Simplification and Factorisation

8/15/2019 Simplification and Factorisation

1/22

Ucaphcflebtcl`

b`d Gbetlrcsbtcl`

2.5C`trlduetcl`

C` tncs Uketcl` wk kxphbc` wnbt cs akb`t oy tnk pnrbsk „hcjk tkras– b`d snlw nlw hcjk tkras brkelhhketkd tlmktnkr b`d scaphcflkd.

@kxt wk el`scdkr rkalvc`m orbejkts. C` lrdkr tl scaphcgy b` kxprksscl` wncen el`tbc s̀ orbejkts ctcs lgtk` `kekssbry tl rkwrctk tnk kxprksscl` c` b` kqucvbhk`t glra out wctnlut b`y orbejkts. Tncs

prlekss lg rkalvc`m orbejkts aust ok ebrrckd lut beelrdc`m tl pbrtceuhbr ruhks wncen brk dksercokd c`tncs Uketcl .̀

Gc`bhhy, gbetlrcsbtcl`, wncen eb` ok el`scdkrkd bs tnk rkvkrsk lg tnk prlekss, cs dkbht wctn. Ct cskssk`tcbh tnbt ylu nbvk nbd phk`ty prbetcek c` rkalvc`m orbejkts okglrk ylu study gbetlrcsbtcl`.

VrkrkqucsctksOkglrk stbrtc`m tncs Uketcl` ylu snluhd . . .• ok gbachcbr wctn bhmkorbce `ltbtcl`

• nbvk elapktk`ek c` rkalvc`m orbejkts

Hkbr`c`m Lutelaks

L` elaphktcl` ylu snluhd ok bohk tl . . .

• usk tnk hbws lg c`dceks

• scaphcgy kxprksscl`s oy elhhketc`m hcjk tkras

• usk tnk hbws lg c`dceks

• cdk`tcgy elaal` gbetlrs c` b` kxprksscl`

• gbetlrcsk scaphk kxprksscl`s

• gbetlrcsk qubdrbtce kxprksscl`s

46 NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb

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2/22

2. Bddctcl` b`d suotrbetcl` lg hcjk tkras

Hcjk tkras brk auhtcphks lg tnk sbak qub`tcty. Glr kxbaphk ;y, 2=y b`d 2?

y brk bhh auhtcphks lg yb`d sl brk hcjk tkras. Ucachbrhy, 5x?, ∟;x? b`d 2

4x? brk bhh auhtcphks lg x? b`d sl brk hcjk tkras.

Gurtnkr kxbaphks lg hcjk tkras brk9jx b`d x wncen brk oltn auhtcphks lg x,

x?y, 7x?y, ∟25x?y, ∟?yx?, wncen brk bhh auhtcphks lg x?y

boe?, ∟=boe?, jboe?, brk bhh auhtcphks lg boe?

Hcjk tkras eb` ok bddkd lr suotrbetkd c` lrdkr tl scaphcgy kxprksscl`s.

Kxbaphk ?=Ucaphcgy ;x ∟ 25x + ??x.

Ulhutcl`

Bhh tnrkk tkras brk auhtcphks lg x b`d sl brk hcjk tkras. Tnk kxprksscl` eb` ok scaphcflkd tl 24x.

Kxbaphk ?>Ucaphcgy ;z + ?x.

Ulhutcl`

;z b`d ?x brk `lt hcjk tkras. Tnky brk `lt auhtcphks lg tnk sbak qub`tcty. Tncs kxprksscl` eb``ltok scaphcflkd.

Tbsj sjUcaphcgy ;b + ?o ∟ =b ∟ 3o.

Slur slhutcl`

;b + ?o ∟ =b ∟ 3o 1

B`swkr

∟?b ∟ =o

NKHA (?66;)9Uketcl` 2.59 Ucaphcflebtcl` b`d Gbetlrcsbtcl`

42



3/22

Kxbaphk ?3Ucaphcgy ?x? ∟ =x + 22x? + x.

Ulhutcl`

?x? b`d 22x?, oltn okc`m auhtcphks lg x?, eb` ok elhhketkd tlmktnkr b`d bddkd tl mcvk 25x?.

Ucachbrhy, ∟=x b`d x eb` ok bddkd tl mcvk ∟7x.

]k mkt ?x? ∟ =x + 22x? + x 1 25x? ∟ 7x wncen eb``lt ok scaphcflkd gurtnkr.

Tbsj sjUcaphcgy

2

?x + 5

4x ∟ ?y.

Slur slhutcl`2

?x + 5

4x ∟ ?y 1

B`swkr;

4

x ∟ ?y

Kxbaphk 56Ucaphcgy 5b?o ∟ =b?o ∟ ?o? + b?.

Ulhutcl`

@ltk tnbt 5b?

o b`d =b?

o brk oltn auhtcphks lg b?

o b`d sl brk hcjk tkras. Tnkrk brk `l ltnkr hcjktkras. Tnkrkglrk

5b?o ∟ =b?o ∟ ?o? + b? 1 ∟4b?o ∟ ?o? + b?

4? NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



4/22

Kxkrecsks

2. Ucaphcgy, cg plsscohk,

(b) ;x + ?x + 5x, (o) 5q ∟ ?q + 22q , (e) =x? + 22x?, (d) ∟22v? + ?v?, (k) ; p + 5q

?. Ucaphcgy, cg plsscohk, (b) ;w + 5r ∟ ?w + r, (o) ;w? + w + 2, (e) 7w? + w? ∟ 5w?


(b) =x + ? + 5x + >x ∟ 22, (o) ?x? ∟ 5x + 7x ∟ ?, (e) ∟;x? ∟ 5x? + 22x + 22,

(d) 4q ? ∟ 4r? + 22r + 7q , (k) b? + ob + bo + o?, (g) 5x? + 4x + 7x + >,

(m) s5 + 5s? + ?s? + 7s + 4s + 2?.

4. Kxphbc` tnk dcstc`etcl`, cg b`y, oktwkk` kben lg tnk glhhlwc`m kxprksscl`s, b`d scaphcgy cg plsscohk.

(b) 2>x ∟ 3x, (o) 2>x(3x), (e) 2>x(∟3x), (d) ∟2>x ∟ 3x, (k) ∟2>x(3x);. Kxphbc` tnk dcstc`etcl`, cg b`y, oktwkk` kben lg tnk glhhlwc`m kxprksscl`s, b`d scaphcgy cg plsscohk.

(b) 4x ∟ ?x, (o) 4x(∟?x), (e) 4x(?x), (d) ∟4x(?x), (k) ∟4x ∟ ?x, (g) (4x)(?x)


(b) ?

5x? +

x?

5 , (o) 6.;x? + 5

4x? ∟ 22

? x, (e) 5x5 ∟ 22x + 5yx + 22,

(d) ∟4βx? + οx? wnkrk β b`d ο brk el`stb`ts.

B`swkrs

2. (b) 26x, (o) 2?q , (e) 2>x?, (d) ∟3v?, (k) eb``lt ok scaphcflkd.

?. (b) 5w + 4r, (o) eb``lt ok scaphcflkd, (e) 4w?

5. (b) 2>x ∟ 3, (o) ?x? + 5x ∟ ?, (e) ∟>x? + 22x + 22, (d) eb``lt ok scaphcflkd,

(k) b? + ?bo + o?, (g) 5x? + 26x + >, (m) s5 + ;s? + 26s + 2?

4. (b) 3x, (o) 27?x?, (e) ∟27?x?, (d) ∟?=x, (k) ∟27?x?

;. (b) 4x ∟ ?x 1 ?x, (o) 4x(∟?x) 1 ∟>x?

, (e) 4x(?x) 1 >x?

, (d) ∟4x(?x) 1 ∟>x?

,(k) ∟4x ∟ ?x 1 ∟7x, (g) (4x)(?x) 1 >x?

7. (b) x?, (o) 2.?;x? ∟ 22

? x, (e) eb``lt ok scaphcflkd, (d) (ο ∟ 4β)x?


45



5/22


6/22

Tnk eruecbh dcstc`etcl` oktwkk` tnk rlhk lg tnk gbetlr 3 c` tnk twl kxprksscl`s 3(? + y) b`d 3(?y) c`Kxbaphk 52 snluhd ok `ltkd.

Kxbaphk 5?_kalvk tnk orbejkts grla 3(x + ?y).

Ulhutcl`

C` tnk kxprksscl` 3(x + ?y) tnk 3 aust auhtcphy oltn tnk x b`d tnk ?y c` tnk orbejkts. Tnus

3(x + ?y) 1 3x + 3(?y)

1 3x + 2>y

Tbsj sj_kalvk tnk orbejkts grla 3(?x + 5y).

_kakaokr tnbt tnk 3 aust auhtcphy oltn tnk tkra ?x b`d tnk tkra 5y9

Slur slhutcl`

3(?x + 5y) 1

B`swkr

2>x + ?=y

Kxbaphk 55_kalvk tnk orbejkts grla ∟5(;x ∟ z ).

Ulhutcl`

Tnk `uaokr ∟5 aust auhtcphy oltn tnk ;x b`d tnk z .

∟5(;x ∟ z ) 1 (∟5)(;x) ∟ (∟5)(z )

1 ∟2;x + 5z


4;



7/22


8/22

Tbsj sj_kalvk tnk orbejkts grla a(a ∟ `).

C` tnk kxprksscl` a(a ∟ `) tnk flrst a aust auhtcphy oltn tkras c` tnk orbejkts9

Slur slhutcl`

a(a ∟ `) 1

B`swkr

a? ∟ a`

Kxbaphk 5;_kalvk tnk orbejkts grla tnk kxprksscl` ;x∟ (5x + 2) b`d scaphcgy tnk rksuht oyelhhketc`m hcjk tkras.

Ulhutcl`

Tnk orbejkts c` ∟(5x + 2) wkrk rkalvkd c` Kxbaphk 54 l` pbmk 47.

;x ∟ (5x + 2) 1 ;x ∟ 2(5x + 2)

1 ;x ∟ 5x ∟ 21 ?x ∟ 2

Kxbaphk 57

Unlw tnbt ∟x ∟ 2

4 , ∟(x + 2)

4 b`d ∟

x + 2

4 brk bhh kqucvbhk`t kxprksscl`s.

