03. algebraic expressions factorization

8/12/2019 03. Algebraic Expressions Factorization

1/39

ehios`luehrp`ysobs.oi

2B@HT]NZ

AKNJZ OB NPTZNRROCIRG B]CZOV ]OCI

Bcisthit 6H syejca `hvoik h goxnm iuenrobha

vhaun os bhaanm h bcisthit.

Whrohjan 6 H syejca wòb` thlns vhrocus iuenrobha

vhauns os bhaanm h vhrohjhan.

Nx.8 [n licw t`ht t`n pnroentnr T cg h squhrn cg somn

s os kovni jy T 5 = s. @nrn, = os h bcisthit him

T him s hrn vhrohjans.

Nx.: ]`n pnroentnr T cg h rnbthikan cg somns a him j

os kovni jy T 5 : (a+ j). @nrn, : os h bcisthit

him a him j hrn vhrohjans.

Haknjrhob Nxprnssocis 6 H bcejoihtoci cg

bcisthits him vhrohjans bciinbtnm jy t`n sokis cg

guimhenitha cpnrhtoci cg hmmotoci, sujtrhbtoci,

euatopaobhtoci him movosoci os bhaanm hi haknjrhob

nxprnssoci.

]nres 6 Whrocus phrts cg hi haknjrhob nxprnssociwòb` hrn snphrhtnm jy t`n sokis cg + cr hrn bhaanm

t`n tnres cg t`n nxprnssoci.

Nx.2 :x: 2xy + >y: os hi haknjrhob nxprnssoci

bcisostoik cg t`rnn tnres, ihenay, :x:,2xy him

>y:.

Nx.= ]`n nxprnssoci :x2 2x: + =x ; os hi haknjrhob

nxprnssoci bcisostoik cg gcur tnres, ihenay,

:x2, 2x: , =x him ;.

BCI]NI]R

Bcisthit & Whrohjan

Haknjrhob Nxprnssocis

Eciceoha & Joiceoha

Omnitotons

Ghbtcrozhtoci

Tcayiceohas

Eciceoha 6 Hi haknjrhob nxprnssoci bcithoioik

ciay cin tnre os bhaanm h eciceoha.

Nx.> >,2y,;xy,2

:x:yz,

2

>h:jb2 ntb. hrn haa eciceohas.

Joiceoha 6 Hi haknjrhob nxprnssoci bcithoioik twc

tnres os bhaanm h joiceoha.

Nx.7 ]`n nxprnssoci :x 2, 2x + :y, xyz > ntb. hrn haajoiceohas.

]roiceoha 6 Hi haknjrhob nxprnssoci bcithoioik

t`rnn tnres os bhaanm h troiceoha.

Nx.; ]`n nxprnssocis h j + :x: + y: xy,

x2 :y2 2x:y:z ntb. hrn troiceoha.

Ghbtcrs 6 Nhb` tnres oi hi haknjrhob nxprnssoci os

h prcmubt cg cin cr ecrn iuejnrs (s) him /cr aotnrha

(s). ]`nsn iuejnr(s) him aotnha(s) hrn licwi hs t`n

ghbtcrs cg t`ht tnres.

H bcisthit ghbtcr os bhaanm h iuenrobha ghbtcr, wòan

h vhrohjan ghbtcr os licwi hs h aotnrha ghbtcr.

Bcnggobonit 6 Oi h tnre cg hi haknjrhob nxprnssoci

hiy cg t`n ghbtcrs wot` t`n soki cg t`n tnre os bhaanm

t`n bcnggobonit cg t`n ct`nr ghbtcrs.

Nx.0 Oi >xy, t`n bcnggobonit cg x os >y< t`n bcnggobonit

cg y os >x him t`n bcnggobonit cg xy os >.

Nx.? Oi x, t`n bcnggobonit cg x os 8.

Bcisthit ]nre 6 H tnre cg t`n nxprnssoci `hvoik

ic aotnrha ghbtcr os bhaanm h bcisthit tnre.

Nx.88 Oi t`n haknjrhob nxprnssoci x: xy + yz =, t`n

bcisthit tnre os =.

Aoln him îaoln ]nres 6 ]`n tnres `hvoik t`n

shen aotnrha ghbtcrs hrn bhaanm aoln cr soeoahr tnres,

ct`nrwosn t`ny hrn bhaanm uiaoln tnres.

Nx.8: Oi t n haknjrhob nxsprnssoci :h:j + 2hj: ;hj =jh:,

wn `hvn : h:j him =jh: hs aoln tnres, w`nrnhs

2hj: him ;hj hrn uiaoln tnres.


2/39


NPHETANRNx.82 Hm m 6 ;x: =x + >, 2x: + : x 8 h i m

>x: x + ?.

Rca. [n `hvn,

Znquornm sue

5 (;x: =x +>) + (2x: + :x8)

+ (>x: x+?)

5 ;x: 2x: + >x: =x + :x x + > 8+?

\Bcaanbtoik aoln tnresU

5 (; 2 + >)x: + (= + : 8)x +(> 8+ ?)

\Hmmoik aoln tnresU

5 ?x: 2x + 82

Nx.8= Hmm 6 >x: 2

8x +

:

>,

:

8x: +

:

8x

2

8him

:x: +

>

8x

7

8.

Rca. Znquornm sue

5

:

>x

2

8x> : +

2

8x

:

8x

:

8 :

+

7

8x

>

8x: :

5 >x: :

8x: :x:

2

8x +

:

8x +

>

8x +

:

>

2

8

7

8\Bcaanbtoik aoln tnresU

5

::

8> x: +

>

8

:

8

2

8 x +

7

8

2

8

:

>

\Hmmoik aoln tnreU

5

:

=884x: +

24

78>84x +

7

8:8>

5:

>x: +

24

88x + :

Euatopaobhtoci cg Haknjrhob Nxprnssocis

(o) ]`n prcmubt cg twc ghbtcrs wot` aoln sokis ospcsotovn him t`n prcmubt cg twc ghbtcrs wot`

uiaoln sokis os inkhtovn

o.n., (h) (+) (+) 5 +

(j) (+) () 5

(b) () (+) 5

him, (m) ( ) () 5 +

(oo) Og h os hiy vhrohjan him e, i hrn pcsotovn

oitnknrs, t`ni

he hi 5 he+i

Gcr nxhepan , h2 h> 5 h2+> 5 h0,

y= y 5 y=+8 5 y> ntb.

Nx.8> Goim t`n prcmubt cg t`n gcaacwoik phors cg

pcayiceohas 6

(o) =, ;x (oo) =h, ;h

(ooo) =x,;xy (ov) =x2, 2xy

(v) =x, 4

Rca. [n `hvn,(o) = ;x 5 (= ;) x 5 :0 x 5 :0 x

(oo) (=h) (;h) 5 (= ;) (hh) 5 :0h:

(ooo) (=x) (;xy) 5 (= ;) (x xy) 5 :0x 8+8y

5 :0x:y

(ov) (=x2) (2xy) 5 (=2) (x2 xy)

5 8: (x2+8y) 5 8:x=y

(v) =x 4 5 (= 4) x 5 4 x 5 4

Nx.87 Goim t`n hrnhs cg rnbthikans wot` t`n gcaacwoik

phors cg eciceohas hs t`nor anikt` him jrnhmt`

rnspnbtovnay 6

(o) (x, y) (oo) (84x, >y)

(ooo) (:x:, >y:) (ov) (=h, 2h:)

(v) (2ei, =ip)

Rca. [n licw t`ht t`n hrnh cg h rnbthikan os t`n

prcmubt cg ots anikt` him jrnhmt`.

Anikt` Jrnhmt` Anikt` Jrnhmt` 5 Hrnh

(o) x y x y 5 xy

(oo) 84 x >y 84x >y 5 >4xy

(ooo) :x: >y: :x: >y: 5 (: >)

(x: y:) 5 8 4x:y:

(ov) =h 2h: =h 2h: 5 (= 2)

(h h:)

5 8: h2

(v) 2ei =ip 2ei =ip 5 (2 =)

(e i i p)

5 8: ei:p

Nx.8; Euatopay 6

(o) 2hj:b2 jy >h2j:b

(oo) =x:yz jy :

2x:yz:

(ooo) >

0x:yz2 jy

=

2xy:z

(ov)8=

2x:y jy yx

:

; =

(v) :.8h:jb jy =hj:

Rca. (o) [n `hvn,

(2hj:b2) (>h2j:b)

5 (2 >) (h h2 j: j: b2 b)

5 8>h8+2j:+:b2+8

5 8>h=j=b=

(oo) [n `hvn,

(=x:yz)

::yzx:

2


3/39


5

:

2= (x: x: y y z z:)

5 7x:+:y8+8z8+: 5 7x=y:z2

(ooo) [n `hvn,

2:yzx

>

0

zxy

=

2

:

5

=

2

>

0 (x: x y y: z2 z)

5 82:88:

zyx>

7 5

=22zyx

>

7

(ov) [n `hvn,

yx

8=

2 :

yx

:

; =

5

:

;

8=

2 (x: x= y y)

5=

2x:+=y8+8 5

=

2x7y:

(v) [n `hvn,

(:.8h:jb) (=hj:)

5 (:.8 =) (h: h j j: b)

5 0.=h:+8j8+:b 5 0.=h2j2b

Nx.80 Euatopay 6

(o) 7h:jb, :h:j him =

8

(oo)?

=h>j:, 84h2j him 7

(ooo) 2.8>x him :2x:y

(ov) x, x:yz him ;

2xyz:

Rca. (o) [n `hvn,

( 7h:jb) (:h:j)

=

8

5

=

8

:7 (h: h: j j b)

5 2h:+:j8+8b 5 2h=j:b

(oo) [n `hvn,

:>jh?

= (84h2j) (7)

5

784

?

= (h> h2 j: j)

52

04h>+2j:+8 5

2

04h0j2

(ooo) [n `hvn,

(2) (8>x) (:2x:y)

5 (2 8> :2) (x x: y)

5 842>x8+:y 5 842>x2y.

(ov) [n `hvn,

(x) (x:yz)

:xyz;

2

5

;

28 (x x: x y y z z:)

5;

2x8+:+8y8+8z8+: 5

;

2x=y:z2

Nx.8? Euatopay nhb` cg t`n gcaacwoik eiceohas 6

(o) 2xyz, >x, 4 (oo)

>

7hj,

7

>jb,

?

8:hjb

(ooo)=

2x:yz:, 4.>xy:z:, 8.87x:yz2, :xyz

(vo) :4x84y:4z24, (84xyz):

(v) (2x:y), (=xy:z), (xy:z:) him

z

>

=

Rca. (o) [n `hvn,

(2xyz) (>x) 4

5 (2>4) (xxyz)

5 4 x:yz 5 4

(oo) [n `hvn,

hj

>7

jb

7>

hjb

?8:

?

8:

7

>

>

7+ (hhjjjbb)

5?

8:h8+8j8+8+8b8+8 5

2

=h:j2b:

(ooo) [n `hvn,

::yzx=

2 (4.>xy:z:) (8.87x:yz2) (:xyz)

5

:87.8>.4

=

2 (x: x x: x y y:

y y z: z: z2 z)

5

:

844

887

84

>

=

2 (x:+8+:+8 y8+:+8+8

z:+:+2+8)

5844

0;x7y>z0


4/39


(ov)[n `hvn, (:4x84y:4z24) (84xyz):

5 (:4x84y:4z24) (84xyz) (84xyz)

5 (:4 84 84) (x84 x x y:4 y y

z24 z z )

5 :444x84+8+8y:4+8+8z24+8+8

5 :444x8:y::z2:

(v) [n `hvn, (2x:y) (=xy:z) (xy:z:)

z

>

=

5

>

=8=2 (x: x x y y: y:

zz: z)

5>

=0x:+8+8y8+:+:z8+:+8 5

>

=0x=y>z=

Nx.:4 Nxprnss t`n gcaacwoik prcmubt hs h eciceoha6

(x2) (;x>)

:x>

8 (7x=)

Wnrogy t`n prcmubt gcr x 5 8

Rca. [n `hvn,

(x2) (;x>)

:x>

8 (7x=)

5

7

>

8;8 (x2 x> x: x=)

5 >

=:x2+>+:+= 5

>

=:x8=

Wnrogobhtoci 6 Gcr x 5 8, wn `hvn

[email protected]. 5 (x2) (;x>)

:x>

8 (7x=)

5 (8)2 {;(8>)}

:)8(

>

8 {7 (8)=}

5 8 ; >

8 7 5

>

=:

him,

[email protected]. 5 >

=: (8)8= 5 >

=:

[email protected]. 5 [email protected].

Nx.:8 Goim t`n vhaun cg (>h7) (84hj:) (:.8h:j2)

gcr h 5 8 him j 5:

8.

Rca. [n `hvn,

(>h7) (84hj:) (:.8h:j2)

5 (> 84 :.8) (h7 h h: j: j2)

5

84

:884> (h7 h h: j: j2)

5 84> h7+8+:j:+2 5 84>h?j>

Tuttoik h 5 8 him j 5:

8, wn `hvn

84>h?j> 5 84> (8)?

>

:

8

5 84> 82:

85

2:

84>

Euatopaobhtoci cg h Eciceoha & h Joiceoha

Nx.:: Euatopay 6 :x jy (2x + >y)

Rca. [n `hvn,

:x (2x + >y) 5 :x 2x + :x >y 5 7x: + 84xy

Nx.:2 Euatopay 6 (;xy + >y) jy 2xy

Rca. [n `hvn,(;xy + >y) 2xy

5 ;xy 2xy + >y 2xy

5 :8x8+8y8+8+8>xy8+8 5 :8x:y: + 8>xy:

Nx.:= Euatopay 6 >

hj2 :

jy

j

2

h:

Rca. [n `hvn,

>

hj2

:

j

2

h:

5

>hj2

:

2

h:

>

hj2

: j

5 >

2

2

:h8+8j:+

>

2hj:+8 5

>

:h:j: +

>

2hj2

Nx.:> Euatopay 6

xy

>

=x2 : jy

:

8xy..

Rca. @crozcitha ent`cm Bcauei ent`cm

[n hvn, [n hvn,

xy

>

=x2 :

:

8xy 2x

>

=y:x

5 2x :

8xy

>

=y:x

:

8xy

:

8xy

5

:

82 xxy yx

:

2 : 2:yx>

:

:

8

>

= y:yxx

5 yx:

2 :

>

:y2x: 5

:

2x:y

>

:x:y2


5/39


Nx.:7 Mntnreoin nhb` cg t`n gcaacwoik prcmubts him

goim t`n vhaun cg nhb` gcr x 5 :, y 5 8.8>, z 5 4.48.

