Chapter6ModuleAlgebraicExpressionsIII.ppt

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02/20/16Copyright Reserved

1

UNGKAPAN ALGEBRA III..

UNGKAPAN ALGEBRA I

UNGKAPAN ALGEBRA II


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Permudahkan ungkapan

algebra yang berikut


2

a.) 2g 5a + 3g

= 2g + 3g 5a

= 5g 5a

Sebutan tak

serupaSebutan serupa


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3

b.) w + 2u 5w 3u

= w 5w + 2u 3u

= - 4w u

c.) 5h + 1 + h +

= 5h + h + 1 +

= !h + 1"


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4

CUBA

1.)#$ 3% 2$ 2%

= #$ 2$ 3% 2%

= 5$ 5%

2.) 3a 2b + 5a !b

= 3a + 5a 2b !b

= &a &b

YEH..BERJAYA


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5

a.) 2 ' - 3b )

= 2 $ -3b

= - ! b

b.) 2a ' - 4b )

= - 2a $ -4b

= & ab

c.) 3 a ' 3 b )

= 3a $ 3b

= ab

d.) 4 c ' 3 c )

= - 4 c $ 3c

= - 12 c(


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6

1.) 5 ' - 4 w )

= - 2" w

2.) &b ' 3c )

= 24 bc

YEH..BERJAYA

Try u best

3.) 4w ' - 3 w )

= - 12 w

3.) - 5a ' - 2 a )

= 1" a(


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7

Memahami dan menggunakan konsep kembanganMemahami dan menggunakan konsep kembangan

BAB 6

Ungkapan Algebra III

Memahami dan menggunakan konsep pemfaktoranMemahami dan menggunakan konsep pemfaktoranungkapan algebra untuk menyelesaikan masalah.ungkapan algebra untuk menyelesaikan masalah.

Melakukan penambahan dan penolakan ke atasMelakukan penambahan dan penolakan ke ataspecahan algebra.pecahan algebra.

Melakukan pendaraban dan pembahagian ke atasMelakukan pendaraban dan pembahagian ke ataspecahan algebra.pecahan algebra.

6.1

6.2

6.3

6.4

(3x+2)(5x1)

7y-

143x15

3

X2+12x+8

(8x+3y)-

(x+5y)

x2 y2

x + y 2 5y

3 !


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8

Kembangan Ungkapan Algebra6.1Kembangan ialah pendaraban

a.) satu ungkapan algebra dalam tanda kurungdengan satu sebutan algebra atau satu nombor

a) !" # !" $ %y)

& '"( *("y

Mendarab setiap sebutandalam tanda kurung dengannombor atau sebutan di luar

tanda kurung.

Contoh +

Kembangkan setiap yang berikut.

a) ! # !" $ %y)

& '" *(y


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Contoh :


9

a) !" # !" $ %y)

3x

3x - y

3x ! 3x

" #x$

# x$ - %&xy

3x ! - y

" - %&xy

a) !" # !" $ %y) & '", *("y


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10

*,, 1 / 2 ' 3 5 )


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11

*,, 2 / d ' e + 4d )


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12

*,, 3 / - 4m ' 2 m )


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13

*,, 4 / - 3p ' p 50 )


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14

*,, 5 /- 5 ' 2 y 3 )


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15

*,, ! / 2e ' 3e + 2g )


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16

*,, /


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17

Kembangan Ungkapan Algebra6.1b.) dua ungkapan algebra dalam tanda

kurung.

b)#" $ y) #!" -y)

& !"( -"y !"y -y(

ebutan serupa

& !"( ("y $ -y(

Mendarab setiapsebutan dalam tanda

kurung pertama

dengan tiaptiapsebutan dalam tanda

kurung kedua.

Contoh ( +

Kembangkan setiap yang berikut.

& " # !" -y )$y # !" -y )


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Contoh 3 :


18

x

3x ' (y

3 x$ '(xy

#" $ y) #!" -y)

- y - 3 xy - (y$

& !", -"y $ !"y $ -y,

& !", ("y $ -y,

#" $ y) #!" -y)


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19

/A0A1 * + # " ( ) # " ! )


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20

/A0A1 ( + # " $ ! ) # " $ % )


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21

/A0A1 ! + # (" $ - ) # !" ( )


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22

/A0A1 % + # !" $ % ) # (" $ ! )


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23

/A0A1 - + # 2 $ e ) # ( e )


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/A0A1 6 + # -h $ !k ) # (h $ k )


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25

/A0A1 3 + # (" $ -y ) # (" !y )


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26

/A0A1 2 + # " - ), &


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27

/A0A1 ' + # (" $ ! ), &


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28

/A0A1 *4 + # !" $ y ), &


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29

/A0A1 +


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)enye*esa+kan )asa*ah yan,

)e*+batkan uas%. Raah /+ ba0ah 1enunukkan pe*an ruan,

ta1u ru1ah En2+k )an+a1.