Ulhutcl`

El`scdkr ∟(x + 2). _kalvc`m tnk orbejkts wk lotbc` ∟x ∟ 2 b`d sl

∟x ∟ 2

4 cs kqucvbhk`t tl

∟(x + 2)

4

B `kmbtcvk qub`tcty dcvcdkd oy b plsctcvk qub`tcty wchh ok `kmbtcvk. Nk`ek

∟(x + 2)

4 cs kqucvbhk`t tl ∟

x + 2

4

Slu snluhd study bhh tnrkk kxprksscl`s ebrkguhhy tl rkelm`csk tnk vbrckty lg kqucvbhk`t wbys c` wncen

wk eb` wrctk b` bhmkorbce kxprksscl`.


4=



9/22

Ulaktcaks tnk orbejktkd kxprksscl` eb` bppkbr l` tnk hkgt, bs c` (b + o)e. Tl rkalvk tnk orbejktsnkrk wk usk tnk glhhlwc`m ruhks9

Jky Vlc`t 2;

(b + o)e 1 be + oe

(b ∟ o)e 1 be ∟ oe

@ltk tnbt wnk` tnk orbejkts brk rkalvkd oltn tnk tkras c` tnk orbejkts auhtcphy e.

Kxbaphk 5=_kalvk tnk orbejkts grla (?x + 5y)x.

Ulhutcl`

Oltn tkras c` tnk orbejkts auhtcphy tnk x lutscdk. Tnus

(?x + 5y)x 1 ?x(x) + 5y(x)

1 ?x? + 5yx

Tbsj sj_kalvk tnk orbejkts grla (b) (x + 5)(∟?), (o) (x ∟ 5)(∟?).

Slur slhutcl`

(b) (x + 5)(∟?) 1

B`swkr

Oltn tkras c` tnk orbejkt aust auhtcphy tnk ∟?, mcvc`m ∟?x ∟ 7

Slur slhutcl`

(o) (x ∟ 5)(∟?) 1

B`swkr

∟?x + 7

4> NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



10/22

5. _kalvc`m orbejkts grla kxprksscl`s lg tnk glra(b + o)(e + d)(b + o)(e + d)(b + o)(e + d)

Ulaktcaks ct cs `kekssbry tl el`scdkr twl orbejktkd tkras auhtcphckd tlmktnkr. C` tnk kxprksscl`(b + o)(e + d), oy rkmbrdc`m tnk flrst orbejkt bs b sc`mhk tkra wk eb` usk tnk rksuht c` Jky Vlc`t24 tl wrctk ct bs (b + o)e + (b + o)d. _kalvc`m tnk orbejkts grla kben lg tnksk tkras prldueks

be + oe + bd + od. Alrk el`ecskhy9

Jky Vlc`t 27

(b + o)(e + d) 1 (b + o)e + (b + o)d 1 be + oe + bd + od

]k skk tnbt kben tkra c` tnk flrst orbejktkd kxprksscl` auhtcphcks kben tkra c` tnk skel`d orbejktkdkxprksscl`.

Kxbaphk 5>

_kalvk tnk orbejkts grla (5 + x)(? + y)

Ulhutcl`

]k fl`d (5 + x)(? + y) 1 (5 + x)(?) + (5 + x)y

1 (5)(?) + (x)(?) + (5)(y) + (x)(y) 1 7 + ?x + 5y + xy

Kxbaphk 53_kalvk tnk orbejkts grla (5x + 4)(x + ?) b`d scaphcgy ylur rksuht.

Ulhutcl`

(5x + 4)(x + ?) 1 (5x + 4)(x) + (5x + 4)(?)

1 5x? + 4x + 7x + > 1 5x? + 26x + >


43



11/22

Kxbaphk 46_kalvk tnk orbejkts grla (b + o)? b`d scaphcgy ylur rksuht.

Ulhutcl`

]nk` b qub`tcty cs squbrkd ct cs auhtcphckd oy ctskhg. Tnus

(b + o)? 1 (b + o)(b + o) 1 (b + o)b + (b + o)o

1 b? + ob + bo + o? 1 b? + ?bo + o?

Jky Vlc`t 2=

(b + o)? 1 b? + ?bo + o?

(b ∟ o)? 1 b? ∟ ?bo + o?

Tbsj sj_kalvk tnk orbejkts grla tnk glhhlwc`m kxprksscl`s b`d scaphcgy tnk rksuhts.

(b) (x + =)(x + 5), (o) (x + 5)(x ∟ ?),

Slur slhutcl`

(b) (x + =)(x + 5) 1

B`swkr

x? + =x + 5x + ?2 1 x? + 26x + ?2

Slur slhutcl`

(o) (x + 5)(x ∟ ?) 1

B`swkr

x? + 5x ∟ ?x ∟ 7 1 x? + x ∟ 7

;6 NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



12/22

Kxbaphk 42Kxphbc` tnk dcstc`etcl` oktwkk` (x + 5)(x + ?) b`d x + 5(x + ?).

