )3 , 4( , )1 , 4( {

X R1 :

)1 , 2( R1 , )2 , 3( R1 )1 , 3( R1 )1 , 2( R1 , )2 , 4( R1 )1 , 4( R1 )2 , 3( R1 , )3 , 4( R1 )2 , 4( R1 )1 , 3( R1 , )3 , 4( R1 )1 , 4( R1R2 = { )1,1( , )2,2( , )3,3( , )4,4( {

R2 . .

} R X= { a , b , c X : 3 R = { )a , a( , )c , c( {

R :

b , b( R( X . .

42

)2 - 2(

} X= { 4 , 3 , 2 X 1 . R1 = { )4 , 4( , )2 , 2( , )4 , 2( , )2 , 4( {R2 = { )3 , 4( , )2 , 3( {R3 = { )2 , 3( {

2 . N .

)b a ( a / b ( a < b (a > b ) a+b = 8 (

X= { 2 , 3 , 5 , 7 , 9 , 11 { 3 . R . 5 X R

.

4 . R a , b( R( } X= { 1 , 2 , 3 , 4 R X

a,b X a+b =

} X= { 1 , 3 , 4 , 5 . 5 .

(R1 = { )1 , 1( , )3 , 3( , )4 , 4( , )5 , 5( {

(R2 = { )1 , 1( , )3 , 3( , )4 , 4( , )5 , 5( , )1 , 3( , )3 , 1( {

43

(R3 = { )1 , 1( , )3 , 3( , )4 , 4( , )5 , 5( , )1 , 3( {

(R4 = { )3 , 4( , )3 , 1( , )4 , 3( {

(R5 = { )1 , 3( , )4 , 5( , )5 , 3( {

(R6 = { )4 , 4( {

A = { 1 , 2 , 4 , 6 { 6 . a , b( H( ))a b(( , A H

R = { )x , y( : x+y = 2 , x , y N{ . 7

N R ) N R ) R = { )1 , 1( { ( ( R

} R X= { c , d , e , f X . 8 . ) (

e

d c c

f

44

9 . .( ( (

( } X= { 2 , 4 , 6 : 10 .

R1 = { )2 , 2( , )4 , 2( , )4 , 4( , )4 , 6( {R2 = { )2 , 2( , )4 , 6( , )6 , 4( {R3 = { )2 , 2( {R4 = { )2 , 4( {R5 = { )2 , 2( , )4 , 4( , )6 , 6( {

11 . )10( R2R5 R3R4 )

(.

44

45

CHAPTER 3

Qperations on the Rational Numbers

]1-3 [ Q 2-3[ [

]3-3[ ]4-3[

Q 5-3[ []6-3[

]7-3[ ]8-3[

46

: Q a a,b b0

b

.

]1-3[

79

. 2

15

.

7 2 75 23 35 6 41

= + = + = + 9 15 9 5 15 3 45 45 45

1

:

25

1+ 2(5

+0.3 ( 146

5-6

+ ( 24

14

2(+ (9

16

+

:

2 1 2+1 3

( = = + 4 4 4 4

= = = + 3 9 5+14- 14 5- ( 6 6 6 6 2

47

2 1 22 13 4 3 4 + 3 7

+ = + = + ( = = 9 6 92 63 18 18 18 18

2 3 : . . = 18 .

: 3 = 6 18 2 = 9 18

2 1 26 + 19 12+9 21

: = = = + 9 6 9 6 54 54

21 3 7

= =54 3 18

2 1 2 15 + 21

= + = + 1 ( 5 1 5 15

5+2

=5

7

=5

2 2 3 22 3 4 3

+ = + = + = 0.3 + 5 5 10 52 10 10 10

= 710

(

: :

* .* . . .

.

48

:

a c ad + cb = + b d bd

:

2 7 2 1

14 21 7 7

( + ( + - 3

1 5 - 13 5 3

2 2 2 6 4

( + ( + +

3 3 5 2 1 ( + ( + + 4 3 6 4 5

( + ( + 3.4 ( 3- + 13.1 -

7 2 5 5 4 6 5 8

Q 2-3[ [ :

1(

+ = + 3 1 1 35 7 7 5

1 :

35 35 35 7 5 = + = + 26 5 21 1 3:

= + = + 26 21 5 3 17 5 35 35 35

1 Q

a c c a + = + b d d b

a

b

, c d Q

b 0 , d 0

a c c a + = + a c c a + = + a c c ab d d b

a

b , c

d Q

b 0 , d 0

+ = + + = +

:

) (

49

5 4 2 5 4 2 2(

2 : ) + ( + = + ) + ( 9 9 9 9 9 9

5 4 2 9 2 11 : = + ) ( = + ) + (

9 9 9 9 9 9

5 4 2 5 6 11 = ) ( + = ) + ( +

9 9 9 9 9 9 )2( Q

a c e a c e

) + ( + = + ) + ( b d f b d f b d f b d f b d f b d f

a c e , , , , Q

b d f b d f b d f

.