H+tun, *uas /a*a1 1$ ruan, ta1u ru1ah

En2+k )an+a102/20/16

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' 2a b) m

' a 3b) m


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)enye*esa+kan )asa*ah yan,

)e*+batkan uas&. Sebuah pa/an, seko*ah berbentuk se,+

e1pat tepat yan, berukuran 4 y 5 6 7 1

panan, /an 4 &y ' 3 7 1 *ebar. H+tun, *uaspa/an, +tu.


31

' 2y + 3) m

' y !) m

E ! B k #


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32

Expan!ng Bra"ke#$6.1

5"ample 3+ind the product of #a b)(.

#a b)(

& #a b) #a b)

& a(

ab ba b(

& a( (ab b(

7he e"pansion of t8o similar linear algebraice"pressions can be found in the follo8ing methods.

#a b)( means

the s9uare of

#a b)

0ike terms can be simplified

Method (+

9uare the first term a: in the

bracket

Multiply all the terms: (: a and b

s9uare the second term b: in the

bracket

#a b)(

Method *+

& a(

& a( (ab b(

#( " a " b) b(

%&mpare #'e an$(er$

E ! B k #


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33


5"ample 2+ind the product of #a b)(.

#a b)(

& #a b) #a b)

& a(

ab ba b(

& a( (ab b(


#a b)( means

the s9uare of

#a b)


Method (+

9uare the first term a: in the

bracket

Multiply all the terms: (: a and b

s9uare the second term b: in the

bracket

#a b)(

Method *+

& a(

& a( (ab b(

#( " a " b) b(

%&mpare #'e an$(er$

E ! B k #


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34


5"ample '+ind the product of #(r !s)(.

#(r !s)(

& #(r !s) #(r !s)

& %r

(

6rs 6rs 's(

& %r( *(rs 's(


#(r !s)( means

the s9uare of

#(r !s)


Method (+

9uare the first term (r: in the

bracket

Multiply the terms: (: (r and !s

s9uare the second term !s: in

the bracket

#(r !s)(

Method *+

& #(r)(

& %r(*(rs 's(#("(r"!s) #!s)(

%&mpare #'e an$(er$

E ! B k #


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35


5"ample *4+ind the product of #m $ (n)(.

#m $ (n)(

& #m $ (n) #m $ (n)

& m(

(mn (mn %n(

& m($ %mn %n(


#m $ (n)( means

the s9uare of

#m $ (n)


Method (+

9uare the first term m: in the

bracket

Multiply the terms: (: m and (n

s9uare the second term (n: in

the bracket

#m $ (n)(

Method *+

& m(

& m( %mn %n(#(xmx(n)#(n)(

%&mpare #'e an$(er$


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36

7hree ;dentities 7hat7hree ;dentities 7hat

Must Be RememberedMust Be Remembered #a b)( & #a b) #a b) & a( (ab b(

#a b)( & #a b) #a b) & a( (ab b(

a(

b(

& # a b ) # a $ b )

t i ti f Al b i 5 i


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37

actorisation of Algebraic 5"pressions6.2

actors of a term is the number or terms that candivide a term e"actly.

A$ an example 3p*

!p9 & * " !p9

& ! " p9

& p " !9

& 9 " !p

7o find the factors of a term:state the term as a product of

t8o different terms first.

7hus: thefactors of !p9 &

*: !: p: 9: !p:

!9: p9: and !p9

t i ti f Al b i 5 i6 2


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38



A$ an example5"pand+

!#!a b)

& 'a !b

actorising is

the reverse ofe"panding.

actorise+

& 'a !b

& !#!a b)

'a !b!#!a b)

'a and !b can be

divided by !. o: !

is a common factor

of'a !b

1umber ! must be

multiplied to each term

in the bracket:

5"pand

actorise



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39



Example 1

#a) %ab($ 6ac 2a

& (#(ab($ !ac %a)

& (a#(b( !c %)

7aking out the

common factor (

7aking out the

common

factors of eachterm.