Ulhutcl`

C` tnk flrst kxprksscl` rkalvc`m tnk orbejkts wk fl`d

(x + 5)(x + ?) 1 x? + 5x + ?x + 7

1 x? + ;x + 7

C` tnk skel`d kxprksscl` wk nbvk

x + 5(x + ?) 1 x + 5x + 7 1 4x + 7

@ltk tnbt c` tnk skel`d kxprksscl` tnk tkra (x + ?) cs l`hy auhtcphckd oy 5 b`d `lt oy x.

Kxbaphk 4?_kalvk tnk orbejkts grla (s? + ?s + 4)(s + 5).

Ulhutcl`

Kben tkra c` tnk flrst orbejkt aust auhtcphy kben tkra c` tnk skel`d. ]lrjc`m tnrlumn bhh elaoc-`btcl`s systkabtcebhhy wk nbvk

(s? + ?s + 4)(s + 5) 1 (s? + ?s + 4)(s) + (s? + ?s + 4)(5)

1 s5 + ?s? + 4s + 5s? + 7s + 2?

1 s5 + ;s? + 26s + 2?


;2



13/22

K`mc`kkrc`m Kxbaphk 2

_khcbochcty c` b elaau`cebtcl` `ktwlrj

C`trlduetcl`

Tnk rkhcbochcty lg b elaau c̀ebtcl` `ktwlrj dkpk`ds l` tnk rkhcbochcty lg cts elapl`k`t pbrts. Tnkrkhcbochcty lg b elapl`k`t eb` ok rkprksk`tkd oy b ùaokr oktwkk` 6 b`d 2 wncen rkprksk`ts tnkprlobochcty tnbt ct wchh guètcl` lvkr b mcvk` pkrcld lg tcak.

B vkry scaphk systka wctn l`hy twl elapl`k`ts E 2 b`d E ? eb` ok el`flmurkd c` skrcks lr c` pbrbhhkh.Cg tnk elapl`k`ts brk c` skrcks tnk` tnk systka wchh gbch cg l`k elapl`k`t gbchs (skk Gcmurk 4)

E 2 E ?

Gcmurk 49 Oltn elapl`k`ts 2 b`d ? aust guètcl` glr tnk systka tl guètcl`

Cg tnk elapl`k`ts brk c` pbrbhhkh tnk` l`hy l`k elapl`k`t `kkd guètcl` prlpkrhy (skk Gcmurk ;)b`d wk nbvk oucht-c` rkdu`dbèy.

E 2

E ?

Gcmurk ;9 Kctnkr elapl`k`t 2 lr ? aust guètcl` glr tnk systka tl guètcl`

Tnk rkhcbochcty lg b systka wctn twl u`cts c` pbrbhhkh cs mcvk` oy 2 ∟ (2 ∟ _2)(2 ∟ _?) wncen cs tnksbak bs _2 + _? ∟ _2_?, wnkrk _c cs tnk rkhcbochcty lg elapl`k`t E c. Tnk rkhcbochcty lg b systkawctn 5 u`cts c` pbrbhhkh, bs c` Gcmurk 7, cs mcvk` oy

2 ∟ (2 ∟ _2)(2 ∟ _?)(2 ∟ _5)

E 2

E 5

E ?

Gcmurk 79 Bt hkbst l`k lg tnk tnrkk elapl`k`ts aust guètcl` glr tnk systka tl guètcl`

;? NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



14/22

Vrlohka c` wlrds

(b) Unlw tnbt tnk kxprksscl` glr tnk systka rkhcbochcty glr tnrkk elapl`k`ts c` pbrbhhkh cs kqubhtl _2 + _? + _5 ∟ _2_? ∟ _2_5 ∟ _?_5 + _2_?_5

(o) Gc`d b` kxprksscl` glr tnk rkhcbochcty lg tnk systka wnk` tnk rkhcbochcty lg kben lg tnkelapl`k`ts cs tnk sbak c.k. _2 1 _? 1 _5 1 _

(e) Gc`d tnk systka rkhcbochcty wnk` _ 1 6.=;

(d) Gc`d tnk systka rkhcbochcty wnk` tnkrk brk twl pbrbhhkh elapl`k`ts kben wctn rkhcbochcty_ 1 6.=;.

Abtnkabtcebh stbtkak`t lg tnk prlohka

(b) Unlw tnbt 2∟(2∟_2)(2∟_?)(2∟_5) ≢ _2+_?+_5∟_2_?∟_2_5∟_?_5+_2_?_5

(o) Gc`d 2 ∟ (2 ∟ _2)(2 ∟ _?)(2 ∟ _5) c` tkras lg _ wnk` _2 1 _? 1 _5 1 _

(e) Gc`d tnk vbhuk lg (o) wnk` _ 1 6.=;

(d) Gc`d 2 ∟ (2 ∟ _2)(2 ∟ _?) wnk` _2 1 _? 1 6.=;.