(o) :;x: (8 2x) (oo) xz(x: + y:)

(ooo) z:(x y) (ov) (:z 2x) ( =y)

Rca. (o) [n `hvn,

:;x: (8 2x)5 :;x: (8 2x)

5 :;x: 8 :;x: 2x \Nxphimoik t n jrhblntU

5 :;x: 08x2

Tuttoik x 5 :, wn `hvn

:;x: (8 2x)

5 :; (:): (8 2:) 5 :; = (8 7)

5 :; = > 5 >=4

(oo) [n `hvn,

xz(x: + y:)

5 xz (x: + y:)

5 xz x: + xz y: 5 x2z + xy:zTuttoik x 5 :, y 5 8.8> him z 5 4.48, wn knt

xz (x: + y:)

5 : 4.48 {(:): + (8.8>):}

5 4.4: (= + 8.2::>) 5 4.4: >.2::> 5 4.847=>4

(ooo) [n `hvn,

z:(x y)

5 z: (x y)

5 z: x z: y 5 z:x z:y

Tuttoik x 5 : y 5 8.8> him z 5 4.48, wn knt

z:(x y)

5 (4.48): (: 8.8>)5 (4.4448) (4.0>) 5 4.44440>

(vo) [n `hvn,

(:z 2x) ( =y)

5 (:z) ( =y) 2x ( =y) 5 0zy + 8:xy

Tuttoik x 5 :, y5 8.8> him z 5 4.48, wn `hvn

(:z 2x) =y

5 \(: 4.48) (2 :)U (= 8.8>)

5 (4.4: 7) (=.7) 5 >.?0 =.7 5 :;.>40

Nx.:; Roepaogy t`n nxprnssoci him nvhauhtn t`ne hs

mornbtnm 6

(o) x(x 2) + : gcr x 5 8

(oo) 2y (:y ;) 2(y=) 72 gcr y 5 :

Rca. (o) [n `hvn,

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

Gcr x 5 8, wn `hvn

x: 2x + : 5 (8): 2 8 + : 5 8 2 + :

5 2 2 5 4

(oo) [n `hvn,

2y(:y 7) 2 (y =) 72

5 (7y: :8y) (2y =) 72

5 7y: :8y 2y + 8: 72

5 7y: :=y >8

Gcr y 5 :, wn `hvn

7y: :=y >8 5 7 (:): :=(:) >8

5 7 =+:=: >85:=+=0 >85 ;: >85 :8

Nx.:0 Rujtrhbt 2pq (p q) grce :pq(p + q)Rca. (o) [n `hvn,

2pq (p q) 5 2p:q 2pq:

him, :pq (p + q) 5 :p:q + :pq:

Rujtrhbtoci 6

:p:q + :pq:

2p:q 2pq:

+ +

p:q + >pq:

Nx.:? Hmm 6 (o) p(p q), q (q r) him r(r p)

(oo) :x (z x y) him :y(z y x)Rca. (o) [n `hvn,

p (p q) + q (q r) + r (r p)

5 p: pq + q: qr + r: rp

5 p: + q: + r: pq qr rp

(oo) [n `hvn,

:x(z x y) + :y (z y x)

5 :xz :x: :xy + :yz :y: :xy

5 :xz :x: =xy + :yz :y:

Nx.24 Roepaogy nhb` cg t`n gcaacwoik nxprnssocis 6

(o) 8>h: 7h (h :) + h(2 + ;h)(oo) x:(8 2y:) + x(xy: :x) 2y(y =x:y)

(ooo) =st(s t) 7s:(t t:) 2t:(:s: s) + :st(s t)

Rca. (o) [n `hvn,

8>h: 7h(h :) + h(2 + ;h)

5 8>h: 7h: + 8:h + 2h + ;h:

5 8>h: 7h: + ;h: + 8:h + 2h 5 87h: + 8>h

(oo) [n `hvn,

x:(8 2y:) + x(xy: :x) 2y(y =x:y)

5 x: 8 2y: x: + x xy: x :x 2y

y + 2y = x:y

5 x: 2x:y: + x:y: :x: 2y: + 8:x:y:

5 (x: :x:) + (2x:y: + x:y: + 8:x:y:) 2y:

5 x: + 84x:y: 2y:

(ooo) =st(s t) 7s:(t t:) 2t:(:s: s) + :st(s t)

5 =st s =st t 7s: t + 7s: t:

2t: :s: + 2t: s + :st s :st t

5 =s:t =st: 7s:t + 7s:t: 7s:t:

+ 2st: + :s:t :st:

5 (=s:t 7s:t + :s:t) + (=st: + 2st: :st:)

+ (7s:t: 7s:t:)

5 2st:


6/39


Euatopaobhtoci cg ]wc Joiceohas

Nx.28 Euatopay (2x + :y) him (>x + 2y).

Rca. [n `hvn,

(2x + :y) (>x + 2y)

5 2x (>x + 2y) + :y (>x + 2y)5 (2x >x + 2x 2y) + (:y >x + :y + 2y)

5 (8>x: + ?xy) + (84xy + 7y:)

5 8>x: + ?xy + 84xy + 7y:

5 8>x: + 8?xy + 7y:

Nx.2: Euatopay (:x + 2y) him (=x >y)

Rca. [n `hvn,

(:x + 2y) (=x >y)

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

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

5 (0x: 84xy) + (8:xy 8>y:)

5 0x: 84xy + 8:xy 8>y:

5 0x: + :xy 8>y:

Nx.22 Euatopay (;h + 2j) him (:h + 2j) jy bcauei

ent`cm.

Rca. [n `hvn,

::

:

:

j?hj:;h8= tnreaolnt`nHmmoik

j2jyj2h;kEuatopayoi

h:jyj2h;kEuatopayoi

j?hj:8

hj7h8=

j2h:

j2h;

Nx.2= Euatopay (;x2y) jy (=x>y) jy bcauei ent`cm.

Rca. [n `hvn,

::

:

:

y8>xy=;x:0 tnresaolnt`nHmmoik

y>jyy2x;kEuatopayoi

x=jyy2x;kEuatopayoi

y8>xy2>

xy8:x:0

y>x=

y2x;

Nx.2> Euatopay (4.>x y) jy (4.>x + y)

Rca. @crozcitha Ent`cm6

[n `hvn,

(4.>x y) (4.>x + y)

5 4.>x (4.>x + y) y (4.>x + y)

5 4.>x 4.>x + 4.>x y y 4.>x y y

5 4.:>x: + 4.>xy 4.>xy y:

54.:>x: y:

Bcauei ent`cm6

[n `hvn,

tnresaolnt`nHmmoikyx:>.4

yjyyx>.4kEuatopayoi

x>.4jyyx>.4kEuatopayoi

yxy>.4

xy>.4x:>.4

yx>.4

yx>.4

::

:

:

Nx.27 Euatopayoik

>

y2x= him

>

y=x2

Rca. @crozcitha Ent`cm 6

>

y2x=

>

y=x2

5 =x

>

y=x2 +

>

y2

>

y=x2

5 =x 2x =x >

y=+

>

y2 2x

>

y2

>

y=

5 8:x: >

87xy +

>

?xy

:>

8:y:

5 8:x: >

;xy

:>

8:y:

Bcauei ent`cm6

[n `hvn,

tnresaolnt`nHmmoiky:>

8:xy

>

;x8:

.>

y=jy

>

y2x=kEuatopayoi

.x2jy>

y2x=kEuatopayoi

y:>

8:xy

>

87

xy>

?x8:

>

y=x2

>

y2x=

::

:

:

Nx.2; Goim t`n vhaun cg t`n gcaacwoik prcmubts6

(o) (x + :y) (x :y) ht x 5 8, y 5 4

(oo) (2e :i) (:e 2i) ht e 5 8, i 5 8

(ooo) (=h: + 2j) (=h: + 2j) ht h 5 8, j 5:

Rca. (o) [n `hvn,

(x + :y) (x :y)

5 x(x :y) + :y (x :y)

5 x x x :y + :y x :y :y

5 x: :xy + :yx =y:

5 x: =y:

[`ni x 5 8, y 5 4, wn knt

(x + :y) (x :y)

5 x: =y: 5 (8): = (4): 5 8 4 5 8.


7/39


8/39


Nx.=8 Nvhauhtn 6

(o) (:x + 2y):

(oo) (:x 2y):

(ooo) (:x + 2y) (:x 2y)

Rca. (o) [n `hvn,

(:x: + 2y): 5 (:x): + :(:x) (2y) + (2y):

\^soik6 (h + j): 5 h: + :hj + j:U

5 =x: + 8:xy + ?y:

(oo) [n `hvn,

(:x 2y): 5 (:x): : (:x) (2y) + (2y):

\^soik6 (h j): 5 h: :hj + j:U

5 =x: 8:xy + ?y:

(ooo)[n `hvn,

(:x + 2y) (:x 2y) 5 (:x): (2y):

\^soik 6 (h + j) (h j) 5 h: j:U

5 =x: ?y:.

Nx.=: [rotn mcwi t`n squhrns cg nhb` cg t`n gcaacwoik

joiceohas 6

(o)

:

hx (oo)

:

8j> (ooo)

:

yy

:

Rca. (o) [n `hvn,

:

:

hx

5 x: + : x

:

h+

:

:

h

\^soik 6 (h + j): 5 h: + :hj + j:U

5 x: + xh +=

h:

(oo) [n `hvn,

:

:

8j>

5 (>j): : >j

:

8+

:

:

8

\^soik 6 (h j): 5 h: :hj + j:U

5 :>j: >j +=

8

(ooo)[n `hvn,

::

:

y

y

5 y: + : y :

y:

+

::

:

y

5 y: + y2 +=

y=

Nx.=2 Goim t`n prcmubt cg t`n gcaacwoik joiceohas 6

(o)

2x

2

= :

2x

2

= :

(oo)

:: y>x

2

:

:: y>x

2

:

Rca. (o) [n `hvn,

2x

2

= :

2x

2

= :

5

:

: 2x2

=

\ h.h 5h:U

5

::x

2

=

+ :

2

=x: 2 + (2):

\^soik 6 (h + j): 5 h: + :hj + j:U

5?

87x= + 0x: + ?

(oo) [n `hvn,

:: y>x

2

:

:: y>x

2

:

5

:::

y>x2

:

\ h.h 5 h:U

5

::

x2

:

+ :

2

:x: >y: + (>y:):

\^soik 6 (h + j): 5 h: + :hj + j:U

5?

=x= +

2

:4x:y: + :>y=

Nx.== Og x +x

85 =, goim t`n vhauns cg

(o) x: + :x

8(oo) x= + =

x

8

Rca. (o) [n `hvn,

x +x

85 =

Ci squhroik jct` somns, wn knt

:

x

8

x

5 =:

x: + : x x

8+

:

x

8

5 87

x: + : + :x

85 87

x: + :x

85 87 : \Ci trhispcsoik : ci Z@RU

x: + :x

85 8=


9/39


(oo) [n `hvn,

x: + :x

85 8=


:

::

x

8x

5 8=:

(x:): +

:

:x

8

+ : x: :

x

85 8?7

x= + =x

8+ : 5 8?7

x= + =x

85 8?7 : \Ci trhispcsoik : ci Z@RU

x= + =x

85 8?=

Nx.=> Og x x

85 ?, goim t`n vhaun cg x: + :

x

8.

Rca. [n `hvn,

x x

85 ?


:

x

8x

5 08

x: : x x8 +

:

x8 5 08

x: : + :x

85 08

x: + :x

85 08 + : \Ci trhispcsoik : ci Z@RU

x: + :x

85 02

Nx.=7 Og x: + :x8 5 :;, goim t`n vhauns cg nhb` cg t`n

gcaacwoik 6

(o) x +x

8(oo) x

x

8

Rca. (o) [n `hvn,

:

x

8x

5 x: + : x

x

8+ :

x

8

:

x

8x

5 x: + : + :

x

8

:

x

8x

5 x: + :

x

8+ :

:

x

8x

5 :; + :

)kovni(:;

x

8x

:

:

:

x

8x

5 :?

x +x

85 :?

\]hloik squhrn rcct cg jct` somnsU

(oo) [n `hvn,

:

x

8x

5 x: : x

x

8+

:

x

8

:

x8x

5 x: : +

:x8

:

x

8x

5 x: + :

x

8 :

:

x

8x

5 :; :

)kovni(:;

x

8x

:

:

:

x

8x

5 :>

:

x8x

5 >:

x x

85 >

Nx.=; Og 2x + :y 5 8: him xy 5 7, goim t`n vhaun cg

?x: + =y:.

Rca. [n `hvn,

(2x + :y): 5 (2x): + (:y): + : 2x :y

(2x + :y): 5 ?x: + =y: + 8:xy

8:: 5 ?x: + =y: + 8: 7

\Tuttoik 2x + :y 5 8: him xy 5 7U

8== 5 ?x: + =y: + ;:

8== ;: 5 ?x: + =y:

?x: + =y: 5 ;:

Nx.=0 Og =x: + y: 5 =4 him xy 5 7, goim t`n vhaun cg

:x + y.

Rca. [n `hvn,

(:x + y): 5 (:x): + y: + : :x y

(:x + y): 5 (=x: + y:) + =xy

(:x + y): 5 =4 + = 7

\^soik =x: + y: 5 =4 him xy 5 7U


10/39


(:xy + y): 5 7=

:x + y 5 7=

:x + y 5 0 \]hloik squhrn rcct cg jct` somnsU

Nx.=? Goim t n bcitoiunm prcmubt 6

(o) (x + :) (x :) (x: + =)

(oo) (:x + 2y) (:x 2y) (=x: + ?y:)

(ooo) (x 8) (x + 8) (x: + 8) (x= + 8)

(ov)

x

8x

x

8x

:

:

x

8x

=

=

x

8x

(v)

8

>

yx

8

>

yx

Rca. (o) [n `hvn,

(x + :) (x :) (x: + =)

5 {(x + :)(x :)} (x: + =)\Jy hsscbohtovoty cg euatopaobhtociU

5 (x: ::) (x: + =) \ (h + j) (h j) 5 h: j:U

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

5 (x:): =: \ (h + j) (h j) 5 h: j:U

5 x= 87

(oo) [n `hvn,

(:x + 2y) (:x 2y) (=x: + ?y:)

5 {(:x + 2y) (:x 2y)} (=x: + ?y:)

5 {(:x + 2y) (:x 2y)} (=x: + ?y:)

5 :: )y2()x:( (=x: + ?y:)\^soik 6 (h + j) (h j) 5 h: j:U

5 (=x: ?y:) (=x: + ?y:)

5 (=x:): (?y:):

\^soik 6 (h + j) (h j) 5 h: j:U

5 87x= 08y=.