+ + 5 7- 3 4 6 8

3 :

3 -7 5 18 -28 15

+ ) + ( = + ) + ( 4 6 8 24 24 24

-10 15 5

= + 24 24 24

50

Q

: Z Q

= + 0 = 0 + 4 4 4 5 5 5

3= 3 + 0 = 0 + 13 1 1 2 2 2

a a a = + 0 = 0 +

b b bab

Q

: : 3 3- 5- 5

0 0

:

3-4

34

5 3

5- 3

5 2 6

5 2- 6

-ab

ab

:

51

)1-( :ab

)-1 ( = -ab

:

4

) ( + 3- 3 ( 4 4

1 1 5 + 5- (

2 2 :

0 = = = + 0 3- + 3 3- 3 ( 4 4 4 4

0 = = = + = 5 + 5- 0 11+11- 11 11 - 1 1 ( 2 2 2 2 2 2

) (

]3-3[

: a - b = a + ) - b (

:

1

( -

+ )- ( )

14

45

14

45

52

:

= = + = + = + = - 11 16-5 5- 16 51 - 44 1- 4 1 4 ( 5 4 5 4 54 45 20 20 20 20

4 -1 44 -15 16 -5 16+)-5( 11

= = ) (+ =) (+ = ) (+ ( 5 4 54 45 20 20 20 20

4 1 4 -1

)( + = - 5 4 5 4

.

: a c ,

b d

a c a - c ) ( + = -

b d b d

2 :

- - ( 2- 3 ( 5 3- 4 6 7 5

:

= + = ) ( + = - ( 29 14 15 2 3 2- 3 7 5 7 5 35 35 35

= = ) ( + = ) ( + = - ( 19- 10- + 9- 10- 9- 5- 3- 5 3-4 6 4 6 12 12 12 12

53

)1 - 3(

: :

- + - 7 ( 1 3 ( 7 11 ( 12

3 6 6 8 4

3 8 4 3 7 - ) ( + 5 ) ( - ( 1- 7 ( 3- ( 1- 4-

( 4- 1 7- ( 1 2 ( 6 3 1 + + 5 + - 7 1 + - 3

2 4 5 3 4 3 6 3

: : 3 5

= + 1( 4 7

41 45 8 ( ( (

28 28 11 5

2( : 7

( ( ( 5- 7 7- 5 5 7

3( : 2 1 4 3

( ( ( 3 3 4

3 4 5

54

: cm 40 cm 10

cm 14

: ) 8 (

) 3- (

38

18

27

1 4

12

38

12

54

: m 20 m 6 m 5

1: m 10 m 6 .2

14

15

- 34

:

:

4 , 6.2m2 58

m2

2 58

m2

55

]4-3[

1 2

2 3

.

1 2

3 6

1 2 1 2 2 = =

2 3 2 3 6

1

= 3

:

1 :

2 4 1 2 ( ( 2 3

5 3 5 7 :

( 2 2 1 2 1 = =

5 7 5 7 35

2 4 17 2

10 ( 34 = = 2 3

5 3 51

3 3

56

)1( :

.

a c a c

b d b d

: .

2 1 2

3

4 3

: 1 13 1 13 2 13 2 1 2 = = = = 3

4 3 4 3 2 3 6 6

3 4 3 5

5

: 19 19 5 19 5 19 4 19 = = = = 5 = 3 5

5 5 1 5 1 5 1

Q 5-3[ [

Z N :

57

1( :

:

( 2 3 3 2 =

5 7 7 5

2 3 = 3 2 ( 3 1 1 3 4 5 5 4

Q

:

a c c a = b d d b

a

b , c

d Q

b 0 , d b 0 , d b 0 , d 0

2( :

:

( 1 3 2 1 3 2 ) ( = ) (

5 8 2 5 8 2

( 1 2 1 2 ) 3 1( 7 = 3 ) 1 7(

3 4 3 4 Q

58

:

a c e a c e ) ( = ) (

b d f b d f b d f b d f b d f b d f

a c e Q

b d f b d f b d f

.