Abtnkabtcebh b`bhyscs

(b) 2 ∟ (2 ∟ _2)(2 ∟ _?)(2 ∟ _5) ≢ 2 ∟ (2 ∟ _2 ∟ _? + _2_?)(2 ∟ _5)

1 2 ∟ ((2 ∟ _2 ∟ _? + _2_?) Ý 2 ∟ (2 ∟ _2 ∟ _? + _2_?) Ý _5)

1 2 ∟ (2 ∟ _2 ∟ _? + _2_? ∟ (_5 ∟ _2_5 ∟ _?_5 + _2_?_5))

1 2 ∟ (2 ∟ _2 ∟ _? + _2_? ∟ _5 + _2_5 + _?_5 ∟ _2_?_5)

1 2 ∟ 2 + _2 + _? ∟ _2_? + _5 ∟ _2_5 ∟ _?_5 + _2_?_5

1 _2 + _? + _5 ∟ _2_? ∟ _2_5 ∟ _?_5 + _2_?_5

(o) ]nk` _2 1 _? 1 _5 1 _ tnk rkhcbochcty cs

2 ∟ (2 ∟ _)5 wncen cs kqucvbhk`t tl 5_ ∟ 5_? + _5

(e) ]nk` _2 1 _? 1 _5 1 6.=; wk mkt

2 ∟ (2 ∟ 6.=;)5 1 2 ∟ 6.?;5 1 2 ∟ 6.62;7?; 1 6.3>45=;

(d) 2 ∟ (6.?;)? 1 6.35=;

Tnk abtnkabtcebh b`bhyscs el`flras tnk kxpketbtcl` tnbt tnk alrk elapl`k`ts tnkrk brk c` pbr-bhhkh tnk` tnk alrk rkhcbohk tnk systka okelaks (2 elapl`k`t9 6.=;: ? elapl`k`ts9 6.35=;: 5elapl`k`ts9 6.3>45=;). ]ctn tnrkk elapl`k`ts c` pbrbhhkh, bs c` pbrt (e), bhtnlumn kben c`dcvcdubhelapl`k`t cs rkhbtcvkhy u`rkhcbohk (_ 1 6.=; caphcks b l`k c` glur enbèk lg gbchurk lg b` c`dcvcdubhelapl`k`t) tnk systka bs b wnlhk nbs b` lvkr 3>% prlobochcty lg guètcl`c`m (u`dkr 2 c` ;6 enbèklg gbchurk).


;5



15/22

Kxkrecsks

2. _kalvk tnk orbejkts grla kben lg tnk glhhlwc`m kxprksscl`s9

(b) ?(a`), (o) ?(a + `), (e) b(a`), (d) b(a + `), (k) b(a ∟ `),

(g) (ba)`, (m) (b + a)`, (n) (b ∟ a)`, (c) ;( pq ), (i) ;( p + q ),(j) ;( p ∟ q ), (h) =(xy), (a) =(x + y), (`) =(x ∟ y), (l) >(? p + q ),

(p) >(? pq ), (q) >(? p ∟ q ), (r) ;( p ∟ 5q ), (s) ;( p + 5q ) (t) ;(5 pq ).

?. _kalvk tnk orbejkts grla kben lg tnk glhhlwc`m kxprksscl`s9

(b) 4(b + o), (o) ?(a ∟ `), (e) 3(x ∟ y),

5. _kalvk tnk orbejkts grla kben lg tnk glhhlwc`m kxprksscl`s b`d scaphcgy wnkrk plsscohk9

(b) (? + b)(5 + o), (o) (x + 2)(x + ?), (e) (x + 5)(x + 5), (d) (x + ;)(x ∟ 5)

4. _kalvk tnk orbejkts grla kben lg tnk glhhlwc`m kxprksscl`s9

(b) (= + x)(? + x), (o) (3 + x)(? + x), (e) (x + 3)(x ∟ ?), (d) (x + 22)(x ∟ =),(k) (x + ?)x, (g) (5x + 2)x, (m) (5x + 2)(x + 2), (n) (5x + 2)(?x + 2),(c) (5x + ;)(?x + =), (i) (5x + ;)(?x ∟ 2), (j) (; ∟ 5x)(x + 2) (h) (? ∟ x)(2 ∟ x).

;. _kalvk tnk orbejkts grla (s + 2)(s + ;)(s ∟ 5).

B`swkrs

2. (b) ?a`, (o) ?a + ?`, (e) ba`, (d) ba + b`, (k) ba ∟ b`, (g) ba`, (m) b` + a`,

(n) b` ∟ a`, (c) ; pq , (i) ; p + ;q , (j) ; p ∟ ;q , (h) =xy, (a) =x + =y, (`) =x ∟ =y,

(l) 27 p + >q , (p) 27 pq , (q) 27 p ∟ >q , (r) ; p ∟ 2;q , (s) ; p + 2;q , (t) 2; pq

?. (b) 4b + 4o, (o) ?a ∟ ?`, (e) 3x ∟ 3y

5. (b) 7 + 5b + ?o + bo, (o) x? + 5x + ?, (e) x? + 7x + 3, (d) x? + ?x ∟ 2;

4. L` rkalvc`m orbejkts wk lotbc`9

(b) 24 + 3x + x?, (o) 2> + 22x + x?, (e) x? + =x ∟ 2>, (d) x? + 4x ∟ ==(k) x? + ?x, (g) 5x? + x, (m) 5x? + 4x + 2 (n) 7x? + ;x + 2

(c) 7x? + 52x + 5;, (i) 7x? + =x ∟ ;, (j) ∟5x? + ?x + ;, (h) x? ∟ 5x + ?