7aking out the

common factor a



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40



Example 2 +a"#&r!$e &, #'e ,&ll&(!ng.#a) !"y( *("(y

& !#"y( %"(y)

& !"#y( %"y)

& !"y#y %")7aking out thecommon factor !

-ak!ng #'e

"&mm&n ,a"#&r$ &,

ea"' #erm.

7aking out the

common factor y

7aking out the

common factor "

actorisation of Algebraic 5 pressions6 2


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41

actorisation of Algebraic 5"pressionsactors of a term is the number or terms thatcan divide a term e"actly.

actorisation /f 7he


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42

actorisation of Algebraic 5"pressionsactors of a term is the number or terms that candivide a term e"actly.actorisation /f 7he


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43

actorisation of Algebraic 5"pressionsactorisation /f 7he


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44



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45



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actorisation /f 7he


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actorisation /f 7he


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actorisation /f 7he


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actorisation /f 7he


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actorisation /f 7he


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actorisation /f 7he


52/79



actorisation of algebraic e"pressions 8hichconsist of three terms.

"( ("y y(

& #" y) #" y)

& #" y)(

Example 1 +a"#&r!$e &, #'e ,&ll&(!ng.

7he identities that must be remembered.

"( ("y y(

& #" y) #" y)

& #" y)(

a) "( *4" (-

& "( (#-") -(

& #" - )(

b) "( *%" %'

& "( (#3") 3(

& #" 3)(

actorisation of Algebraic 5"pressions6 2


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actorisation of algebraic e"pressions 8hichconsist of three terms.

"( ("y y(

& #" y) #" y)

& #" y)(


7he identities that must be remembered.

"( ("y y(

& #" y) #" y)

& #" y)(

a) 'p( 6p9 9(

& #!p)( (#!p9) 9(

& #!p 9)(

b) "( %"y %y(

& "( (#("y) #(y)(

& #" $ (y)(



54/79



actorisation of algebraic e"pressions 8hichconsist of four terms.


a" ay b" by

& #a" ay) #b" by)

& a#" y) b#" y)

& #a b) #" y)

roup the term

8ith the same

common factor

7ake out the

common factor

8ithin the group

7aking out the

common factor

#" y)



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%m !n mn *(

& #%m mn) #!n *()

& m#% n) !#n %)

& m#% $ n) $ !#n %)& #% n) #m !)

roup the term

8ith the same

common factor

7ake out the

common factor

8ithin the group

7aking out the

common factor

#% $ n)



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p9 9r ps rs

& #p9 9r) #ps rs)

& 9#p r) s#p r)

& #p r) #9 s)

roup the term

8ith the same

common factor

7ake out the

common factor

8ithin the group

7aking out the

common factor

#p r)



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2ab $ *4a *(b *-

& #2ab $ *4a) #*(b *-)

& (a#%b -) !#%b -)

& #%b -) #(a !)

roup the term

8ith the same

common factor

7ake out the

common factor

8ithin the group

7aking out the

common factor

#%b $ -)



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Example +a"#&r!$e &, #'e ,&ll&(!ng.

!6" $ %y $ !"y (3"(

& #!6" $ %y) #!"y $ (3"

(

)& %#'" y) $ !"#y $ '")

& %#'" $ y) !"#y '")

& #% !")#'" $ y)

roup the term

8ith the same

common factor

7ake out the

common factor

8ithin the group

7ake out the

common factor

#'" $ y)

actorisation /f Algebraic 5"pressions6.2


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actorisation /f Algebraic 5"pressionsactorisation of algebraic e"pressions 8hichconsist of four terms.


"(

3" *(

& "( %" !" *(

& #"( %") #!" *()

& "#" %) !#" %)

& #" !)#" %)

%'ange #'e

expre$$!&n !n#&

,&r #erm$

roup the term 8ith

the same common

factor

7ake out the

common factor

#" %)

7ips+

% ! & 3

% " ! & *(



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"(

*4" !'

& "( *!" !" !'

& #"( *!") #!" !')

& "#" *!) !#" *!)

& #" !)#" *!)

%'ange #'e

expre$$!&n !n#&

,&r #erm$

roup the term 8ith

the same common

factor

7ake out the

common factor

#" *!)

7ips+

*! ! & *4

*!"#!) & !'