4

2 1 3 7 1 3 8 4

: 2 1 3 7 13 8 4

3 = = ) ( = ) ( = ) ( 5 77 11 7 11 7 1 11 7 4 3 8 4 3 2 4 6 4 24 24

Q Z

a Z , 1 a = a 1 = a Q

Q :

a a a = 1 = 1

b b b

a Q

b

59

5

7 ( 7 7 13

1 = 1 = 13 13

5( 5- 5- 13

1 13 13 = 1 =

:

30 = 03 = 0 )1

3 320 2( 0 = 0 =

2

-4 470 3( 0 = 0 =

7

5 2 5

60 4( 0 = 2 0 =

6

=

:

a a a 0 = 0 = 0 Q

b b b

60

: :

5 5-

1 - 1 3 3

:

1 = 2 3 2 3 2 3 2 3

-3 -8 -8 -3 1 =

8 3 3 8 :

b a

a b

a b b a 1 = =

b a a b

a Q

b

61

a c e , , , , Q

b d f b d f b d f

a c e a c a e ) ( + ) ( = ) + (

b d f b d b f b d f b d b f b d f b d b f

6

) + ( 5 2 34 5 6

3 2 5 3 2 3 5 : ) ( + ) ( = ) + (

4 5 6 4 5 4 6

= + = + = + = 37 25 12 5 3 15 6 20 24 10 8 40 40 40

1 7 3 7

4 :

1 1 1 ) 7 ( + ) 3 7 ( = ) + 3 ( 7 = 3 7

4 4 4

7 84 7 91 3 22 = = + = + 21 =

4 4 4 4 4

62

8

3.5 )24( )32(

:

:

+ = 32 3 + 24 3 1 12 2

+ = 32 + 24 7 72 2

84 + 112 = = 196 .

: 3 12

= )32 + 24 ( 72

) 56 ( = = 196 .

62

= 196 .

63

]6-3[ : 1

5

9 10

910

1 5

9

10 1

5

: :

1

2 3 4 3 -

5 8 :

4 3 - 3 2 136- 8 17- 35 17- 5 8

= = = 5 8 5 35 175

: .

1 2 4 2

) ( 3 5 3

1 2 3 1 3 :

) ( = ) ( 3 5 4 3 10

=

13

103

= 109

= 1 19

64

)2 - 3(

1. :

4 27

123

( 4- 7

3 4

( ) 3- 4

( 23

(

1 13

1 12

7 4 ( 32

7 ( 25

8 13

(

3 1 3-( ( )5 1 2- ( )

4-7

5 1-( 7

( )

3 6 ( 5.2- 3.44

)-1 12

3 1 14.3 ( )7

(

2. :

1-2

1 2( 2

1 34

1- ( ) 5

) 32

3- 4

( )

) 32

1 - 4

( 35

2 ( 3

) 12

3 + 4

( )

1- -2

)2 34

+ 1 35

( ( ) 52

1 - 3

( ) 13

+ 24

( )

3.

34 4 ( 1 =

23 9 9 ( = ( 1 =

35 =

53

4 ( 5

7 = 15 (

85

( 8 =

65

13.5 8 3 4

m .4

. 4 12

m

.

5. ) (

1 1 3 .

12

1 2 . 5

.

2( ) 14

6. :

1 2 34

( 9 7 12

(

7 . 35 cm 7.

2 34 20

58

kg 8.

32 9.

( 78

( ) 14

10. )3( )

11. )6( 56.9 40.25 . .

66

12. 12.75 40 .

( ( .

( .

]7-3[

1 - 225

- 225 2 - 225

:225 = 33 55 = )35( )35( = )15( ) 15( = )15(2

15 225 ) (

:)-15-15( = 225 , 1515 = 225

225 15- 15 :

255 15 =

67

2

49 81

- - 2

4981

- :

= 2) (= = = 7 7 7 77 49 81 99 9 9 9

79

49 81

49 7 =

81 9

64100

1

: 64 8 8 8 2 = ) ( = =

100 10 10 10

:

64

100=

64100

=8

10

: :

ab=

ab

, ab

Q .a 0 , b> 0

68

64 . 17

2 1764

100=

176410

1764

: 1764 = 223377

= )237()237(= )42()42(

223377

17648824411474971

2{3{7{

= )42(2

1764100

=4210

= 4.2

171625

3

17 1625

=44125

=3 3 7 7

55=

(3 7)(3 7)55

=215

215

3377

4411474971

3{7{

55

2551

5{

= (215

)2

44125

=215= 4 1

5

69

]8-3[ :

273 = 3 83 = 2 273 = 3 83 = 2

3 ) (

. a3 = a3 ,a z

ab

3 =a3

b3, ab Q , b 0

827

3

1

:

827

3 =83

273=

23

222

8421

2{333

27931

3{

64125

3

2

:

555

1252551

5{222222

6432168421

2{2{

64125

3 = 64

1253 =

643

1253=

45

70

)3 - 3(

1. :

( 49 ( 0.81 ( 1 7

9 225

( 15 ( 11 ( 4 1 1 5

9 25 49

2. :

( 125 ( 7 ( 1- 42 343 8 216

( 3.375- ( 25 ( 81 11- 10

125 64

3 33

333

3. 20.25km2

.