(ooo) [n `hvn,

(x 8) (x + 8) (x: + 8) (x= + 8)

5 {(x 8) (x + 8)} (x: + 8) (x= + 8)

5 (x: 8) (x: + 8) (x= + 8)

5 )8x)(8x( :: + (x= + 8)

5 ::: 8)x( (x= + 8)5 (x= 8) (x= + 8)

5 ::= 8)x(5 x0 8

(ov) [n `hvn

x

8x

x

8x

:

:

x

8x

=

=

x

8x

5

x

8x

x

8x

:

:

x

8x

=

=

x

8x

5

:

:

x

8x

:

:

x

8x

=

=

x

8x

5

:

:

::

x

8)x(

=

=

x

8x

5

=

=

x

8x

=

=

x

8x

5 (x=):

:

=x

8

5 x0 0x

8

(v) [n `hvn,

8

>

yx

8

>

yx

5

8

>

yx

8

>

yx

5 x: +

:

8>

y

5 x:

8

>

y:

:>

y:

5 x: :>

y:

>

y: 8

Nx.>4 Trcvn t`ht6

:h: + :j: + :b: :hj :jb :bh

5 (h j): + (j b): + (b h):

Rca. [n `hvn,

A@R 5 :h: + :j: + :b: :hj :jb :bh

5 (h: :hj + j:) + (j: :jb + b:)

+ (b: :bh + h:)

\Zn-hrrhikoik t`n tnresU5 (h j): + (j b): + (b h):

5 [email protected].

@nibn , :h: + :j: + :b: :hj :jb :bh

5 (h j): + (j b): + (b h):

Nx.>8 Og h: + j: + b: hj jb bh 5 4, prcvn t`ht

h 5 j 5 b.

Rca. [n `hvn,

h: + j: + b: hj jb bh 5 4

:h: + :j: + :b: :hj :jb :bh 5 : 4

\Euatopayoik jct` somns jy :U


11/39


(h: :hj + j:) + (j: :jb + b:)

+ (b: :hb + h:) 5 4

(h j): + (j b): + (b h): 5 4

h j 5 4, j b 5 4, b h 5 4

znrcosquhitotynhb`ogciayhimogznrcosquhitotonspcsotovncgRue

h 5 j , j 5 b h im b 5 h

h 5 j 5 b.

Nx.>: ^soik t`n gceuahn gcr squhroik h joiceoha,

nvhauhtn t`n gcaacwoik 6

(o) (848): (oo) (??): (ooo) (?2):

Rca. [n `hvn,

(o) (848): 5 (844 + 8):

5 (844): + : 844 8+ (8):

\^soik 6 (h + j:) 5 h: + :hj + j:U

5 84444 + :44 + 8

5 84:48

(oo) (??): 5 (844 8):

5 (844): : 844 8 + (8):

\^soik 6 (h j): 5 h: :hj + j:U

5 84444 :44 + 8

5 ?048

(ooo) (?2): 5 (?4 + 2):

5 (?4): + : ?4 2t (2):

5 0844 + >=4 + ? 5 07=?

Nx.>2 Goim t`n vhaun cg x, og(o) 7x 5 :2: 8;:

(oo) =x 5 ?0: 00:

(ooo) :>x 5 >27: 827:

Rca. (o) [n `hvn,

7x 5 :2: 8;:

7x 5 (:2 + 8;) (:2 8;)

\^soik 6 h: j: 5 (h + j) (h j)U

7x 5 =4 7

7

x75

7

7=4\Movomoik jct` somns jy 7U

x 5 =4

(oo) [n `hvn,

=x 5 ?0: 00:

=x 5 (?0 + 00) (?0 00)

\^soik 6 h: j: 5 (h + j) (h j)U

=x 5 807 84

=

x=5

=

84807\Movomoik jct` somns jy =U

x 5=

8074

x 5 =7>

(ooo)[n `hvn,

:>x 5 >27: 827:

:>x 5 (>27 + 827) (>27 827)

\^soik 6 (h: j:) 5 (h + j) (h j)U

:>x 5 7;: =44

:>

x:>5

:>

=447;:\Movomoik jct` somns jy :>U

x 5 7;: 87

x 5 84;>:

Nx.>= [`ht eust jn hmmnm tc ?x: :=x + 84 tc ehln

ot h w`can squhrn 3

Rca. [n `hvn,

?x: :=x + 84 5 (2x): : 2x = + 84

Ot os nvomnit grce t`n hjcvn nxprnssoci t`ht

Gorst tnre 5 2x him , Rnbcim tnre 5 =

]c ehln t`n kovni nxprnssoci h w`can squhrn,wn eust `hvn (=): 5 87 oi pahbn cg 84.

@nibn, wn eust hmm 7 tc ot tc ehln h pnrgnbt

squhrn.

Hmmoik him sujtrhbtoik 7, wn knt

?x: :=x + 84 + 7 7 5 ?x: :=x + 87 7

5 (2x =): 7

Nx.>> Goim t`n gcaacwoik prcmubts6

(o) (x + :) (x + 2)

(oo) (x + ;) (x :)

(ooo) (y =) (y 2)

(ov) (y ;) (y + 2)

(v) (:x 2) (:x + >)

(ov) (2x + =) (2x >)

Rca. ^soik t`n omnitotoy 6

(x + h) (x + j) 5 x: + (h + j)x + hj, wn `hvn

(o) (x + :) (x + 2) 5 x: + (: + 2)x + : 2

5 x: + >x + 7

(oo) (x + ;) (x :) 5 (x + ;) {x + ( :)}

5 x: + {; + (:)}x + ; :

5 x: + >x 8=

(ooo) (y =) (y 2) 5 {y + ( =)}{y + (2)}

5 y: + {(=) + (2)y + (=) (2)}

5 y: ;y + 8:

(ov) (y ;) (y + 2) 5 {y + ( ;)} (y + 2)

5 y: + {(;) + 2} y + (;) 2

5 y: =y :8

(v) (:x 2) (:x + >) 5 (y 2) (y + >),

w`nrn y 5 :x

5 {y + (2)} (y + >)

5 y: + {(2) + >}y + (2) >

5 y: + :y 8>


12/39


Nx.>7 Nvhauhtn t`n gcaacwoik6

(o) 84; 842 (oo) >7 =0 (ooo) ?> ?;

Rca. ^soik t`n omnitoty 6

(x + h) (x + j) 5 x: + (h + j)x + hj wn `hvn

(o) 84; 842 5 (844 + ;) (844 + 2)

5 (844): + (; + 2) 844 + ; 2

5 84444 + 84 844 + :8

5 84444 + 8444 + :8

5 884:8

(oo) >7 =0 5 (>4 + 7) (>4 :)

5 (>4 + 7) {>4 + (:)}

5 (>4): + {7 + (:)} >4 + 7 :

5 :>44 + = >4 8:

5 :>44 + :44 8:

5 :;44 8:

5 :700

(ooo) ?> ?; 5 (844 >) (844 2)5 {844 + (>)} {844 + (2)}

5 (844): + {(>) + (2)} 844 + (>) (2)

5 84444 0 844 + 8>

5 84444 044 + 8>

5 ?:8>

GHB]CZOVH]OCI

Ghbtcrs 6Og hi haknjrhob nxprnssoci os wrottni

hs t`n prcmubt cg iuejnrs cr haknjrhob

nxprnssocis, t`ni nhb` cg t`nsn iuejnrs him

nxprnssocis hrn bhaanm t`n ghbtcrs cg t`nkovni haknjrhob nxprnssoci him t`n haknjrhob

nxprnssoci os bhaanm t`n prcmubt cg t`nsn

nxrprnssocis.

Ghbtcrozhtoci 6 ]`n prcbnss cg wrotoik h

kovni haknjrhob nxprnssoci hs t`n prcmubt cg

twc cr ecrn ghbtcrs os bhaanm ghbtcrozhtoci.

Nx.>; [rotn mcwi haa pcssojan ghbtcrs cg 2x:y.

Rca. [n `hvn,

2x:y 5 8 2x:y 5 2 x:y 5 2x xy 5 2xy x

5 x:

2y 5 y 2x:

]ùs, t`n pcssojan ghbtcrs cg 2x:y hrn

8, 2x: y, 2, x: y, 2x, xy, 2xy, x, x:, 2y, y, 2x:

Nx.>0 [rotn mcwi haa pcssojan ghbtcrs cg 8:x:.

Rca. [n `hvn,

8:x: 5 8 8:x: 5 8: x: 5 2 =x: 5 = 2x:

5 : 7x: 5 7 :x:

5 2x =x 5 7x :x 5 : 2x :x 5 2 :x :x

]ùs, t`n pcssojan ghbtcrs cg 8:x: hrn

8, 8:x:, 8:, x:, 2, =x:, =, 2x:, :, 7x:, 7, :x:, 2x, =x,

7x, :x

Krnhtnst Bceeci Ghbtcr (KBG) Cr

@ok`nst Bceeci Ghbtcr (@BG)

]`n krnhtnst bceeci ghbtcr cg kovni eciceohas

os t`n bceeci ghbtcr `hvoik krnhstnst bcnggobonit

him òk`nst pcwnr cg t`n vhrohjans.]`n gcaacwoik stnp-wosn prcbnmurn woaa jn `napgua

tc goim t`n KBG cg twc cr ecrn eciceohas.

Rtnp O Cjthoi t`n kovni eciceohas.

Rtnp OO Goim t`n iuenrobha bcnggobonit cg t`n

n h b` e c ic eo h a h i m t `n o r k rnh tn st

bceeci ghbtcr (KBG/@BG).

Rtnp OOO Goim t`n bceeci aotnrhas hppnhroik oi

t`n kovni eciceohas.

Rtnp OW Goim sehaanst pcwnr cg nhb` bceeci

aotnrha.

Rtnp W [rotn h eciceoha cg bceeci aotnrhas wot`sehaanst pcwnrs cjthoinm oi stnp OW.

Rtnp WO ]`n rnquornm KBG os t`n prcmubt cg t`n

bcnggobonit cjthoinm oi stnp OO him t`n

eciceoha cjthoinm oi stnp W.

Nx.>? G o im t ` n k rn ht n st b c ee ci ghb t crs c g t ` n

eciceohas :8h2j; him 2>h>j>.

Rca. ]` n i u en rob ha b c nggob o ni t s c g t `n k ov n i

eciceohas hrn :8 him 2>

]`n krnhtnst bceeci ghbtcr cg :8 him 2> os ;

]`n bceeci aotnrhas hppnhroik oi t`n kovni

eciceohas hrn h him j

]`n sehaanst pcwnr cg h oi t`n twc eciceohas

5 2

]`n sehaanst pcwnr cg j oi t`n twc eciceohas

5 >

]`n eciceoha cg bceeci aotnrhas wot` sehaanst

pcwnrs 5 h2j>

]`n krnhtnst bceeci ghbtcr 5 ;h2j>

Nx.74 G o im t ` n k rn ht n st b c ee ci ghb t crs c g t ` n

eciceohas 8=x:y2, :8x:y:, 2>x=y>z.

Rca. ]` n i u en rob ha b c nggob o ni t s c g t `n k ov n ieciceohas hrn 8=, :8, him 2>

]`n krnhtnst bceeci ghbtcr cg 8=, :8 him 2> os ;

]`n bceeci aotnrhas hppnhroik oi t`n t`rnn

eciceohas hrn x him y

]`n sehaanst pcwnr cg x oi t`n t`rnn eciceohas

5 :

]`n sehaanst pcwnr cg y oi t`n t`rnn eciceohas

5 :

]`n eciceoha cg bceeci aotnrhas wot` sehaanst

pcwnrs 5 x:y:

@nibn, t`n krnhtnst bceeci ghbtcr 5 ;x:y:


13/39


Ghbtcrozhtoci cg Haknjrhob Nxprnssoci

w`ni h bceeci Eciceoha Ghbtcr

Cbburs oi nhb` ]nre

Oi crmnr tc ghbtcrozn haknjrhob nxprnssocis

bcisostoik cg h bceeci eciceoha ghbtcrs cg nhb`tnre wn usn t`n gcaacwoik stnp-wosn prcbnmurn.

Rtnp O Cjthoi t`n haknjrhob nxprnssoci.

Rtnp OO Goim t`n krnhtnst bceeci ghbtcr

(KBG/@BG) cg ots tnres.

Rtnp OOO N xp rn ss n hb ` t nr e c g t `n k ov ni

nxprnssoci hs t`n prcmubt cg t`n KBG

him t`n quctonit w`ni ot os movomnm jy

t`n KBG.

Rtnp OW ^ sn t `n m os tr oj ut ov n p rc pn rt y c g

euatopaobhtoci cvnr hmmotoci tc nxprnss

t`n kovni haknjrhob nxprnssoci hs t`nprcmubt cg t`n KBG him t`n quctonit cg

t`n kovni nxprnssoci jy t`n KBG.

Nx.78 Ghbtcrozn nhb` cg t`n gcaacwoik haknjrhob

nxprnssocis6

(o) 2x + 8> (oo) :x: + >x

(ooo) 2x:y 7xy: (ov) 7x2 + 0x:y

Rca. (o) ]`n krnhtnst bceec ghbtcr cg t n tnres ihenay,

2x him 8> cg t`n nxprnssoci 2x + 8> os 2. Hasc,

2x 5 2 x him 8> 5 2 >.

2x + 8> 5 2 (x + >)

(oo) ]`n krnhtnst bceeci ghbtcr cg t`n tnres :x:

him >x cg t`n nxprnssoci :x: + >x os x. Hasc,

:x: 5 :x x him >x 5 > x.

:x: + >x 5 :x x + > x

5 (:x + >)x

(ooo) Banhray, 2xy os t`n krnhtnst bceeci ghbtcr cg

t`n tnres 2x:y him 7xy: cg t`n joiceoha 2x:y

7xy:. Hasc, 2x:y 5 2xy x him 7xy: 5 2xy :y

2x:y 7xy: 5 2xy x 2xy :y

5 2xy(x :y)

(ov) Banhray, :x: os t`n KBG cg t`n tnres 7x2 him

0x:y cg t`n kovni joiceoha 7x2 + 0x:y. Hasc,7x2 5 :x: 2x him 0x:y 5 :x: =y.