;. s5 + 5s? ∟ 25s ∟ 2;




16/22

4. Gbetlrcsbtcl`B ùaokr cs sbcd tl ok gbetlrcskd wnk` ct cs wrcttk` bs b prlduet. Glr kxbaphk, ?2 eb` ok gbetlrcskdc`tl = Ý 5. ]k sby tnbt = b`d 5 brk gbetlrs lg ?2.

Bhmkorbce kxprksscl`s eb` bhsl ok gbetlrcskd. El`scdkr tnk kxprksscl` =(?x + 2). _kalvc̀ m tnk

orbejkts wk eb` rkwrctk tncs bs

=(?x + 2) 1 =(?x) + (=)(2) 1 24x + =.

Tnus 24x + = cs kqucvbhk`t tl =(?x + 2). ]k skk tnbt 24x + = nbs gbetlrs = b`d (?x + 2). Tnkgbetlrs = b`d (?x + 2) auhtcphy tlmktnkr tl mcvk 24x + =. Tnk prlekss lg wrctc`m b` kxprksscl` bs bprlduet lg cts gbetlrs cs ebhhkd gbetlrcsbtcl`. ]nk` bsjkd tl gbetlrcsk 24x + = wk wrctk

24x + = 1 =(?x + 2)

b`d sl wk skk tnbt gbetlrcsbtcl` eb` ok rkmbrdkd bs rkvkrsc`m tnk prlekss lg rkalvc`m orbejkts.

Bhwbys rkakaokr tnbt tnk gbetlrs lg b` bhmkorbce kxprksscl` brk auhtcphckd tlmktnkr.

Kxbaphk 45Gbetlrcsk tnk kxprksscl` 4x + ?6.

Ulhutcl`

Oltn tkras c` tnk kxprksscl` 4x + ?6 brk kxbac`kd tl skk cg tnky nbvk b`y gbetlrs c` elaal`.Ehkbrhy ?6 eb` ok gbetlrcskd bs (4)(;) b`d sl wk eb` wrctk

4x + ?6 1 4x + (4)(;)

Tnk gbetlr 4 cs elaal` tl oltn tkras l` tnk rcmnt: ct cs ebhhkd b elaal` gbetlr b`d cs phbekd bttnk grl`t b`d lutscdk orbejkts tl mcvk

4x + ?6 1 4(x + ;)

@ltk tnbt tnk gbetlrcskd glra eb` ok enkejkd oy rkalvc`m tnk orbejkts bmbc`.

Kxbaphk 44Gbetlrcsk z ? ∟ ;z .

Ulhutcl`

@ltk tnbt scèk z ? 1 z Ý z wk eb` wrctk

z ? ∟ ;z 1 z (z ) ∟ ;z

sl tnbt tnkrk cs b elaal` gbetlr lg z . Nkèk

z

?

∟ ;z 1 z (z ) ∟ ;z 1 z (z ∟ ;)


;;



17/22

Kxbaphk 4;Gbetlrcsk 7x ∟ 3y.

Ulhutcl`

Oy loskrvbtcl`, wk skk tnbt tnkrk cs b elaal` gbetlr lg 5. Tnus 7x ∟ 3y 1 5(?x ∟ 5y)

Tbsj sjGbetlrcsk 24z + ?2w.

(b) Gc`d tnk gbetlr elaal` tl oltn 24z b`d ?2w9

Slur slhutcl`

B`swkr

=

(o) @lw gbetlrcsk 24z + ?2w9

Slur slhutcl`

24z + ?2w 1

B`swkr=(?z + 5w)

@ltk9 Cg ylu nbvk b`y dluot, ylu eb` enkej ylur b`swkr oy rkalvc`m tnk orbejkts bmbc`.

Tbsj sjGbetlrcsk 7x ∟ 2?xy.

Gcrst cdk`tcgy tnk twl elaal` gbetlrs9

Slur slhutcl`

B`swkr

7 b`d x

@lw gbetlrcsk 7x ∟ 2?xy9

Slur slhutcl`

7x ∟ 2?xy 1

B`swkr7x(2 ∟ ?y)




18/22

Kxkrecsks

2. Gbetlrcsk

(b) ;x + 2;y, (o) 5x ∟ 3y, (e) ?x + 2?y, (d) 4x + 5?z + 27y, (k) 2?

x + 24

y.

C` kben ebsk enkej ylur b`swkr oy rkalvc`m tnk orbejkts bmbc`.

?. Gbetlrcsk

(b) b? + 5bo, (o) xy + xyz , (e) 3x? ∟ 2?x

5. Kxphbc` wny b cs b gbetlr lg b + bo out o cs `lt. Gbetlrcsk b + bo.

4. Kxphbc` wny x? cs b gbetlr lg 4x? + 5yx5 + ;yx4 out y cs `lt.

Gbetlrcsk 4x? + 5yx5 + ;yx4.

B`swkrs

2. (b) ;(x + 5y), (o) 5(x ∟ 5y), (e) ?(x + 7y), (d) 4(x + >z + 4y), (k) 2?

(x + 2?

y)

?. (b) b(b + 5o), (o) xy(2 + z ), (e) 5x(5x ∟ 4).