70

11 10125 64

20.25km2 .

71

CHAPTER 4

Polynomials

]1-4[ ]2-4[

]3-4[ ]1-3-4[

]2-3-4[ ]4-4[

]5-4[

72

]1 4 [ :

:)) 3 3(( ..... 32 = 3 . 3

:)) 4 4(( ...... 43 = 4.4.4

3 )32( )2( :

{a.a.a.a...= an

n )a( )n(

n . 57 5 .

a4 a2 1 ( :

{

a2 . a4 = )a.a( . )a.a.a.a({

{= a.a.a.a.a.a :

nm a an am = an+m :

) ( .

53 . 54 . 52 = 5 3+4+2 = 59

8 . 85 . 86 = 8 1+5+6 = 812

73

: : 35 . 23

2( : 37 34

{

37

34=

3.3.3.3.3.3.33.3.3.3

= 3.3.3 = 33

{

) ( :

37

34= 374 = 33

:an

am= anm a0

: ) (.

:

y9

y3 , 5

4

5 , z

-1

z , m

-10

m-11

/y9

y3=y9-3 =y6 , 5

4

5=54-1 =53

z-1

z=z-1-1 =z-2 , m

-10

m -11=m -10-(-11) =m -10+11) =m1 =m

74

Power Rule 3(

: 4)23(

23 4

23( )4 = 23.23.23.23 = 23+3+3+3 = 212 = 234

an( )m = anm :

: ) ()a5 : )3 = a53 = a15

32( )7 = 327 = 314 p3( )2 = p32 = p6

: 4)43(

3 4( )4 = 3 4( ) 3 4( ) 3 4( ) 3 4( ) = 3 3 3 3( ) 4 4 4 4( ) = 34.44

:5 2( )5 = 55.25

ab( )n = an .bn )A( :4( :

32

5

32

5

=32

32

32

32

32

=

3.3.3.3.32.2.2.2.2

=35

25

ab

n

=an

bn, b 0 )B( :

.

75

:

a( 2y( )4 b( 3m n( )3 c(

-3x2y3( )2 d(

2xy3

3

a( 2y( )4 = 24.y4 = 16y4

b( 3m n( )33mn( )3= 3( )3 m3n3 = 27m3n3

c( 3x2y3( )2 = 3( )2 z2( )2 y3( )2 = 9 4 y6

d(

2xy3

=

2x( )3

y3( )3=8x3

y9 , y 0

a0 = 1 , a 0 5( 0:

0 1.a( 2

0 = 1 0 3( )0 = 1b( 2

0 = 1 0 3( )0 = 1c( a b( )

0= 1, a b 0

d( xy

0

= 1, xy 0 x

y

0

= 1, xy 0

Negative Exponeut 6 (

an = 1

an , a 0

)a 23 = 1

23

b( 5( )2 = 15( )2

c(

d( e(

6p5

= 6.p5 = 6p5

21 + 51 = 12+

15=

5 + 210

=710

34

43=

43

34

3m n( )33x2y3( )2 = 3( )2 z2( )2 y3( )2 = 9 4 y6

76

:1(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

33.34 = 33+4 = 37

2(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

45

4 3= 4 53 = 4 2

3(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

52( )2 = 54 = 154

4(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

6.7( )5 = 65.75

5(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

25

4

=24

54=

5 4

2 4

6(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0

3x( )0=1 , x 0

7(

an .am = an+m

an

am= anm

an( )m = anm

ab( )m = am .bm

ab

m

=am

bm , b 0

a0 =1 , a 0

am = 1am , a 0 2( )3 =

12( )3

=18

=18

)4-1(

1/ :a( 1

10

1

b( 23

3

c( 41 +51 d( 5100 2101

e( y3y3

y4 y6 f( 2m

3 4m5( ) 9m7( ) g( 6p+1( )100

6p+1( )101

2/

a( rr5( )4 b( 33 k10( )1 c( m2

n3

4

d( 4x5 z

3

5y4

2

e( 5a1

8b1a11b

4

c5

2

f( z4+m( )2 .zm

z2( )m

77

Polynomial ][Multiplying Binomials 2 4 [ [

: 6 (5+7 ) = (6 5)+ (6 7 )6 (5+7 ) = (6 5)+ (6 7 )

2 y.(x2 + 4x 6) = (2xy.x2 )+ (2xy.4x) (2xy.6)

= 2x3y+ 8x2y12xy

: X) + 5), (X + 2)

x +2

x +5

= )4-1(

(X + 2) (X +5)(X +2) (X + 5) = A , B , C , D

)A( = ........ )B( = ........ )C( = ........