7x2 + 0x:y 5 :x: 2x + :x: =y

5 :x:(2x + =y)

Nx.7: Ghbtcrozn 6

(o) ;(:x + >) + 2 (:x + >)

(oo) (x + :)y (x + :)x

(ooo) >h(:x + 2y) :j (:x + 2y)

(ov) 0(>x + ?y): + 8: (>x + ?y)

Rca. [n `hvn,

(o) ;(:x + >) + 2 (:x + >) 5 (; + 2) (:x + >)

\]hloik (:x + >) bceeciU

5 84(:x + >)

(oo) (x + :) y + (x + :) x 5 (x + :) (y + x)

\]hloik (x + :) bceeciU(ooo) >h (:x + 2y) :j (:x + 2y)

5 (:x + 2y) (>h :j)

\]hloik (:x + 2y) bceeciU

(ov) 0(>x + ?y): + 8:(>x + ?y) 5 =(>x + ?y)

{:(>x + ?y) + 2}

5 =(>x + ?y) (84x + 80y + 2)

Nx.72 Ghbtcrozn 6

(o) (y x) h + (x y)j

(oo) ?(h :j): + 7(:j h)

(ooo) (x :y): =x + 0y

(ov) :h + 7j 2 (h + 2j):

Rca. [n `hvn,

(o) (y x)h + (x y)j 5 (x y)h + (x y)j

\]hloik (8) bceeci grce (y x)U

5 (x y) (h + j) \]hloik (x y) bceeciU

5 (x y) (j h) \h + j 5 j hU

(oo) ?(h :j): + 7(:j h)

5 ?(h :j): 7(h :j)

\ :j h 5 (h :j)U

5 2(h :j) {2 (h :j) :}

\]hloik 2(h :j) bceeciU

5 2(h :j) (2h 7j :)

(ooo) (x :y): =x + 0y 5 (x :y): =(x :y)

y0x=grce

bceeci=]hloik

5 (x :y) {(x :y) =}

\]hloik (x :y) bceeciU

5 (x :y) (x :y =)

(ov) :h + 7j 2(h + 2j): 5 :(h + 2j) 2 (h + 2j):

\]hloik : bceeci grce :h + 7jU

5 (h + 2j) {: 2 (h + 2j)}

\]hloik (h 2j) bceeciU

5 (h + 2j) (: 2h ?j)

Ghbtcrozhtoci jy Krcupoik t`n ]nres

Nx.7= Ghbtrcozn 6

(o) hx + jx + hy + jy

(oo) hx: + jy: + jx: + hy:

(ooo) h: + jb + hj + hb

(ov) hx hy + jx jy


14/39


Rca. [n `hvn,

(o) hx + jx + hy + jy 5 (hx + jx) + (hy + jy)

\Krcupoik t`n tnresU

5 (h + j)x + (h + j)y

5 (h + j) (x + y)\]hloik (h + j) bceeciU

(oo) hx: + jy: + jx: + hy:

5 hx: + jx: + hy: + jy:

\Zn-hrrhikoik t`n tnresU

5 (h + j)x: + (h + j)y:

5 (h + j)(x: + y:) \]hloik (h + j) bceeciU

(ooo) h: + jb + hj + hb 5 (h: + hj) + (hb + jb)

\Zn-krcupoik t`n tnresU

5 h (h + j) + (h + j)b

5 (h + j) (h + b)\]hloik (h + j) bceeciU

(ov) hx hy + jx jy 5 h (x y) + j (x y)

5 (h + j) (x y) \]hloik (x y) bceeciU

Nx.7> Ghbtcrozn nhb` cg t`n gcaacwoik nxprnssoci6

(o) h2x + h: (x y) h(y + z) z

(oo) (x: + 2x): >(x: + 2x) y (x: + 2x) + >y

Rca. (o) [n `hvn,

h2x + h: (x y) h (y + z) z

5 h2x + h:x h:y hy hz z

5 (h2x + h:x) (h:y + hy) (hz + z)

5 h:x (h + 8) hy(h + 8) z(h + 8)

5 (h + 8)(h:x hy z)

(oo) (x: + 2x): > (x: + 2x) y (x: + 2x) + >y

5 (x:

+ 2x) >)x2x( :

y >)x2x( :

5 (x: + 2x >) (x: + 2x y)

Ghbtcrozhtoci cg Joiceoha Nxprnssocis

nxprnssojan hs t`n moggnrnibn cg twc squhrns

Nx.77 Ghbtcrozn 6

(o) ?h: 87j:

(oo) 27h: (x y):

(ooo) 04h: =>j:

(ov) (2h j): ?b:

Rca. [n `hvn,

(o) ?h: 87j: 5 (2h): (=j):

5 (2h + =j) (2h =j)

\^soik 6 (h: j:) 5 (h + j) (h j)U

(oo) 27h: (x y): 5 (7h): (x y):

5 {7h + (x y)} {7h (x y)}

\^soik 6 h: j: 5 (h + j) (h j)U

5 (7h + x y) (7h x + y)

(ooo) 04h: =>j: 5 >(87h: ?j:)

5 >{(=h): (2j):}

5 > (=h + 2j) (=h 2j)

\^soik 6 h: j: 5 (h + j) (h j)U

(ov) (2h j): ?b: 5 (2h j): (2b):

5 {(2h j) + 2b}{(2h j 2b)}

5 (2h j + 2b) (2h j 2b)

Nx.7; Ghbtcrozn 6

(o) 87h: :h=

:>

(oo) 87h:j :h87

j

(ooo) 844(x + y): 08 (h + j):

(ov) (x 8): (x :):

Rca. [n `hvn,

(o) 87h::

h=

:>5 (=h):

:

h:

>

h:

>h=

h:

>h=

(oo) 87h:j :h87

j 5 j

:: h87

8h87

5 j

:

:

h=

8)h=( 5 j

h=

8h=

h=

8h=

(ooo) 844(x + y): 08 (h + j):

5 {84 (x + y)}: {?(h + j)}:

5 {84(x + y) + ? (h + j)} {84 (x + y) ? (h + j)}

5 (84x + 84y + ?h + ?j) (84x + 84y ?h ?j)

(ov) (x 8): (x :):

5 {(x 8) + (x :)}{(x 8) (x :)}

5 (:x 2) (x 8 x +:)

5 (:x 2) 8

5 :x 2

Nx.70 Ghbtcrozn nhb` cg t`n gcaacwoik haknjrhob

nxprnssoci6

(o) x= 08y= (oo) :x> :x

(ooo) 2x= :=2 (ov) : >4x:

(v) x0 y0 (vo) h8:x= h=x8:

Rca. (o) x= 08y= 5 (x:): (?y:):

5 (x: ?y:) (x: + ?y:)

5 :: )y2(x (x: + ?y:)

5 (x 2y) (x + 2y) (x: ?y:)

(oo) :x> :x 5 :x (x= 8)U

5 :x ::: 8)x(

5 :x (x: 8) (x: + 8)

5 :x (x 8) (x + 8) (x: + 8)

(ooo) 2x= :=2 5 2 (x= 08)

5 2 ::: ?)x( 5 2(x: ?) (x: + ?)

5 2(x: 2:) (x: + ?)

5 2(x + 2) (x 2) (x: + ?)


15/39


(ov) : >4x: 5 :{8 :>x:}

5 :{8: (>x):}

5 :(8 >x) (8 + >x)

(v) x0 y0 5 {(x=): (y=):}

5 (x= y=) (x= + y=)

5 {(x:): (y:):}(x= + y=)

5 (x: y:) (x: + y:) (x= + y=)

5 (x y) (x + y) (x: + y:) (x= + y=)

5 (x y) (x + y) (x: + y:)

{(x:): + (y:): + :x:y: :x:y:}

5 (x y) (x + y) (x: + y:)

:::: xy:)yx(

5 (x y) (x + y) (x: + y:) xy:yx :: xy:yx ::

(vo) h8:

x=

h=

x8:

5 h=

x=

00

xh5 h=x= :=:= )x()h(

5 h=x= (h= + x=) (h= x=)

5 h=x= (h= + x=) :::: )x()h(5 h=x= (h= + x=) (h: + x:) (h: x:)

5 h=x= (h= + x=) (h: + x:) (h + x) (h x)

Ghbtcrozhtoci cg Haknjrhob Nxprnssocis

nxprnssojan hs h pnrgnbt squhrn

(o) h: + :hj + j: 5 (h + j): 5 (h + j) (h + j)

(oo) h: :hj + j: 5 (h j): 5 (h j) (h j)

Nx.7? Ghbtcrozn 6

(o) x: + 0x + 87 (oo) =h: =h + 8

Rca. [n `hvn,

(o) x: + 0x + 87 5 x: + : x = + =:

5 (x + =): \^soik 6 h: + :hj + j: 5 (h + j):U

5 (x + =) (x + =)

(oo) =h: =h + 8 5 (:h): : :h 8 + (8):

5 (:h 8): \^soik 6 h: :hj + j: 5 (h j):U

5 (:h 8) (:h 8)

Nx.;4 Ghbtcrozn 6

(o) =x: + 8:xy + ?y:

(oo) x= 84x:y: + :>y=

(ooo) h= :h:j: + j=

Rca. [n `hvn,

(o) =x: + 8:xy + ?y: 5 (:x): + : :x 2y + (2y):

5 (:x + 2y):

5 (:x + 2y) (:x + 2y)

(oo) x= 84x:y: + :>y=

5 (x:): : x: >y: + (>y:):

5 (x: >y:):

5 (x: >y:) (x: >y:)

(ooo) h= :h:j: + j= 5 (h:): : h: j: + (j:):

5 (h: j:):

5 {(h j) (h + j)}: 5 (h j): (h +j):

Nx.;8 Ghbtcrozn nhb` cg t n gcaacwoik nxprnssocis6

(o) x: :xy + y: x + y

(oo) =h: + 8:hj + ?j: 0h 8:j

(ooo) h: + j: : (hj hb + jb)

Rca. (o) x: :xy + y: x + y 5 (x: :xy + y:)

5 (x y): (x y)

5 (x y) {(x y) 8}

5 (x y) (x y 8)

(oo) =h: + 8:hj + ?j: 0h 8:j

5 (:h): + : :h 2j + (2j): =(:h + 2j)

5 (:h + 2j):

=(:h + 2j)5 (:h + 2j) (:h + 2j =)

(ooo) h: + j: : (hj hb + jb)

5 h: + j: :hj + :hb :jb

5 (h j): + :b (h j)

5 (h j) {(h j) + :b}

5 (h j) (h j + :b)

Nx.;2 Ghbtcrozn nhb` cg t n gcaacwoik nxprnssocis6

(o) x: + :xy + y: h: + :hj j:

(oo) :>x: 84x + 8 27y:

(ooo) 8 :hj (h: + j:)

Rca. (o) x: + :xy + y: h: + :hj j:

5 (x: + :xy + y:) (h: :hj + j:)

5 (x + y): (h j):

5{(x + y) + (h j)} {(x + y) (h j)}

5 (x + y + h j) (x + y h + j)

(oo) :>x: 84x + 8 27y:

5 (>x): : >x 8 + 8: (7y):

5 (>x 8): (7y):

5 (>x 8 + 7:) (>x 8 7y)

(ooo) 8 :hj (h: + j:) 5 8 (:hj + h: +j:)

5 8 (h + j):

5 {8 + (h + j)} {8 (h + j)}

5 (8 + h + j) (8 h j)

Nx.;= Ghbtcrozn6

(o) x: + 0x + 8> (oo) x= + x: + 8 (ooo) x= + =

Rca. [n `hvn,

(o) x: + 0x + 8> 5 (x: + 0x + 87) 8

\Znpahboik 8> jy 87 8U

5 {(x): + : x = + = :} 8

5 (x + =): 8:

5 {x + = + 8} {(x + =) 8}

5 (x + >) (x + 2)


16/39


(oo) x= + x: + 8 5 x= + :x: + 8 x:

\Hmmoik him sujtrhbtoik x:U

5 (x= + :x: + 8) x:

5 :::: 88x:)x( x:

5 (x: + 8): x:

5 {(x: + 8) + x} {(x: + 8) x}

5 (x: + x + 8) (x: x + 8)

(ooo) x= + = 5 x= + =x: + = =x:

\Hmmoik him sujtrhbtoik =x:U

5 {(x:): + : x: : + ::} =x:

5 (x: + :): (:x):

5 {(x: + :) + :x} {(x: + :) :x}

5 (x: + :x + :) (x: :x + :)

Ghbtcrozhtoci cg Xuhmrhtob Tcayiceohas oi

cin vhrojhan

Hakcrot`e 6

Rtnp O Cjthoi t`n quhmrhtob pcayiceoha

x: + px + q.

Rtnp OO Cjthoi p 5 bcnggobonit cg x him,

q 5 bcisthit tnre.

Rtnp OOO Goim t`n twc iuejnrs h him j sub` t`ht

h + j 5 p him hj 5 q.

Rtnp OW Rpaot up t`n eomman tnre hs t`n sue cg

twc tnres hx him jx.

Rtnp W Ghbtcrozn t`n nxprnssoci cjthoinm oi stnpOW jy krcupoik t`n tnre.

Nx.;> Ghbtcrozn nhb` cg t`n gcaacwoik nxprnssocis6

(o) x: + 7x + 0

(oo) x: + =x :8

(ooo) x: ;x + 8:

Rca. (o) Oi crmnr tc ghbtcrozn x: + 7x + 0, wn goim twc

i u e j n rs p h i m q su b ` t ` h t p + q 5 7 h i m

pq 5 0.

Banhray, : + = 5 7 him : = 5 0

[n icw spaot t`n eomman tnre 7x oi t`n kovni

quhmrhtob hs :x + =x.

x: + 7x + 0 5 x: + :x + =x + 0

5 (x: + :x) + (=x + 0)

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

5 (x + :) (x + =)

(oo) Oi crmnr tc ghbtcrozn x: + =x :8, wn `hvn tc goim

twc iuejnrs p him q sub` t`ht

p + q 5 = him pq 5 :8

Banhray, ; + (2) 5 = him ; 2 5 :8.

[n icw spaot t`n eomman tnre =x cg x: + =x :8

hs ;x 2x.

x: + =x :8 5 x: + ;x 2x :8

5 (x: + ;x) (2x + :8)

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

5 (x + ;) (x 2)

(ooo) Oi crmnr tc ghbtcrozn x:

;x + 8 : wn `hvn tc goimtwc iuejnrs p him q sub` t`ht p + q 5 ; him

pq 5 8:.