5. b(2 + o).

4. x?(4 + 5yx + ;yx?).

;. Gbetlrcsc`m qubdrbtce kxprksscl`sRubdrbtce kxprksscl`s elaal`hy leeur c` ab`y brkbs lg abtnkabtces, pnysces b`d k`mc`kkrc`m. Ab`yqubdrbtce kxprksscl`s eb` ok wrcttk` bs tnk prlduet lg twl hc`kbr gbetlrs b`d, c` tncs Uketcl`, wkkxbac`k nlw tnksk gbetlrs eb` ok kbschy glu`d.

Jky Vlc`t 2>

B` kxprksscl` lg tnk glra

bx? + ox + e b 1 6

wnkrk b, o b`d e brk ùaokrs cs ebhhkd b qubdrbtce kxprksscl` (c` tnk vbrcbohk x).

Tnk ùaokrs o b`d e aby ok zkrl out b aust `lt ok zkrl (glr, tnk`, tnk qubdrbtce rkdueks tl b

hc`kbr kxprksscl` lr el`stb`t). Tnk ùaokr b cs ebhhkd tnk elksteck`t lg x?

, o cs tnk elksteck`t lg x b`d e cs ebhhkd tnk el`stb`t tkra.


;=



19/22

Ebsk 2

El`scdkr tnk prlduet (x +2)(x + ?). _kalvc`m orbejkts yckhds x? + 5x + ?. El`vkrskhy, wk skk tnbttnk gbetlrs lg x? + 5x + ? brk (x + 2) b`d (x + ?). Nlwkvkr, cg wk wkrk mcvk` tnk qubdrbtce kxprksscl`flrst, nlw wluhd wk gbetlrcsk ct < Tnk glhhlwc`m kxbaphks snlw nlw tl dl tncs out `ltk tnbt `lt bhh

qubdrbtce kxprksscl`s eb` ok kbschy gbetlrcskd.Tl k`bohk us tl gbetlrcsk b qubdrbtce kxprksscl` c` wncen tnk elksteck`t lg x? kqubhs 2, wk `ltk tnkglhhlwc`m kxpb`scl`9

(x + a)(x + `) 1 x? + ax + `x + a` 1 x? + (a + `)x + a`

Ul, mcvk` b qubdrbtce kxprksscl` wk eb` tnc`j lg tnk elksteck`t lg x bs a + ` b`d tnk el`stb`t tkrabs a`. L`ek tnk vbhuks lg a b`d ` nbvk okk` glu`d tnk gbetlrs eb` ok kbschy lotbc`kd.

Kxbaphk 47Gbetlrcsk x? + 4x ∟ ;.

Ulhutcl`

]rctc`m x? + 4x ∟ ; 1 (x + a)(x + `) 1 x? + (a + `)x + a` wk skkj `uaokrs a b`d ` suentnbt a + ` 1 4 b`d a` 1 ∟;. Oy trcbh b`d krrlr ct cs `lt dcsteuht tl fl`d tnbt a 1 ; b`d ` 1 ∟2(lr, tnk ltnkr wby rlu`d, a 1 ∟2 b`d ` 1 ;). Ul wk eb` wrctk

x?

+ 4x ∟ ; 1 (x + ;)(x ∟ 2)Tnk b`swkr eb` ok enkejkd kbschy oy rkalvc`m orbejkts.

Tbsj sjGbetlrcsk x? + 7x + >.

Bs tnk elksteck`t lg x? cs 2, wk eb` wrctk

x? + 7x + > 1 (x + a)(x + `) 1 x? + (a + `)x + a`

sl tnbt a + ` 1 7 b`d a` 1 >.

Gcrst, fl`d suctbohk vbhuks glr a b`d `9

Slur slhutcl`

B`swkr

a 1 4, ` 1 ? lr, tnk ltnkr wby rlu`d, a 1 ?, ` 1 4

Gc`bhhy gbetlrcsk tnk qubdrbtce9

;> NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



20/22

Slur slhutcl`

x? + 7x + > 1

B`swkr(x + 4)(x + ?)

Ebsk ?

]nk` tnk elksteck`t lg x? cs `lt kqubh tl 2 ct aby ok plsscohk tl kxtrbet b ùakrcebh gbetlr. Glrkxbaphk, `ltk tnbt 5x? + 2>x + ?4 eb` ok wrcttk` bs 5(x? + 7x + >) b`d tnk` gbetlrcskd bs c`tnk prkvclus Tbsj c` Ebsk 2. Ulaktcaks `l ùakrcebh gbetlr eb` ok glu`d b`d b shcmnthy dcstkrk`tbpprlben aby ok tbjk`. ]k wchh dkal`strbtk b tken`cquk wncen eb` bhwbys ok uskd tl trb s̀glratnk mcvk` kxprksscl` c`tl l`k c` wncen tnk elksteck`t lg tnk squbrkd vbrcbohk kqubhs 2.

Kxbaphk 4=Gbetlrcsk ?x? + ;x + 5.