)D(= ........

A B

C D

x

2

5 x

x

2

)4-2(

= .......... + ..............+ ................+ .............= x2 +7x+10

+x = 5( ) x+ 2( ) :

= x2 + 2x+ 5x+10

= x2 +7x+10

) ( :

78

x+5 x+2 x2+5x x)x+5( 2x+10 2)x+5( x2+7x+10 )2x+10( )x2 +5x(

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

)x - 1()x - 2( = x2-2x-x+2 = x2-3x+2

x2 1( ) x+ 4( )x2 1( ) x+ 4( ) 2 x2-1 x+4 x3-x 4x2-4 x3-x+4x2-4 = x3+4x2-x-4

:3

+2x 3( ) x2 5( )2x+ 3( ) x2 5( ) )a

2x2 +1( ) 3x 2( )2x2 +1( ) 3x 2( ) )b3A B( ) 2A+ B( )3A B( ) 2A+ B( ) )c

:a(

)2x +3()x2 - 5( = 2x3-10x+3x2-15 = 2x3+3x2-10x-15

79

b( 2x2+1 3x-2 6x3+3x -4x2-2 6x3+3x-4x2-2 = 6x3-4x2+3x-2

c(

)3A -B()2A + B( = 6A2+3AB-2BA-B2

= 6A2+3AB-2AB-B2

= 6A2+AB-B2

3A-B 2A+B 6A2-2AB 3AB-B2

6A2+AB-B2

80

2x 5x2( ) x2 3( ) 4

/ :

)2x -5x2()x2 -3( = 2x3-6x-5x4+15x2

= -5x4+2x3+15x2-6x

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

/ 2X) = + 4) (X + 3)(2X + 4) (X + 3)

= 2x2 +6x+ 4x+12

= 2x2 +10x+12 cm2

2x 3x2( ) 3x+2x3( )2x 3x2( ) 3x+ 2x3( )

81

)4-2(

1 . :-

(X 2 m)(X + n)) (X 4)(X 2) (

(Xy1)(ny+ 2) (3A1)(2A+1) ( ( (2X + 3y)(3X + 2y) ( (y3 +1)(y 3) (

(2X 3)(25

X +1) n2) ( y2 )(X y) (

2 . :-

(3X + 2)(X +1) = 3X 2 + ......+ 2X + ( .......(X 2 + 4)(X 2 1) = ...... X 2 + ...... 4 (

(X + 2)(X + ....) = X 2 + 5X + ....+10 ( (X + .....)(..... 4) = X 2 .....+ 3X ( ......

-: x 3 .

6x+2

2x+1x+ 3

x+ 34X 1

X+

2

81

82

Squaring a Binomal 3 - 4 [ []1-3-4[

Q a,b (a+b)2

)a+ b(2=)a +b( )a+b( = a2+ab+ba+b2

= a2+ab+ab+b2

= a2+2ab+b2

ab

ab

a2

b2

a

a

b

b

a+b( )2 = a2 + 2ab+b2

+ +

2a+b 1 /

2a+b( )2 = 2a( )2 +2 2a( )bb2a+b( )2= a2 +2ab+b2

= 4a2 + 4ab+b2

3x+ 4y 2 /

= 9x2 + 24xy+16y2

(3X + 4y)2 = (3X)2 + 2(3X).(4y)+ (4y)2

83

3

/

]2-3-4 [ Q a,b )a-b(2 :

13

x+2y

13

x + 2y

2

=13

x

2

+ 2 13

x

. 2y( ) + 2y( )2

=19

x2 + 43

xy+ 4y2

)a- b(2=)a -b( )a-b( = a2-ab-ba+b2

= a2-ab-ab+b2

= a2-2ab+b2

ab( )2 = a2 2ab+b2

.

+ .

2x-5 1

/(2X 5)2 = (2X)2 2(2X)5+ 52

= 4x2 20x+25

84

3x2-2y

2

3x 2y( )2 = 3x( )2 2 3x( ) 2y( )+ 2y( )2 = 9x2 12xy+ 4y2

a+b( )2 = a2 + 2ab+b2

ab( )2 = a2 2ab+b2

.