Banhray, 2 = 5 ; him 2 = 5 8:.

[n icw spaot t`n eomman tnre ;x cg t`n kovni

quhmrhtob hs 2x =x.

x: ;x + 8: 5 x: 2x =x + 8:

5 (x: 2x) (=x 8:)

5 x(x 2) = (x 2)

5 (x 2) (x =)

Nx.;7 Ghbtcrozn nhb` cg t`n gcaacwoik quhmrhtob

pcayiceohas 6(o) x: :2x + 82:

(oo) x: :8x + 840

(ooo) x: + >x 27

Rca. (o) Oi crmnr tc ghbtcrozn x: :2x + 82:, wn `hvn tc

goim iuejnrs p him q sub` t`ht p + q 5 :2 him

pq 5 82:.

Banhary, 8: 88 5 :2 him 8: 88 5 82:.

[n i cw s pa ot t `n e om ma n t nr e : 2x c g

x: :2x + 82: hs 8:x 88x

x: :2x + 82: 5 x: 8:x 88x + 82:

5 (x: 8:x) (88x 82:)

5 x (x 8:) 88 (x 8:)

5 (x 8:) (x 88)

(oo) Oi crmnr tc ghbtcrozn x: :8x + 840, wn `hvn tc

goim twc iuejnrs sub` t`ht t`nor sue os :8 him

t`n prcmubt 840.

Banhray, :8 5 8: ? him 8: ? 5 840

Rc, wn spaot t`n eomman tnre :8x hs 8:x ?x

x: :8x + 840 5 x: 8:x ?x + 840

5 (x: 8:x) (?x 840)

5 x(x 8:) ?(x 8:)

5 (x 8:) (x ?)

(ooo) Oi crmnr tc ghbtcrozn x: + >x 27, wn `hvn tc goim

twc iuejnrs p him q sub` t`ht p + q 5 > him

pq 5 27.

Banhray, ? + (=) 5 > him ? = 5 27.

Rc, wn wrotn t`n eomman tnre >x cg x: + >x 27

hs ?x =x.

x: >x 27 5 x: + ?x =x 27

5 (x: + ?x) (=x + 27)

5 x (x + ?) = (x + ?)

5 (x + ?) (x =)


17/39


Ghbtcrozhtoci cg Xuhmrhtob Tcayiceohas

cg ]`ncrne hx: + j x + b , h 8

Trcbnmurn6

Rtnp O Cjthoi t n quhmrhtob troiceoha

hx: + jx + b

Rtnp OO Cjthoi h 5 bcnggobonit cg x:,

j 5 bcnggobonit cg x him b 5 bcisthit tnres.

Rtnp OOO Goim t`n prcmubt cg t`n bcnggobonit cg

x: him t`n bcisthit tnre o.n. hb.

Rtnp OW Rpaot up t`n bcnggobonit cg x o.n. j oitc

twc phrts w`csn sue os j him prcmubt

hb him wrotn t`n eomman tnre hs t`n sue

cg twc tnres.

Rtnp W Ghbtcrozn t`n nxprnssoci cjthoinm oi stnp

OW jy krcupoik t`n tnre. Ghbtcrs sc

cjthoinm woaa jn t`n rnquornm ghbtcrs cg

t`n kovni quhmrhtob troiceoha.

Nx.;; Ghbtcrozn6

(o) :x: + >x + 2 (oo) 7x: + >x 7

(ooo) 7x: 82x + 7 (ov) :x: 2x + :

Rca. (o) ]`n kovni nxprnssoci os :x: + >x + 2

@nrn, bcnggobonit cg x: 5:, bcnggbonit cg x 5 >, him

bcisthit tnre 5 2.

[n s`haa icw spaot up t`n bcnggobonit cg t`n

eomman tnre o.n. > oitc twc phrts sub` t`ht t`nor

sue os > him prcmubt nquha tc t`n prcmubt cg

bcnggobonit cg x: him bcisthit tnre o.n. : 2 5 7.Banhray, : + 2 5 > him : 2 5 7. Rc, wn rnpahbn

t`n eomman tnre >x jy :x + 2x.

]ùs, wn `hvn

:x: + >x + 2 5 :x: + :x + 2x + 2

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

5 :x(x + 8) + 2(x + 8)

5 (x + 8) (:x + 2)

(oo) ]`n kovni nxprnssoci os 7x: + >x 7

@nrn, bcnggobonit cg x: 5 7, bcnggobonit cg x 5 >,

bcisthit tnre 5 7

[n s`haa icw spaot up t`n bcnggobonit cg x o.n., >oitc twc phrts sub` t`ht t`nor sue os nquha tc

bcnggobonit cg x o.n., > him prcmubt nquha tc t`n

prcmubt cg bcnggobonit cg x: him bcisthit tnre

o.n., 7 7 5 27.

Banhray, ? + (=) 5 > him ? = 5 27. Rc, wn

rnpahbn t`n eomman tnre >x jy ?x =x.

]ùs, wn `hvn

7x: + >x 7 5 7x: + ?x =x 7

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

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

(ooo) ]`n kovni nxprnssoci os 7x: 82x + 7.

@nrn, bcnggobonit cg x: 5 7, bcnggobonit cg x 5 82,

him bcisthit tnre 5 7.

[n s`haa icw spaot up t`n bcnggobonit cg x o.n.

82 oitc twc phrts w`csn sue os 82 him prcmubt

nquha tc prcmubt cg t`n bcnggobonit cg x: himbcisthit tnre o.n, 7 7 5 27. Banhray, =, ? 5

82 him = ? 5 27. Rc, wn wrotn t`n eomman tnre

82x hs =x ?x

]ùs, wn `hvn

7x: 82x + 7 5 7x: =x ?x + 7

5 :x (2x :) 2 (2x :)

5 (2x :) (:x 2)

(ov) ]`n kovni nxprnssoci os :x: 2x + :.

@nrn, bcnggobonit cg x: 5 :, bcnggobonit cg

x 5 2 him bcisthit tnre 5 :.

[n s`haa icw spaot up t`n bcnggobonit cg t`neomman tnre o.n. 2 oitc twc phrts sub` t`ht t`nor

sue os : him t`n prcmubt os nquha tc t`n prcmubt

cg t`n bcnggobonit cg x: him bcisthit tnre o.n.

: : 5 =.

Banhray, = + 8 5 2 him = 8 5 =.

Rc, wn wrotn t`n eomman tnre 2x hs =x + x.

]ùs, wn `hvn

:x: 2x + : 5 :x: =x + x + :

5 :x (x + :) + 8 (x + :)

5 (x + :) (:x + 8)

Nx.;0 Ghbtcrozn6(o) 8:x: :2xy + 84y:

(oo) 8:x: + ;xy 84y:

Rca. (o) ]`n kovni nxprnssoci os 8:x: :2xy + 84y:

@nrn, bcnggobonit cg x: 5 8:, bcnggobonit cg

x 5 :2y, him bcisthit tnre 5 84y:.

Icw, wn spaot up t`n bcnggobonit cg t`n eomman

tnre o.n., :2y oitc twc phrts w`csn sue os :2y

h im p r cm ub t n qu ha t c t `n p rc mu bt c g t `n

b c nggo b on it c g x: h i m b c ist hi t t nre o . n. ,

8: 84y: 5 8:4y:.

Banhray, 8>y 0y 5 :2y him 8>y 0y 5 8:4y :

Rc,wn rnpahbn t n eomman tnre :2xy jy 8>xy 0xy.

]ùs, wn `hvn

8:x: :2xy + 84y: 5 8:x: 8>xy 0xy + 84y:

5 2x (=x >y) :y (=x >y)5 (=x >y) (2x :y)

(oo) ]`n kovni nxprnssoci os 8:x: + ;xy 84y:

@nrn, bcnggobonit cg x: 5 8:, bcnggobonit cg x 5 ;y

him bcisthit tnre 5 84y:.

[n s`haa icw spaot up t`n bcnggobonit cg t`n

eomman tnre o.n. ;y oitc twc phrts w`csn sue os

;y him prcmubt nquha tc t`n prcmubt cg t`n

b c nggo b on it c g x: h im b c ist hi t t n re o . n.

8: 84y:

5 8:4y:

.


18/39


Banhray, 8>y 0y 5 ;y him 8>y 0y 5 8:4y:

Rc, wn rnpahbn t`n eomman tnre ;xy jy 8>xy 0xy

]`us, wn `hvn

8:x: + ;xy 84y: 5 8:x: + 8>xy 0xy 84y:

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

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

Ghbtcrozhtoci cg Xuhmrhtob Tcayiceohas

jy usoik t`n ent`cm cg bcepantoik

t`n pnrgnbt squhrn

Trcbnmurn6

Rtnp O Cjthoi t`n quhmrhtob pcayiceoha. Ant t`n

pcayiceoha jn hx: + jx + b, w`nrn h4.

Rtnp OO Ehln t`n bcnggobonit cg x: uioty jy

movomoik him euatopayoik t`rcuk`cut jy

ot, og ot os ict uioty o.n., wrotn

hx: + jx + b 5 h

h

bx

h

jx

:

Rtnp OOO Hmm him sujtrhbt squhrn cg `hag cg t`n

bcnggobonit cg x o.n., wrotn

hx: + jx + b 5 h

h

bx

h

jx:

5 h

h

b

h:

j

h:

jx

h:

j:x

:::

Rtnp OW [rotn gorst t`rnn tnres hs t`n squhrn cg

h joiceoha him soepaogy ahst twc tnres

o.n., wrotn

hx: + jx + b

5 h

h

b

h:

j

h:

jx

h:

j:x

:::

5 h

:

::

h=

hb=j

h:

jx

Rtnp W Ghbtcrozn ahst stnp cjthoinm oi stnpOW jy usoik h: j: 5 (h j) (h + j) tc

knt mnsornm ghbtcrs.

Nx.;? Ghbtcrozn y: + 7y + 0 jy usoik t`n ent`cm cg

bcepantoik t`n squhrn.

Rca. @nrn, bcnggobonit cg y: os uioty. Rc, wn hmm him

sujtrhbt t`n squhrn cg t`n `hag cg bcnggobonit cg y.

y: + 7y + 0 5 y: + 7y + 2: 2: + 0

::

2:

7ksujtrhbtoihimHmmoik

5 (y: + 7y + 2:) 8

5 (y + 2): 8: \Jy bcepantoik t`n squhrnU

5 {(y + 2) 8} {(y + 2) + 8}

\^soik 6 h: j: 5 (h j) (h + j)U

5 (y + :) (y + =)

Nx.04 Ghbtcrozn6 =y: 0y + 2

Rca. [n `hvn, =y: 0y + 2

5 =

=

2y:y

:\Ehloik bcnggobonit cg y: hs 8U

5 =

=

288y:y :::

:

:

8.,n.o

ycg.Bcngg

:


5 =

=

8)8y:y(

::

5 =

:

:

:

8)8y( \Bcepantoik t`n squhrnU

5 =

:

8)8y(

:

8)8y(

\^soik h: j: 5 (h j) (h + j)U

5 =

:

88y

:

88y

5 =

:

2y

:

8y 5 =

:

2y:

:

8y:

5 (:y 2) (:y 8)

Nx.08 Ghbtcrozn 6 7 x :x:

Rca. [n hvn, 7 x :x: 5 :x: x + 7

5 :

2x

:8x

:

:

xcg.bcnggt`n.,n.o

:jykeuatopayoihimMovomoik

5 :

2

=

8

=

8x

:

8x

:::

::

h

8.,n.oxcg.Bcngg

:



19/39


5 :

2

87

8

=

8x

=

8:x

::

5 :

87

=?=

8x

:

5 :

::

=

;

=

8x

5 :

=

;

=

8x

=

;

=

8x

5 :

=

;

=

8x

=

;

=

8x

5 :

:

2x (x + :)

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

TCA_ICEOHAR

Tcayiceohas 6 Hi haknjrhob nxprnssoci oi wòb`

t`n vhrohjans oivcavnm `hvn ciay ici-inkhtovn

oitnkrha pcwnrs, os bhaanm h pcayiceoha.

Mnkrnn cg h pcayiceoha oi cin vhrohjan6 Oi h

pcayiceoha oi cin vhrohjan, t`n òk`nst pcwnr cgt`n vhrohjan os bhaanm mnkrnn.

Mnkrnn cg h pcayiceoha oi twc vhrohjan6 Oi h

pcayiceoha oi ecrn t`hi cin vhrohjan t`n sue cg

t`n pcwnrs cg t`n vhrohjans oi nhb` tnre os

bceputnm him t`n òk`nst sue sc cjthoinm os

bhaanm t`n mnkrnn cg t`n pcayiceoha.

Bcisthit Tcayiceoha 6 H pcayiceoha bcisostoik

cg h bcisthit tnre ciay os bhaanm h bcisthit

pcayiceoha. ]`n mnkrnn cg h bcisthit pcayiceoha

os znrc.

Aoinhr Tcayiceoha 6 H pcayiceoha cg mnkrnn 8 os

bhaanm h aoinhr pcayiceoha.

Xuhmrhtob Tcayiceoha 6 H pcayiceoha cg mnkrnn

: os bhaanm h quhmrhtob pcayiceoha.

Bujob Tcayiceoha 6 H pcayiceoha cg mnkrnn 2 os

bhaanm h bujob pcayiceoha.

Joquhmrhtob Tcayiceohas 6 H pcayiceoha cg

mnkrnn = os bhaanm h joquhmrhtob pcayiceoha.

Nx.0:2

:x:

:

2x: + x > os h pcayiceoha oi vhrohjans

x w`nrnhs:

8x2 2x: + >x8/: + x 8 os ict h

pcayiceoha, jnbhusn ot bcithois h tnre >x8/:

wòb`

bcithois:

8hs t`n pcwnr cg vhrohjan x, wòb` os

ict h ici-inkhtovn oitnknr.

Nx.02 2 :x: + =x:y + 0y 2

>xy: os h pcayiceoha oi

twc vhrohjans x him y.

Nx.0= (o) :x + 2 os h pcayiceoha oi x cg mnkrnn 8.

(oo) :x: 2x +

>

;os pcayiceoha oi x cg mnkrnn :.

(ooo)2

:h:

:

;h: + = os h pcayiceoha oi h mkrnn 2.