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a(

$ -a 6

& a( *a $ 6a 6

& #a( a) #6a 6)

& a#a *) 6#a *)

& #a 6)#a *)

%'ange #'e

expre$$!&n !n#&

,&r #erm$

roup the term 8ith

the same common

factor

7ake out the

common factor

#a *)

7ips+

* 6 & -

*"#6) & 6



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m($ **m (%

& m($ 2m $ !m (%

& #m( 2m) #!m (%)

& m#m 2) !#m 2)

& #m !)#m 2)

%'ange #'e

expre$$!&n !n#&

,&r #erm$

roup the term 8ith

the same common

factor

7ake out the

common factor

#m 2)

7ips+

2 ! & **

#2)"#!) & (%



63/79




("( '" %

& ("( 2" *" %

& #("( 2") #*" %)

& ("#" %) *#" %)

& #(" *)#" %)

%'ange #'e

expre$$!&n !n#&

,&r #erm$

roup the term 8ith

the same common

factor

7ake out the

common factor

#" %)

7ips+

2 * & '

2 " * & 2("% & 2

actorising And implifying Algebraic ractions6.2


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Algebraic fractionsare fractions 8ith algebraice"pressions as the numeratoror denominator.

%an /& g!8e $&me example$0

"(

y

%a($ b(

a b

nmera#&r

en&m!na#&r

nmera#&r

en&m!na#&r

!" $ (

y

nmera#&r

en&m!na#&r

y($ %y %

y (

nmera#&r

en&m!na#&r

"($ '

#"$!)#"*)

nmera#&r

en&m!na#&r

ood ob



65/79



Example 19!mpl!,/ #'e g!8en ,ra"#!&n.

'ab(

!b

& ' " a " b " b

! " b

& ! " a " b

*

& !ab

*

! *

*

olution+

6ngratulati6n78



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"($ '

" !

& #" $ !) #" !)

" !

& " !

*

& " !

*

olution+

*

actorise the

numerator first.

9ell

d6ne8



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2 $ !(m

( 2m

& 2 #* %m)

( #* %m)

& %#* $ %m)

* %m

& %

olution+%

actorise the

numerator and the

denominator first.

*

:66d ;6b8

*

*



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& :::!p::::

!#(p $ !9)

& :::p::::

(p $ !9

olution+*

actorise the

denominator first.

*!p

6p $ '9

6ngratulati6n78

actorising And implifying Algebraic ractions

6.2


69/79




& #" !) #" ()

#" !) #" !)

& " (

" !

olution+*

actorise the

numerator and the

denominator first.

*"( -" 6

#" !)(

Bra8o9

Addition And ubtraction /f Algebraic ractions6.3


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a) Adding or subtracting t8o algebraicfractions 8ith similar denominators.

Example 1

& " %"

!

& -"

!

olution+

" %"

! !

Maintain the

denominator

Add the

numerator:66d8



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a) Adding or subtracting t8o algebraic fractions 8ithsimilar denominators.

Example 2& (" y #%"!y) (

& (" %" y $ !y

(

& 6" $ (y

(

& ( #!" $ y) (

& !" y

olution+

(" y %" $ !y

( (

Maintain the

denominator

Add or

subtract the

numerators

implify

*

*

e** /one9



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a) Adding or subtracting t8o algebraic fractions 8ithsimilar denominators.

Example 3& !" - #'" 3) " (

& !" '" - 3

" (

& 6" (

" (

& ( 6" " (

olution+

!" - '" $ 3

" ( " (

Maintain the

denominator

Add or

subtract the

numerators

7he positiveterm is al8ays

8ritten in front



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b) Adding or subtracting t8o algebraic fractions 8ithdifferent denominators.

Example 1& -a " ( $ !a 3 " ( *%

& *4a $ !a

*%

& 3a

*%

& a (

olution+

-a !a

3 *%

0et the

denominators

same.

ubtract the

numerators

implifye** ;one9



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m n %m $ -n

m n

& n#m n) m#%m $ -n)

m"n n"m

& mn n( %m( -mn

mn& %m($ %mn n(

mn

& #(m $ n) #(m $ n)

mn

& #(m $ n)(

mn

b) Adding or subtracting t8o algebraic fractions 8ithdifferent denominators.

Example 2 olution+

Multiply the

numerator and

the

denominator

implify

Add or

subtract the

numerators


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! # %a $ -b)

& ! "%a $ ! " -b

& *(a $ *-b

Multiplying algebraic fractions.#a)


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D($ y( " "

(" " y

& #" $ y)#" y) " "

(" #"y)& " $ y

(

Multiplying algebraic fractions.#b)


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r E 3

s (s

& r " (s

s 3

& (r

3


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!y( E y

(" %"

& !y( "%"

(" y& 6y & 6y

*


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7hank7hankFouFou

If at first you don't succeed ... so much for skydiving.

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