2)105(

105( )2 = 100+ 5( )2

= 100( )2 + 2100 5+ 52

=10000+1000+25

=11025

2)19(

19( )2 = 201( )2

= 20( )2 2 201+12

= 400 40+1 = 361

5m+4n 1

4a-4b 2

3 )41(2 , )99(2

85

]4 4 [

)b a+b( )a ( . a+b( ) ab( ) = a2 ab+bab2

= a2 ab+abb2

= a2 b2

a+b( ) ab( ) = a2 b2

.

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

/ 2x+ 3( ) . 2x 3( ) = 2x( )2 3( )2 2x+ 3( ) . 2x 3( ) = 2x( )2 3( )2

= 4x2 9 3x+ 4y( ) 3x 4y( ) 2

/ 3x+ 4y( ) 3x 4y( ) = 3x( )2 4y( )2

= 9x2 16y2

2218 )a+b ) ab( ) 3

/ 2218 = 20+ 2( ) 20 2( )

= 20( )2 2( )2 = 400 4 = 396

)a+b ) ab( ) = a2 b2

a( 52

x+ 32

y

52

x 32

y

(a

b( 101 99 (b

86

)3 - 4(

1. :x+7 )

y-3 )4x2-3 )

a2-1 )2m+3n )

mn+mn2 )

x2-y2 1

3(

ax+by )

. y 2 y 3( )m 2 .

)2x+1 . )m, 2x 1( )m 3 .

4 . :

x+ 6( )2 = x2 + ........+ 36 (a)2x 5( )2 = ........ 20x+25 (b)3+ 2x( )2 = 9+12x+ ........ (c)

5 . : x y( )2 x y( )2 (b) x+ y( )2 + x y( )2 (a)

87

6 . :-(mX + ny)(mX ny) ( x 5y( ) x+ 5y( ) (a )

x2 + y2( ) x2 y2( ) (d ) 1x+

1y

1x

1y

(c )

7 . :66 54 (b 10595 (a ) 6654 (b 105 95 (a )

8 . . a+b( )2 = a2 +b2 (b) x y( ) x+ y( ) = x y( )

2 (a )

c2-2cd+d2 cm2 c d( )cm (

]5 4 [ Comman Factors /

12 8 4 )Greatest Comman Factor ( .

3xy2-6x2y2 :

3xy2 = 3 x yy6x2y2= 3 2 x x yy

3xy2

:

88

7x2 , 21y 1 /

: 7x

2 = 7 x x

21y= 7 3 y 7x2 , 21y 7 .

12ab2, 18bc , 24ab3 2

/

24ab3 = 2 3 2 2 a b b b18bc = 2 3 3 b c12ab2 = 2 3 2 a b b

6 6b b

7 x 3( ) , y x 3( ) 3

)x-3( /

:

2xy2 ,4x2 y,x2 y2 (a)a x+ 5( ) ,b x+ 5( ) (b)

Factoring Polynomial /

ax + bx :ax+bx = x a+b( )

x 3x2 : +12xy

3x2 +12xy = 3x x+ 4y( )

89

1 :

bX by = b(X y) ( aX 2 a2by = a(X 2 aby) (

a2 X a2 y+ 2a2Z = a2 (x y+ 2Z ) (

3a - 6b 3a 6b 3 .

) 3a 6b = 3 a 2b( ) . . ( ) ( 3a6b

.

2 :

7 x 4y( )77x 28y

2ab b 4a( )2ab2ab2 8a2b

5x2y y+ 4x( )5x2y5x2y2 +20x3y

)GCF(

3

6ax2 9bx2 +12x2 /

3x2 : ) . . ( 6ax2 9bx2 +12x2 = 3x2 2a 3b+ 4( )

90

46 36+ 4664 4

/

46 46 36, 4664

46 36 + 46 64 = 46(36+ 64) = 46 100( ) = 4600

)GCF( : 1

3Xy+9y ) 7a-5ab )

5b-cb)15x4 y2 10x6 y4 + 20x3y (d)

24 28 248 2 :

Facoring Perfect - Square /

a+b( )2 = a2 + 2ab+b2 (ab)2 = a2 2ab+b2

2ab = a = b =

x2+6x+9 :

x2 = x( ) 9 = 3( ) 6X = 2(X)(3) x2+6x+9 a2+2ab+b2

:x2+6x+9 =) x+3(2

91

9x2-24x+16 1

3x : =

= 42(3X)(4) =

9x2 24x+16 = 3x 4( )2 :

2=

:

= - = +2 9X 2 16 = 2 9x2 16 = 23x 4 = 24x

Trinomial

2

2 x4b2-4bx2c+4c2 .