Nx.0> 2x= :x2y: + ;xy2 ?x + >y + = os h pcayiceoha

oi x him y cg mnkrnn >, w`nrnhs:

8 2x + ;x:y

=

2x:y: os h pcayiceoha cg mnkrnn = oi x him y..

Nx.07 : =

2x,

:

8+

>

2y, : + 2h ntb. hrn aoinhr pcayiceohas.

Nx.0; :x: 2x + =, : x + x: , :y: :

2y +

=

8hrn

quhmrhtob pcayiceohas.

Nx.00 x2 ;x + :x 2, : +:

8y

:

2y: + =y2 hrn bujob

pcayiceoha.

Nx.0? 2x= ;x2 + x: x + ?, = 2

:x: +

>

2x= hrn

joquhmrhtob pcayiceohas.

Movosoci cg h eciceoha jy h eciceoha

[òan movomoik h eciceoha jy h eciceoha, wn

gcaacw t`n gcaacwoik twc ruans6

Zuan-8 Bcnggobonit cg t`n quctonit cg twc

eciceoha os nquha tc t`n quctonit cg

t`nor bcnggobonits.

Zuan-: ]`n vhrohjan phrt oi t`n quctonit cg twc

eciceohas os nquha tc t`n quctonit cg

t`n vhrohjans oi t`n kovni eciceohas.


20/39


Nx.?4 Movomn 6

(o) 8:x2y2 jy 2x:y (oo) 8>h:jb2 jy 2hj

Rca. (o) [n `hvn,

yx2

yx8::

:2

5yxx2

yyxxx8:

5 = x y 5 =xy

(oo) [n `hvn,

hj2

jbh8> 2:5

jh2

bbbjhh8>

5 >hb2

Movosoci cg h Tcayiceoha jy h Eciceoha

Rtnp O Cjthoi t`n pcayiceoha (movomnim) himt`n eciceoha (movoscr).

Rtnp OO Hrrhikn t`n tnres cg t`n movomnim oimnsbnimoik crmnr cg t`nor mnkrnns. Gcrnxhepan, wrotn;x: + =x 2 + >x2 hs >x: + ;x: + =x 2.

Rtnp OOO Movomnm nhb` tnre cg t`n payiceoha jyt`n kovni eciceoha jy usoik t`n ruansc g m ov os oc i c g h ec ic eo ha j y heciceoha.

Nx.?8 Movomn 6

(o) ?e> + 8:e= 7e: jy 2e:

(oo) :=x2y + :4x:y: =xy jy :xy

Rca. (o) [n `hvn,

:

:=>

e2

e7e8:e? 5 :

>

e2

e?+

:

=

e2

e8: :

:

e2

e7

5 2e2 + =e: :

(oo) [n `hvn,

xy:

xy=yx:4yx:= ::2

5xy:

yx:= 2

+xy:

yx:4 ::

xy:

xy=5 8:x: + 84xy :

Movosoci cg h Tcayiceoha jy h Joiceoha

jy usoik acik movosoci

Rtnp O Hrrhikn t`n tnres cg t`n movomnim himmovoscr oi mnsbnimoik crmnr cg t`normnkrnns.

Rtnp OO Movomn t`n gorst tnre cg t`n movomnim jyt`n gorst tnre cg t`n movoscr tc ctjhoi t`ngorst tnre cg t`n quctonit.

Rtnp OOO Euatopay t`n movoscr jy t`n gorst tnre cgt`n quctonit him sujtrhbt t`n rnsuat grcet`n movomnim tc cjthoi t`n rnehoimnr.

Rtnp OW Bcisomnr t`n rnehoimnr (og hiy) hsmovomnim him rnpnht stnp OO tc cjthoit`n snbcim tnre cg t`n quctonit.

Rtnp W Znpnht t`n hjcvn prcbnss toaa wn cjthoih rnehoimnr wòb` os not nr znrc cr h

pcayiceoha cg mnkrnn anss t`hi t`ht cgt`n movoscr.

Nx.?: Movomn 7 + x =x: + x2 jy x 2.

Rca. [n kc t`rcuk` t n gcaacwoik stnps tc pnrgcre

t`n movosoci6

Rtnp O [n wrotn t`n tnres cg t`n movomnim hs wnaa hs cg

movoscr oi mnsbnimoik crmnr cg t`nor mnkrnss.

]ùs, wn wrotn5 7 + x =x: + x2 hs x2 =x: + x + 7 him x 2

hs x 2

Rtnp OO [n movomn t`n gorst tnre x2 cg t`n movomnim jy t`n

gorst tnre x cg t`n movoscr him cjthoix

x2

5 x: hs

t`n gorst tnre cg t`n quctonit.

Rtnp OOO [n euatopay t`n movoscr x 2 jy t`n gorst tnre x

cg t`n quctonit him sujtrhbt t`n rnsuat grce t`n

movomnim x2 =x: + x + 7. [n cjthoi x: + x +

7 hs t`n rnehoimnr.

x =x + x + 72 :

x 2x2 :

+

x x + 7:

x + 2x:

+ :x + 7 :x + 7+

4

x 2x x :

:

Rtnp OW [n thln x: + x + 7 hs t`n inw movomnim him

rnpnht stnp OO tc cjthoi t`n snbcim tnre

x

x:

x cg t`n quctonit.

Rtnp W [n euatopay t`n movoscr x 2 jy t`n snbcim tnre

x cg t`n quc tonit him suj trhbt t`n rnsuat

x: + 2x grce t`n inw movomnim. [n cjthoi

:x + 7 hs t`n rnehoimnr.

Rtnp WO Icw wn trnht :x + 7 hs t`n inw movomnim him

movomn ots gorst tnre :x jy t`n gorst tnre x cg t`n

movoscr tc cjthoix

x:5 : hs t`n tòrm tnre cg

t`n quctonit.Rtnp WOO [n euatopay t`n movoscr x 2 him t`n tòrm tnre

: cg t` n qu ct on it hi m su jt rh bt t` n rn su at

:x + 7 grce t`n t`n inw movomnim. [n cjthoi

4 hs t`n rnehoimnr.

]ùs, wn bhi shy t`ht

(7 + x =x: + x2) (x 2) 5 x: x :

cr,2x

xx=x7 2

5 x: x :

]`n hjcvn prcbnmurn os mospahbnm ci t`n rok`t

somn cg t`n hjcvn stnp.


21/39


Ictn 6 Oi t`n hjcvn nxhepan, t`n rnehoimnr os znrc. Rc,

wn bh i s hy t `h t ( x 2 ) os h gh bt cr cg

7 + x =x: + x2.

Nx.?2 Movomn 6 x2 7x: + 88x 7 jy x: =x + 2

Rca. Ci movomoik, wn knt

x 7x + 88x 72 :

x =x + 2x2 :

+

:x + 0x 7:

:x + 0x 7:

+ +4

x =x + 2:

x :

x2 7x: + 88x 7 5 (x :) (x: =x + 2)

Nx.?= ^soik movosoci s`cw t ht 2y: + > os ghbtcr cg

7y> + 8>y= + 87y2 + =y: + 84y 2>.

Rca. Ci movomoik 7y> + 8>y= + 87y2 + =y: + 84y 2>

jy 2y: + >, wn cjthoi

7y + 8>y + 87y + =y + 84y 2>> = 2 :

7y + 84y> 2

8>y + 7y + =y + 84y 2>= 2 :

8>y + :>y= :

4

2y + >:

:y + >y + :y ;2 :

7y :8y + 84y 2>2 :

7y + 84y2

:8y 2>:

:8y 2>:

+ +

Roibn t`n rnehoimnr os znrc. ]`nrngcrn, 2y: + > os h

ghbtcr cg 7y> + 8>y= + 87y2 + =y: + 84y 2>.

Movosoci Hkacrotè6

[n licw t`ht og h iuejnr os movomnm jy hict`nr

iuejnr, t`niMovomnim 5 Movoscr Xuctonit + Znehoimnr

Roeoahray, og h pcayiceoha os movomnm jy hict`nr

pcayiceoha, t`ni

Movomnim 5 Movoscr Xuctonit + Znehoimnr

]òs os kninrhaay licwi hs t`n movosoci hakcrotè.

Nx.?> [`ht eust jn sujtrhbtnm grce 0x= + 8=x2 :x:

+ ;x 0 sc t`ht t`n rnsuatoik pcayiceoha os nxhbtay

movosojan jy =x: + 2x :.

Rca. [n licw t`ht

Movomnim 5 Xuctonit Movoscr + Znehoimnr

Movomnim Znehoimnr 5 Xuctonit Movoscr

Banhray, [email protected] cg t`n hjcvn rnsuat os movosjan jy

t`n movoscr. ]`nrngcrn, [email protected]. os hasc movosojan jy

t`n movoscr. ]ùs, og wn sujtrhbt rnehoimnr grcet`n movomnim, t`ni ot woaa jn nxhbtay movosojan jy

t`n movosocr.

Movomoik 0x= + 8=x2 :x: + ;x 0 jy =x: + 2x :,

wn knt

0x + 8=x :x + ;x 0= 2 :

0x + 7x =x= 2 :

+

0x + :x + ;x 02 :

0x + 7x =x2 :

+

=x + 2x ::

:x + :x 8:

=x + 88x 0:

=x 2x + ::

+ + 8=x 84

Xuctonit 5 :x: + :x 8 him, Znehoimnr 5 8=x 84

]ùs, og wn sujtrhbt t`n rnehoimnr 8=x 84 grce

0x= + 8=x2 :x: + ;x 0, ot woaa jn movosojan jy

=x: + 2x :

Nx.?7 Goim t n vhauns cg h him j sc t`ht x= + x2 + 0x:

+ hx + j os movosojan jy x: + 8.

Rca. Og x= + x2 + 0x: + hx + j os nxhbtay movosojan jyx: + 8, t`ni t`n rnehoimnr s`cuam jn znrc.

Ci movomoik, wn knt

x + x + 0x + hx + j= 2 :

x + x= :

x + ;x + hx + j2 :

x + x2

x + 8:

x + x + ;:

;x + x(h 8) + j:

;x + ;

:

x (h 8) + j ;

Xuctonit 5 x: + x + ; him,

Znehoimnr 5 x (h 8) + j ;

Icw, Znehoimnr 5 4

x (h 8) + (j ;) 5 4

x (h 8) + (j ;) 5 4x + 4

h 8 5 4 him j ; 54

\Bcephroik bcnggobonits cg x him bcisthit tnresU

h 5 8 hi m j 5 ;


22/39


Nx.?; Movomn x= x2 + x: + > jy (x + 8) him wrotn t`n

quctonit him rnehoimnr.

Rca. [n `hvn,

x= x2 + x: + > 5 x2 (x + 8) :x:(x + 8)

+ 2x (x + 8) 2 (x + 8) + 0

5 (x + 8) (x2 :x2 + 2x 2) + 0

@nibn, Xuctonit 5 x2 :x: + 2 x 2 h i m ,

Znehoimnr 5 0.

Nx.?0 Movomn 8:x2 0x: 7x + 84 jy (2x :). Hasc,

wrotn t`n quctonit him t`n rnehoimnr.

Rca. [n `hvn,

8:x2 0x: 7x + 84

5 =x: (2x :) : (2x :) + 7

5 {=x: (2x :) : (2x :)} + 7

5 (2x :) (=x: :) + 7

@nibn, Xuctonit 5 =x: : him, Znehoimnr 5 7.

Nx.?? Movomn 7x2 x: 84x 2 jy (:x 2).

Rca. [n `hvn,

7x2 x: 84x 2

5 2x: (:x 2) + =x (:x 2) 8(:x 2) 7

5 {2x: (:x 2)+ =x (:x 2) 8 (:x 2)} 7

5 (:x 2) (2x: + =x 8) 7

@nibn, Xuctonit 5 2x: + =x 8 him, Znehoimnr

5 7.

Movosoci cg Tcayiceohas jy usoik Ghbtcrozhtoci

Nx.844 Movomn6

(o) 2>h: + 2:h ?? jy ;h ?

(oo) hx: + (j + hb) x + jb jy x + b

Rca. (o) [n `hvn,

2>h: + 2:h ??

5 2>h: + ;;h =>h ??

5 ;h (>h + 88) ? (>h + 88) 5 (>h + 88) (;h ?)

...(o)

(2>h: + 2:h ??) (;h ?)

5

?h;

??h2:h2> :

5)?h;(

)?h;)(88h>( 5 >h + 88 \^soik (o)U

htcreoimnichimiuenrhtcroi

)?h;(ghbtcrbceecibhibnawn,iuejnrshsFust

(oo) [n `hvn,

hx: + (j + hb)x + jb

5 (hx: + jx) + (hbx + jb)

5 x (hx + j) + b (hx + j) 5 (hx + j) (x + b) ...(o)

jbx)hbj(hx: (x + b)

5)bx(

jbx)hbj(hx :

5)bx(

)bx)(jhx(

\^soik (o)U

5 hx + j

htcreoimnichimiuenrhtcroi)bx(ghbtcrbceeciBhibnaaoik

Nx.848 Movomn6 h8: + h7j7 + j8: jy h7 h2j2 + j7

Rca. [n `hvn,

h8: + h7j7 + j8:

5 h8: + :h7j7 + j8: h7j7

\Hmmoik him sujtrhbtoik h7j7U

5 (h7 + j7): (h2j2):

5 (h7 + j7 h2j2) (h7 + j7 + h2j2)

5 (h7 h2j2 + j7) (h7 h2j2 + j7) ...(o)

7227

8:778:

jjhh

jjhh

5)jjhh(

)jjhh)(jjhh(7227

72277227

5 h7 + h2j2 + j7

\Bhibnaaoik h7 h2j2 + j7 grce Ir him MrU


23/39


X.8 Hmm t`n gcaacwoik haknjrhob nxprnssocis6

:,2y:

2y> :

+:y> 2

, 2= +

2y: :

:y ,

2

y> 2+ 2y: + 2y +

>

7

X.: Rujtrhbt 6

2y

:

8y: : grce ;y: :y + 84.

X.2 Rujtrhbt6:

2x:y +

>

=y

2

8x:yz g rc e

>

8:x:yz

>

2xyz +

2

:x:y..

X.= Goim t`n vcauen cg t`n rnbthikuahr jcxns wot`

gcaacwoik anikt`, jrnhmt` him `nok`t 6

Anikt` Jrnhmt` @nok`t

(o) :hx 2jy >bz

(oo) e:i i:p p:e

(ooo) :q =q: 0q2

X.> Goim nhb` cg t n gcaacwoik prcmubts6

(o) (:x:) (;h:x;) (7h>x>)

(oo) (=s:t) (2s2t2) (:st=) (:)

X.7 Euatopay 2xy

2

=jy yx

;

7 :him vnrogy ycur

rnsuat gcr x 5 : him y 5 8.

X.; Goim t`n prcmubt cg >x:y , 2

:xy:z,

8>

0xyz: him

=

8 . Wnrogy t`n rnsuat w`ni x 5 8,

y 5 : him y 5 q.

X.0 Goim t`n prcmubt cg :

;s:t him s + t. Wnrogy t`n

rnsuat gcr s 5:

8him t 5 >.

X.? Goim t`n gcaacwoik prcmubts6

(o) 844x (4.48x= 4.48x:)

(oo) 8:8.>hj

84

jhb

(ooo) 4.8h (4.48h 4.448j)

NPNZBORN # 8

X.84 Hmm6

(o) >e(2 e) him 7e: 82e

(oo) =y(2y: + >y ;) him :(y2 =y: + >)

X.88 (o) Rujtrhbt6 2a (a =e + >i) grce

=a(84i 2e + :a)

(oo) Rujtrhbt 6 2h (h + j + b):j (h + j + b)

grce =b(h + j + b)

X.8: Euatopay

y

=

8x

>

8him (>x: =y:)

X.82 Euatopay (2x: + y:) jy (x: + :y:).

X.8= Euatopay6 {:e + (i)} jy {2e + (>)}

X.8> Goim t`n prcmubt cg

:y

;

:y him (;y y:) him

vnrogy t`n rnsuat gcr y 5 2.

X.87 Roepaogy t`n gcaacwoik6

(o)2

8(7x: + 8>y:) (7x: 8>y:)

(oo) ?x=(:x2 >x=) >x7 (x= 2x:)

X.8; Euatopay6 (:x: =x + >) jy (x: + 2x ;)

X.80 Goim t`n prcmubt cg t`n gcaacwoik joiceohas6

(o) (7x: ;y:) (7x: ;y:)

(oo)

y

>

8x

:

8

y

>

8x

:

8

X.8? Goim t`n prcmubt cg t`n gcaacwoik joiceohas6

(o)

y

7

>x

=

2

y

7

>x

=

2

(oo)

j

2h:

j

2h:

(ooo) (h: + j:) (h: + j:)

(ov) (h + b) (h b)

X.:4 Og x+x

85 ? him x: + :

x

85 >2, goim t`n vhaun cg

x x

8.

X.:8 Og x + y 5 8: him xy 5 8=, goim t`n vhaun cg

x: + y:.


24/39


X.:: Roepaogy t`n gcaacwoik prcmubts6

(o) (x: + x + 8) (x: x + 8)

(oo) (x: + :x + :) (x: :x + :)

X.:2 Roepaogy t`n gcaacwoik jy usoik6 (h + j) (h j)

5 h

:

j

:

.(o) 70 ;: (oo) 848 ??

(ooo) 7; ;2 (ov) 8:0: ;;:

X.:= Goim t n krnhtnst bceeci ghbtcr cg t`n eciceohas

7x2h:j:b, 0x:hj2b2 him 8:h2j:b:.

X.:> Ghbtcrozn6

(o) 8:x2y= + 87x:y> =x>y:

(oo) 80h2j: + 27hj= :=h:j2

X.:7 Ghbtcrozn6

(o) (x + y)(:x + 2y) (:x + 2y) (x + y) (x + 8)

(oo) (x + y) (:h + j) (2x :y) (:h + j)

X.:; Ghbtcrozn 6

(o) x: + xy + 0x + 0y

(oo) 8>xy 7x + 84y =

(ooo) i ; + ;ae aei

X.:0 Ghbtcrozn6

(o) h: + :h + hj + :j

(oo) x: xz + xy xz

X.:? Ghbtcrozn nhb` cg t`n gcaacwoik nxprnssocis6

(o) h:

j + hj h(oo) xy hj + jx hy

(ooo) 7hj j: + 8:hb :jb

(ov) h(h + j b) jb

(v) h:x: + (hx: + 8) x + h

(vo) 2hx 7hy 0jy + =jx

X.24 Ghbtcrozn6

(o) x2 :x:y + 2xy: 7y2

(oo) 7hj j: + 8:hb :jb

X.28 Ghbtcrozn 6

(o) x=

y=

(oo) 87x=

08(ooo) x= (y + z)= (ov) :x 2:x>

(v) 2h= =0j= (vo) 08x= 8:8x:

X.2: Ghbtcrozn nhb` cg t`n gcaacwoik haknjrhob

nxprnssocis6

(o) 87(:x 8): :>z:

(oo) =h: ?j: :j 2j

(ooo) x: =x + =y y:

(ov) 2 8: (h j):

(v) x (x + z) y(y + z)

(vo) h: j: h j

X.22 Ghbtcrozn 6

(o) =x: =xy + y: ?z:

(oo) 87 x: :xy y:

(ooo) x= (x z)=

X.2= Ghbtcrozn 6(o) =(x + y): :0y (x + y) + =?y:

(oo) (:h + 2j): + :(:h + 2j) (:h 2j) + (:h 2j):

X.2> Ghbtcrozn nhb` cg t`n gcaacwoik nxprnssocis6

(o) ?x: =y:

(oo) 27x: 8:x + 8 :>y:

(ooo) h: 8 + :x x:

X.27 Ghbtcrozn6

(o) ? h7 + :h2j2 j7

(oo) x87 y87 + x0 + y0

X.2; Ghbtcrzon6 (:x + 2y): >(:x + 2y) 8=

X.20 Ghbtcrozn6 2e: + :=e + 27

X.2? Movomn6

(o) 7x=yz 2xy2z + 0x:yz= jy :xyz

(oo)2

:h:j:b: +

2

=hj:b2

>

8hj2b: jy

:

8hjb

X.=4 Movomn t`n pcayiceoha :x= + 0x2 + ;x: + =x + 2

jy x + 2.

X.=8 Movomn 84x=

+ 8 ; x2

7: x:

+ 2 4 x 2 j y:x: + ;x 8

X.=: Movomn 2y> + 7y= + 7y2 + ;y: + 0 y + ? j y

2y2 + 8 him vnrogy t`ht

Movomnim 5 Movoscr Xuctonit + Znehoimnr

X.=2 Movomn 87x= + 8:x2 84x: + 0x + :4 jy =x 2.

Hasc, wrotn t`n quctonit him rnehoimnr.

X.== Movomn 0y2 7y: + =y 8 jy =y + :. Hasc, wrotn

t`n quctonit him t`n rnehoimnr.

X.=> Movomn6 h= j= jy h j

X.=7 Movomn6 x=h + x:hy:j + =y=j jy x:h + xhyj + y:j


25/39


HIR[NZR (Nx. # 8)

Rca.88>

:0+

7

8?y + :y: +

7

:>y2

Rca.: ?y: y:

>+ 82

Rca.28>

=8x:yz yx

7

> : xyz

>

2 y

>

=

Rca.= (o) 24 hjbxyz (oo) e2i2p2 (ooo) 7=q7

Rca.> (o) 0=x8=h; (oo) =0s7t0 (ooo) 8444x8=y88

Rca.7 =2

yx;

0

Rca.; === zyx

?

=

Rca.0:

;s2t+

:

;s:t:

Rca.? (o) x> x2

(oo) 8:8.>h:jb + 8:.8>hj:

(ooo) 4.448h: + 4.4448hj

Rca.84 (o) :e + e:

Rca.88 (o) :>ai + >a:

(oo) ;hb + 7jb + =b: 2h: hj :j:

Rca.8: x2 :xy

>

= yx

=

> :+ y2

Rca.82 2x= + ;x:y: + :y=

Rca.8> ;y: + y2 =y

;

:

Rca.87 (o) 8:x= ;>y=

(oo) ::>x80 + ?4x8; + 7;>x87 :;4x8>

Rca.8; :x= + :x2 :8x: + =2x 2>

Rca.80 (o) 27x= 0=:y: + =?y= (oo) :=

y:>

8

>

xyx

=

8

Rca.8? (o) :

x87

?

:y

27

:>(oo) =h: :

j

?

(ooo) j= h= (ov) h: b:

Rca.:4 >

Rca.:8 887 Rca.:: (o) x= + x: + 8, (oo) x= :x: + =

Rca.:2 (o) =0?7 (oo) ???? (ooo) =0?8 (ov) 84=>>

Rca.:7 (o) (x + y) (x + 2y 8) (oo) (:x + 2y) (:h + j)

Rca.:0 (o) (h + :) (h + j) (oo) (x + y) (x z)

Rca.:? (o) (h + j) (h 8) (oo) (y + j) (x h)

(ooo) (j + :b) (7h j) (ov) (h + j) (h b)

(v) (x + h) (hx: + 8) (vo) (2h + =j) (x :y)

Rca.24 (o) (x :y) (x: + 2y:) (oo) (7h j)(j + :b)

Rca.28 (o) (x y)(x + y) (x: + y:)

(oo) (:x 2) (:x + 2) (=x: + ?)

(ooo) (x y z) (x + y + z) {x : + (y + z):}

(ov) :x (8 + =x:) (8 :x) (8 + :x)

(v) 2(h :j) (h + :j) (h : + =j:)

(vo) x: (?x 88) (?x + 88)

Rca.2: (o) (0x >z =) (0x + >z =)(oo) (:h + 2j) (:h 2j 8)

(ooo) (x y) (x + y =)

(ov) 2(8 + :h :j) (8 :h + :j)

(v) (x y) (x + y + z)

(vo) (h + j) {(h j) 8}

Rca.22 (o) (:x y + 2z) (:x y 2z)

(oo) (= + x + y) (= x y)

(ooo) (:x: :xz + z:) (:x z)z

Rca.2= (o) (:x >y): (oo) 87h:

Rca.2> (o) (2x + :y) (2x :y)

(oo) (7x >y 8) (7x + >y 8)

(ooo) (h 8 + x) (h + 8 x)

Rca.27 (o) (h2 j2 + 2) (h2 + j2 + 2)

(oo) (x0 + y0) (x0 y0 + 8)

(ooo) (p + q h + j) (p + q + h j + 8)

Rca.2; (:x + 2y ;) (:x + 2y + :)

Rca.20 2(e + :) (e + 7)

Rca.2? (o) 2x2 :

y:

2+ =xz2

(oo) hjb2= + jb

20 bj

>:

Rca.=4 (x + 2) (:x2 + :x: + x + 8)

Rca.=8 (:x: + ;x 8) (>x: ?x + 2)

Rca.== (=y + :)

=

?y

:

>y: :

:

88

Rca.=> (h + j)(h: + j:)

Rca.=7 x:h xhyj + y:j


26/39


X.8 Og

x

8x

5 2, t`ni

:

:

x

8x

os nquha tc -

(H)2

84(J)

?

0:(B) ; (M) 88

X.: Og

x

8x 5

:

8, t`ni

:

:

x

=x= os nquha tc -

(H) ; (J) ; (B) ? (M) ?

X.2 Og

x

8x 5 =, t`ni

=

=

x

8x os nquha tc -

(H) 8?7 (J) 8?= (B) 8?: (M) 8?4

X.= Og

x8x 5 2 , t`ni t`n vhaun cg

2

2

x

8x

os -

(H) 4 (J) 2 2

(B) 2( 2 8) (M) 2( 2 + 8)

X.> Og

x

8x 5 :, t`ni

2

2

x

8x os nquha tc -

(H) 7= (J) 8= (B) 0 (M) :

X.7 Og

:

:

x

8x 5 84:, t`n vhaun cg

x

8x os -

(H) 0 (J) 84 (B) 8: (M) 82

X.; Og

=

=

x

8x 5 2::, t`n vhaun cg

x

8x os -

(H) = (J) 7 (B) 0 (M) 2 :

X.0 Og

2

2

x

8x 5 >:, t`n vhaun cg

x

8x os -

(H) = (J) 2 (B) 7 (M) 82

X.? Og

22

x8x 5 8=, t`n vhaun cg

x8x os -

(H) > (J) = (B) 2 (M) :

X.84 Og x os hi oitnknr sub` t`ht

x

8x 5

=

8;,

t`ni t`n vhaun cg

x

8x os -

(H) = (J)=

82(B)

=

8>(M)

=

8

X.88 Og

=

=

x

8x

5 ;:;, t`n vhaun cg

2

2

x

8x

os

(H) 8:> (J) 8=4 (B) 8>> (M) 8;4

X.8: Og

x

2x: 5 >, t`n vhaun cg

:

:

x

?x= os -

(H) :> (J) 24 (B) 2> (M) =?

X.82 Og

x

8x 5 2, t`n vhaun cg

7

7

x

8x os -

(H) ?:; (J) =8= (B) 27= (M) 2::

X.8= Og t: =t + 8 5 4, t`ni t`n vhaun cg

2

2

t

8t os -

(H) == (J) =0 (B) >: (M) 7=

X.8> Og x + y 5 ; him xy 5 8:, t`n vhaun cg (x: + y:)

os -

(H) :> (J) :? (B) 2; (M) =?

X.87 Og x + y 5 > him xy 5 7, t`n vhaun cg (x2 + y2) os-

(H) ?8 (J) 822 (B) :8; (M) 2=2

X.8; Og x + y 5 > him xy 5 7, t`n vhaun cg (x2 y2) os-

(H) 8? (J) 8? (B) 72 (M) 72

X.80 Og x8/2

+ y8/2

+ z8/2

5 4, t`ni -(H) x + y + z 5 4 (J) (x + y + z)2 5 :; xyz

(B) x + y + z 5 2xyz (M) x2 + y2 + z2 5 4

X.8? Og h + j + b 5 4, t`ni (h2 + j2 + b2) os nquha tc-

(H) 4 (J) hjb

(B) 2hjb (M) (hj + jb + bh)

X.:4 Og h + j + b 5 4, t ni t`n vhaun cg

hj

b

bh

j

jb

h :::

os -

(H) 8 (J) 4 (B) 8 (M) 2X.:8 Og x + y + z 5 ? & xy + yz + zx 5 :2, t ni t`n vhaun

cg (x2 + y2 + z2 2xyz) os -

(H) 840 (J) :4; (B) 77? (M) ;:?