= 2 X 4b2 4c2 = 2(X

2b)(2c) = 4bx2c

x4b2-4bx2c+4c2=)x2b-2c(2

9x2+12x+16 3

= +2 9X 2 16

= 2 3 X 4 24X

92

x2 10x+ 25 (a)

4x2 +12x+ 9 (b)1 2x+ x2 (c)

9x2 +10x+16 (d)

Factoring Difference of Two Squares / a+b( ) ab( ) = a2 b2

a2 b2 = a+b( ) ab( ) a2-b2 .

a+b( ) ab( ) a2-b2

:

) ( .

. x2-25 1 :

x2 25 = )x 5( - )( = (X +5)(X 5)

. 1-49y2 2

: 1 49y2 = 1 7y( ) 1+ 7y( )

1 7y

93

916

x2 254

y2

3

: 9

16x2 25

4y2 = 3

4x+ 5

2y

34

x 52

y

(X 1)2 4y2 4

: (X 1)2 4y2 = [(X 1) 2y][(X 1)+ 2y]

X4-16 5

:16 X 4 = (4 X 2 )(4+ X 2 )

= (2 X)(2+ X)(4+ X 2 ) : x2+4

)x+y(2 - 36 ) X 2 16

81( 64x2-49 (

94

)4 - 4(

1. :

14

X 2 19

b2 ( 25X2-9y2 ) X2-100 )

16-16y2 ) 1-100m2 ) 4a2-36b2 )

4y2-9x2 ) 1-y4 ) x4-4y2 )

2. : ( 2)398(- 2)400( ( 2)170(- 2)180( ( 2)142(- 2)143(

23) ( 2)45(- 2)155( 2

)2 (32

)2 (

3. 9x2 - ....... + 64 ) x2 + ....... + 49 )

.......... - 20x + 25 )9 x2 + 12x + ...... )

4. . :x2 + 2x + 1 ( x2 -12x + 36 (

9x2 +12xy+4y2 ) x2 - 12x + 6 )

5. :3xy2-6xy+3x )3 a2b2-6ab )

32

X 2 32

y2 X)5 ( y) a(X y) (

95

6. :( 365329+635329 ( 39 49 - 2)49(

7. : (

18xy4 12xy2 + 24y2 12xy2 , 6xy , 6y2x , 12y2x( )

( :.

6a2b3c5 +9a3b2c4 +12a2b5

12a3b4 , 21a3b4c5 , 3a2b2 , 3ab( )

95

96

CHAPTER 5

Open sentences

Inequalities )1-5[ )[ properties of Inqulities 2-5[ [

] 3-5[

97

Inequalities )1-5[ )[

)2-5( )5-1(

2 4 3 11 < 3 4 > 2

3 1 0 < 1-3 2 4 0 < 2-4 )(

ax +b > 0 : x 5x - 6 > 0 x 2 0 < 6 - )2(5

2 .

1 0 < 6 - )1(5 1 .

ax > a

ax a

ax < a

a

x a

98

properties of Inequlities 2-5[ [

Addition property 1 - a+c>b+c a > b a,b,cQ :

: . 3+5 5 2+3 2 < 5

Subtraction Property 2 - a-c>b-c a > b a,b,cQ :

: . 7 1 < 4 3 > 43 4 < 7

Multiplication Property 3 - ac > bc c > 0 a > b a,b,cQ : *

: ac < bc c < 0 a > b a,b,cQ : *

:

: 5- < 3 )7( )7-(

-5>3)7( )3()7( )5-(35- < 21

)7-( )3()7-( )5-(35 > 21-

99

Division Property 4 - a,b,cQ :

a b

c > 0 a > b < c c

: 9 6 9 > 6 , c = 3 2 < 3 < 3 3

ac b

:

: 9 6 9 > 6 , c = -3 2- > 3- > -3 -3

]3-5[ : :

ax+b x > 0 ]5-2[

99

]5-2[

100

3x-2

101

:1

x {0,1,2,3,4} x < 3 ( :

)x( x031>132>233>334>4

S = { 0 , 1 , 2 {

-3 -2 -1 0 1 2 3 4

x 5 0,x z ( :

x 5+5 0+5 )5( x 5

S = {... , -2 , -1 , 0 , 1 , 2 , 3 , 4 ,5 {

-2 -1 0 1 2 3 4 5 2x 3 >7,x RQ )

: 2x 3+3 >7 +3 )3(

2x > 10 2x

2>

102

2

x > 5 S={xQ:x> 5 {

x 5 .