X.:: Og8:>

>x

5 8, t`ni x os nquha tc -

(H) > (J) : (B) 4 (M) 2

X.:2 Og 2x 2x8 5 80, t`ni t`n vhaun cg xx os -

(H) 2 (J) 0 (B) :; (M) :87

NPNZBORN # :


27/39


X.:= Og :x :x8 5 87, t`ni t`n vhaun cg x: os -

(H) = (J) ? (B) 87 (M) :>

X.:> Og x him y hrn ici-znrc rhtociha uinquha iuejnrs,

t`ni::

::

xyyx

)yx()yx(

os nquha tc -

(H)xy

8(J)

yx

8

(B)

yx

=

(M)

yx

:

X.:7 Og)h:bj)(bj(

x

5

)j:hb)(hb(

y

5)b:jh)(jh(

z

, t`n vhaun cg (x + y + z) os-

(H) h + j + b (J) h: + j: + b:

(B) 4 (M) oimntnreoihtn

X.:; Antj

h

h

j5 x 6 y. Og (x y) 5

h

j

j

h, t`ni

x os nquha tc -

(H)h

jh(J)

j

jh

(B)h

jh(M) Icin cg t`nsn

X.:0 Og (x :) os h ghbtcr cg (x: + 2qx :q), t`ni t`n

vhaun cg q os -(H) : (J) : (B) 8 (M) 8

X.:? Og x2 + 7x: + =x + l os nxhbtay movosojan jy (x + :),

t`ni t`n vhaun cg l os -

(H) 7 (J) ; (B) 0 (M) 84

X.24 Ant g(x) 5 x2 7x: + 88x 7. ]`ni, wòb` cin cg

t`n gcaacwoik os ict ghbtcr cg g(x) 3

(H) x 8 (J) x : (B) x + 2 (M) x 2

X.28 ]`n pcayiceoha (x= >x2 + >x: 84x + :=) `hs

h ghbtcr hs -

(H) x + = (J) x :

(B) x + : (M) Icin cg t`nsn

X.2: [òb` cg t`n gcaacwoik sthtnenits hrn bcrrnbt 3

8. x + 2 os h ghbtcr cg x 2 + :x: + 2x + 80

:. x + : os h ghbtcr cg x 2 + :x: x :

2. x + 8 os h ghbtcr cg x2 + x: =x =

=. x : os h ghbtcr cg :x2 2x + =

(H) :, 2, = (J) 8, 2, =

(B) 8, :, = (M) 8, :, 2

X.22 (x:? x:> + x82 8) os movosojan jy -

(H) jct` (x 8) & (x + 8)

(J) (x 8) jut ict jy (x + 8)

(B) (x + 8) jut ict jy (x 8)

(M) inot`nr (x 8) icr (x + 8)

X.2= Whaun cg l gcr wòb` (x 8) os h ghbtcr cg

(x2 l) os -

(H) 8 (J) 8 (B) 0 (M) 0

X.2> ]`n ghbtcrs cg (0x2 :;y2) hrn -

(H) (:x 2y) (=x: + ?y: 7xy)

(J) (:x 2y) (=x: + ?y: + 7xy)

(B) (:x 2y) (=x: ?y: 7xy)

(M) (:x 2y) (=x: ?y: + 7xy)

X.27 ]`n ghbtcrs cg (x2 + y2 + :x: :y:) hrn -

(H) (x + y) (x: + y: + xy + :x :y)(J) (x + y) (x: + y: xy + :x :y)

(B) (x + y) (x: + y: + :x :y)

(M) Icin cg t`nsn

X.2; ]`n ghbtcrs cg (x2 >x: + 0x =) hrn -

(H) (x + :) (x :) (x 8)

(J) (x + 8) (x + :) (x :)

(B) (x :): (x 8)

(M) (x :): (x + 8)

X.20 ]`n ghbtcrs cg (x= + =) hrn -

(H) (x: + :):

(J) (x: + :) (x: :)

(B) (x: + :x + :) (x: :x + :)

(M) Icin cg t`nsn

X.2? (x + y)2 (x y)2 bhi jn ghbtcroznm hs -

(H) :y(2x: + y:) (J) :x (2x: + y:)

(B) :y (2y: + x:) (M) :x (x: + 2y:)

X.=4 ]`n ghbtcrs cg (x2 + 0y2) hrn -

(H) (x + :y) (x: :xy + =y:)

(J) (x + :y) (x: + :xy + =y:)

(B) (x + :y) (x :y):

(M) (x + :y)2

X.=8 Og (x + :) him (x 8) hrn t`n ghbtcrs cg

(x2 + 84x: + ex + i), t`n vhauns cg e him i hrn-

(H) e 5 >, i 5 2 (J) e 58;, i 5 0

(B) e 5 ;, i 5 80 (M) e 5 :2, i 5 8?

X.=: Ci movomoik (x2 7x + ;) jy (x + 8), t`n rnehoimnr

os -

(H) : (J) 8: (B) 4 (M) ;


28/39


X.=2 Og (x> ?x: + 8:x 8=) os movomnm jy (x 2), t`n

rnehoimnr os -

(H) 80= (J) >7 (B) : (M) 8

X.== [`ni (x= 2x2 + :x: >x + ;) os movomnm jy

(x :), t`n rnehoimnr os -(H) 2 (J) 2 (B) : (M) 4

X.=> Og >x2 + >x: 7x + ? os movomnm jy (x + 2), t`n

rnehoimnr os -

(H) 82> (J) 82> (B) 72 (M) 72

X.=7 Og (x88 + 8) os movomnm jy (x + 8), t`n rnehoimnr

os -

(H) 4 (J) : (B) 88 (M) 8:

X.=; ]`n vhaun cg nxprnssoci (87x: + :=x + ?) gcr

x 5

=

2os -

(H) : (J) 8 (B) 4 (M) 8

X.=0 Og :x2 + >x: =x 7 os movomnm jy :x + 8, t`n

rnehoimnr os -

(H) :

82(J) 2 (B) 2 (M) 7

X.=? Og x2 + >x: + 84 l anhvns rnehoimnr :x w`ni

movomnm jy x: + :, t`ni t`n vhaun cg l os -

(H) : (J) 8 (B) 8 (M) :

X.>4 [`ni (x2 :x : + p x q ) o s m ov om nm j y

(x: :x 2), t`n rnehoimnr os (x 7 ). ]`n vhauns

cg p him q hrn -

(H) p 5 :, q 5 7 (J) p 5 :, q 5 7

(B) p 5 :, q 5 7 (M) p 5 :, q 5 7

X.>8 ]`n sue cg (x: + 8) him t`n rnboprcbha cg

(x: 8) os -

(H) :x: (J)8x

x:

=

(B)8x

:x:x:

:

(M)

8x

x:=

:

X.>: Gcr ehloik (x= 88x:y: + y=) h pnrgnbt squhrn,

t`n nxprnssoci tc jn hmmnm os -

(H) >x:y: (J) ?x:y:

(B) >x:y: (M) ;x:y:

X.>2 (x= + >x2 + 7x:) os nquha tc -

(H) x(x +2) (x: + :) (J) x: (x + 2) (x + :)

(B) x: (x :) (x 2) (M) x(x: + 2) (x + :)

X.>= Og (x2/: xy8/: + x8/: y y2/:) os movomnm jy

(x8/: y8/:), t`ni t`n quctonit os -

(H) x + y (J) x y

(B) x8/: + y8/: (M) x: y:

X.>> ]`n ghbtcrs cg (x= + 7:>) hrn -

(H) (x: :>), (x: + :>)

(J) (x: + :>), (x: + :>)

(B) (x: 84x + :>), (x: + >x + :>)

(M) mc ict nxost

X.>7 ]`n ghbtcrs cg

08

x

x

= =

= hrn -

(H) ict pcssojan

(J)

?

x

x

: :

:

?

x

x

: :

:

(B)

2

:

?

x

x

: :

:

2

:

?

x

x

: :

:

(M)

2:

?x

x:

:

:

2:

?x

x:

:

:

X.>; ]`n ghbtcrs cg (x: 8 :h h:) hrn -

(H) (x h + 8) (x h 8)

(J) (x + h 8) (x h + 8)

(B) (x + h + 8) (x h 8)

(M) Icin cg t`nsn

X.>0 ]`n ghbtcrs cg (x: 0x :4) hrn -

(H) (x + 84) (x :) (J) (x 84) (x + :)

(B) (x >) (x + =) (M) (x + >) (x =)

X.>? ]`n ghbtcrs cg (x: xy ;:y:) hrn -

(H) (x 0y) (x + ?y) (J) (x ?y) (x + 0y)

(B) (x y) (x + ;:y) (M) (x 7y) (x + 8:y)

X.74 ]`n ghbtcrs cg (x: 88xy 74y:) hrn -

(H) (x + 8>y) (x =y)

(J) (x 8>y) (x + =y)

(B) (8>x + y) (=x y)

(M) Icin cg t`nsn

X.78 ]`n ghbtcrs cg (x= + x: + :>) hrn -

(H) (x:

+ > + 2x) (x:

+ > 2x)(J) (x: + 2x + >) (x: + 2x >)

(B) (x: + x + >) (x: x + >)

(M) Icin cg t`nsn

X.7: ]`n ghbtcrs cg (x= ;x:y: + y=) hrn -

(H) (x: + y: 2xy) (x: + y: + 2xy)

(J) (x: y: 2xy) (x: y: + 2xy)

(B) (x: 2xy + y:) (x: 2xy y:)

(M) Icin cg t`nsn


29/39


X.72 ]`n ghbtcrs cg (x: + =y: + =y =xy :x 0) hrn-

(H) (x :y =) (x :y + :)

(J) (x y + :) (x =y =)

(B) (x + :y =) (x + :y + :)

(M) Icin cg t`nsn

X.7= ]`n ghbtcrs cg (x: + xy :y:) hrn -

(H) (x :y) (x + y) (J) (x + :y) (x y)

(B) (x :y) (x y) (M) (x + :y) (x + y)

X.7> ]`n ghbtcrs cg (x2 x:y xy: + y2) hrn -

(H) (x + y) (x: + y: xy)

(J) (x + y) (x: + y: + xy)

(B) (x + y): (x y)

(M) (x y): . (x + y)

X.77 ]`n ghbtcrs cg (:87x2 7=y2) hrn -

(H) 0(2x :y) (?x: + =y: 7xy)

(J) 0(2x :y) (?x: =y: 7xy)

(B) 0(2x :y) (?x: + =y:)

(M) 0(2x :y) (?x: + =y: + 7xy)

HIR[NZ LN_

X.Ic 8 : 2 = > 7 ; 0 ? 84 88 8: 82 8= 8> 87 8; 80 8? :4

His. B B J H M J H H M B J B M B H B J J B M

X.Ic :8 :: :2 := : > : 7 :; :0 :? 24 2 8 2: 22 2= 2> 27 2; 20 2? = 4

His. H M B M B B M M B B J M J J J J B B H H

X.Ic =8 =: =2 == = > = 7 =; =0 =? >4 > 8 >: >2 >= >> >7 >; >0 >? 7 4

His. B J H J M H B B B B J J J H M M B J J J

X.Ic 78 7: 72 7= 7 > 7 7

His. H H H J M M


30/39


@oits & Rcautoci # 8

Rca.8 Znquornm sue

5 : +2

y:

2y>

:

+:y>

2

2

=+

2y:

:

:

y

+2

y> 2+ 2y: + 2y +

>

7

5 : 2

=+

>

7+

2

y:

:

y+ 2y

2

y> :

+2

y: :

+ 2y: +:

y> 2

+2

y> 2

5

>

7

2

=:

+

2

:

8

2

:

y +

2

2

:

2

>

y:

+

:

>

:

>y2

5

8>

80:424+

7

802=y +

2

?:>y:

+

7

848>y2

58>

:0+

7

8?y + :y: +

7

:>y2

Rca.: ]`n rnquornm moggnrnibn os kovni jy

(;y: :y + 84)

2y

:

8y: :

5 ;y: :y + 84 + :y: :

8y + 2

5 ;y: + :y: :y :

8y + 84 + 2

\Krcupoik aoln tnresU

5 (; + :)y: +

:

8: y + 82

5 ?y: y:

>+ 82

Rca.2 [n `hvn,

yx

2

:xyz

>

2yzx

>

8: ::

yzx

2

8y

>

=yx

:

2 ::

5>

8:x:yz

>

2xyz +

2

:x:y

:

2x:y

>

=y

+ yzx2

8 :

5

>

8:x:yz +

2

8x:yz +

2

:x:y

:

2x:y

>

2xyz

>

=y

\Krcupoik aoln tnresU

5

2

8

>

8:x:yz +

:

2

2

:x:y

>

2xyz y

>

=

58>

=8x:yz yx

7

> : xyz

>

2 y

>

=

Rca.= [n licw t`ht t`n vcauen cg h rnbthikuahr jcx os

kovni jy

Wcauen 5 Anikt` Jrnhmt` @nok`t

(o) Wcauen 5 :hx 2jy > bz

5 (: 2 >) (hx jy bz)5 24 hjbxyz

(oo) Wcauen 5 e:i i:p p:e 5 e:+8i8+:p8+:

5 e2i2p2

(ooo) Wcauen 5 :q =q: 0q2 5 (: = 0)q8+:+2

5 7=q7

Rca.> (o) [n `hvn,

(:x:) (;h:x;) (7h>x>)

5 (: ; 7) (x: x; x> h: h>)

5 0=x:+;+>h:+> 5 0=x8=h;

(oo) [n `hvn,(=s:t) (2s2t2) (:st=) (:)

5 (= 2 : :)(s: s2 s t t2 t=)

5 =0s:+2+8t8+2+= 5 =0s7t0

(ooo) [n `hvn,

(>x7) (84xy=) (:x7y7) (84xy)

5 (> 84 : 84) (x7 x x7 x

y= y7 y)

5 8444x7+8+7+8y=+7+8 5 8444x8=y88

Rca.7 [n `hvn,

2xy2

=

yx;

7 :

5

;

7

2

=

(x x: y2 y)

5 82:8

yx;

0 5

=2yx

;

0

Wnrogobhtoci6 Gcr x 5 : him y 5 8, wn `hvn

[email protected]. 5

2xy2

=

yx

;

7 :

5

2)8(:

2

=

8):(

;

7 :5

2

0

;

:=5

;

7=

him,


31/39


[email protected]. 5 =2

yx;

05

;

0 :2 (8)= 5

;

7=

@nibn, gcr x 5 : him y 5 8, wn `hvn

[email protected]. 5 [email protected].

Rca.; [n `hvn,

(>x:y) >

zxy

2

:

:

zxy

8>

0 :

z

=

8

5

=

8

8>

03. algebraic expressions factorization

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