102

4x+1 3,x R Q )4X +1+ (1) 3+ (1) )1-(

4x 2

4x4

24

)4-(

x 12

S s = x R : x 12

:

Q )6x 3 4 3x( ) > 3,x R6x 12+9x > 315x 12 > 315x 12+12 > 3+1215x >1515x15

>1515

x >1

S

S = x R : x >1{ }

x35 x

21,x R Q )

: ) ( 6 : 2X 30 3X 6

2X 30+30 3X 6+30 )30( 2X 3X +24

2X + (3X) 3X + (3X)+ 24 )3x-( X 24 )1-( X 24

S = {X R : X 24}

Q

Q

Q

103

)3( )80( 2

. x = :

)3( 13

3x 3 < 803x 3+3 < 8+33x < 83

3x( ) 13

< 83( ) 1

3

x < 833

x < 27 23

27

3x 3 < 803x 3+3 < 8+33x < 83

3x( ) 13

< 83( ) 1

3

x < 833

x < 27 23

= 27

)50000( 3

)2000( )1000000( .

X = : 2000X =

2000X + 50000 = 2000X +50000 1000000 2000X + 50000+ (50000) 1000000+ (50000) 2000X 950000 (2000X) 1

2000 (950000) 1

2000

X 475

475

80+3

104

)1 - 5(

1. :

2x+ 320

A)X+3,Y(

X>-3Y>0 )7-19(

)3 - 7(

1. )A)X,Y(, B)X1,Y1 :

. Y= X 1. A . X= Y 2. A

. Y ....... X ....... A 3. . Y ....... X ....... A 4. . Y ....... X ....... A 5.

6. )A)X,0 A . 7. )A)0,Y A .

8. A .

9. A .

10. A .

11. A . . X1>X2 A)X2,5( A)X1,5( 12.

13. )A)C,Y1 )B)C,Y2 Y1 Y2 C . . Y1 Y2 B)X1,Y2( A)X1,Y1( 14.

15. )A)5,10 )B)6,10 .

149

2. )( )( :

1. )A)X1,Y1 )B)X1,Y2 . 2. )A)5,7 )B)5,8 )C)9,4 .

3. )A)5,8 )B)6,8 . 4. )A)5,5 )B)5,7 .

5. )A)7,8 . 6. )A)-7,-8 .

7. )A)5,7 )B)10,14 )C)-5,7 . 8. )A)5,0 )B)5,5 )C)0,5 )O)0,0 .

. Y=0 X A)5,Y( 9. . X=0 Y A)X,Y( 10.

. Y=5 Y A)X+4,Y-5( 11. . Y>5 A)X+4,Y-5( 12.

. X

150

]4-7[

)P2)X2,Y2 )P1)X1,Y1 )20-7( M P1P2 = P1P2 M P1P2 P1,P2

P1)X1,Y1(

P2)X2,Y2(

)7-20(

]1-4-7[ :

) Y V ) X ( )21-7( H ) H A,B,C

V . D,E,F

X

Y

H

V

ABCE

D

F

)7-21(

X

Y

151

:

X Y .

)B)X2,5 )A)X1,5 M AB = | X2-X1|

:

A)X1,5( B)X2,5(

C)6,Y1(

)0,5(

D)6,Y2(

)7-22(

:

Y )H)D,Y2 )C)H,Y1 MCD= | Y2-Y1|

X )B)X2,K )A)X1,K

MAB= | X2-X1 |

:

B,A A,B M AB = M BA

X

Y

152

M AB = M BA |X1-X2| = |X2-X1|

X1,X2Q |Y1-Y2| = |Y2-Y1|

Y1,Y2Q

)B)-57 )A)57 A,B X :

M AB = |X2 - X1| = |-5-5| = 10

X

A)57(B)-57(

)7-23(

Y )O)0,0 )B)3,0 )C)0,4 :

MOB = |3-0 | = 3MOC = | 4 - 0 | = 4

B)3,0(

C)0,4(

)0,0( X )7-24(

BC : OBC O BC = 3( )2 + 4( )2 = 9+16

= 25 = 5 =

4+ 3+5 =12

153

A)1,1( B)10,1( C)1,10(

X AD X

1

B)-8,5+X( A)5,6(Y :

AD X

A,B X

A)56(D)-8,5+X(

)7-25( . X=1 5 +X=6

B)8,X-1( X AB X

2

Y A)7,-3( :

X X AB

B)8,X-1(

A)7,-3( X-1-3

)7-26( X-2

153

XX-- ) )77 ) )7 ) ) ) ) - ) ) 2626 ) ) 26 ) ) ) ) XX

154

)4 - 7(

. C)5,8( B)7,6( A)5,6(