Ulhutcl`

Gcrst `ltk tnk elksteck`t lg x?: c` tncs ebsk ?. Auhtcphy tnk wnlhk kxprksscl` oy tncs ùaokr b`drkbrrb`mk bs glhhlws9

?(?x? + ;x + 5) 1 ?(?x?) + ?(;x) + ?(5) 1 (?x)? + ;(?x) + 7.

]k `lw c`trlduek b `kw vbrcbohk z suen tnbt z 1 ?x Tnus wk eb` wrctk

(?x)? + ;(?x) + 7 bs z ? + ;z + 7

Tncs eb` ok gbetlrcskd tl mcvk (z + 5)(z + ?). _ktur c̀`m tl tnk lrcmc`bh vbrcbohk oy rkphbec`m z oy?x wk fl`d

?(?x? + ;x + 5) 1 (?x + 5)(?x + ?)

B gbetlr lg ? eb` ok kxtrbetkd grla tnk skel`d orbejkt l` tnk rcmnt sl tnbt

?(?x? + ;x + 5) 1 ?(?x + 5)(x + 2)

sl tnbt

?x? + ;x + 5 1 (?x + 5)(x + 2)

Bs b` bhtkr`btcvk tl tnk tken`cquk lg Kxbaphk 4=, kxpkrckèk b`d prbetcek eb` lgtk` nkhp us tlcdk`tcgy gbetlrs. Glr kxbaphk supplsk wk wcsn tl gbetlrcsk 5x? + =x + ?. ]k wrctk

5x? + =x + ? 1 ( )( )

C` lrdkr tl lotbc` tnk tkra 5x?

wk eb` phbek tkras 5x b`d x c` tnk orbejkts tl mcvk5x? + =x + ? 1 (5x + < )(x + < )


;3



21/22

C` lrdkr tl lotbc` tnk el`stb`t ?, wk el`scdkr tnk gbetlrs lg ?. Tnksk brk 2,? lr ∟2,∟?. Oy phbec`mtnksk gbetlrs c` tnk orbejkts wk eb` gbetlrcsk tnk qubdrbtce kxprksscl`. Wbrclus plsscochctcks kxcst9 wkeluhd wrctk (5x +?)(x + 2) lr (5x +2)(x + ?) lr (5x∟?)(x∟2) lr (5x∟2)(x∟?), l`hy l`k lg wncencs elrrket. Oy rkalvc`m orbejkts grla kben c` tur` wk hllj glr tnk gbetlrcsbtcl` wncen prldueks tnkelrrket acddhk tkra, =x. Tnk elrrket gbetlrcsbtcl` cs glu`d tl ok

5x? + =x + ? 1 (5x + 2)(x + ?)

]ctn prbetcek ylu wchh ok bohk tl ebrry lut tncs prlekss quctk kbschy.

Tbsj sjGbetlrcsk tnk qubdrbtce kxprksscl` ;x? ∟ =x ∟ 7.

]rctk ;x? ∟ =x ∟ 7 1 ( )( )

Tl lotbc` tnk qubdrbtce tkra ;x?, c`skrt ;x b`d x c` tnk orbejkts9

;x? ∟ =x ∟ 7 1 (;x + < )(x + < )

@lw fl`d tnk gbetlrs lg ∟79

Slur slhutcl`

B`swkr

5,∟? lr ∟5, ? lr ∟7, 2 lr 7,∟2

[sk tnksk gbetlrs c` tur` tl fl`d wncen pbcr, cg b`y, mcvks rcsk tl tnk acddhk tkra, ∟=x, b`d elaphktktnk gbetlrcsbtcl`9

Slur slhutcl`

;x? ∟ =x ∟ 7 1 (;x + )(x + ) 1

B`swkr

(;x + 5)(x ∟ ?)

L` leebscl`s ylu wchh akkt kxprksscl`s lg tnk glra x?∟y? j`lw` bs tnk dcstkrk`ek lg twl squbrks.

Ct cs kbsy tl vkrcgy oy rkalvc`m orbejkts tnbt tncs gbetlrcsks bs

x? ∟ y? 1 (x + y)(x ∟ y)

Ul, cg ylu eb` hkbr` tl rkelm`csk suen kxprksscl`s ct cs b` kbsy abttkr tl gbetlrcsk tnka.

76 NKHA (?66;)9]lrjollj 29 Obsce Bhmkorb



22/22

Kxbaphk 4>Gbetlrcsk

(b) x? ∟ 57z ?, (o) ?;x? ∟ 3z ?, (e) β? ∟ 2

Ulhutcl`

C` kben ebsk wk brk rkqucrkd tl fl`d tnk dcstkrk`ek lg twl squbrkd tkras.

(b) @ltk tnbt x? ∟ 57z ? 1 x? ∟ (7z )?. Tncs gbetlrcsks bs (x + 7z )(x ∟ 7z ).

(o) Nkrk ?;x? ∟ 3z ? 1 (;x)? ∟ (5z )?. Tncs gbetlrcsks bs (;x + 5z )(;x ∟ 5z ).

(e) β? ∟ 2 1 (β + 2)(β ∟ 2).

Kxkrecsks

2. Gbetlrcsk

(b) x? + >x + =, (o) x? + 7x ∟ =, (e) x? + =x + 26, (d) x? ∟ 7x + 3.

?. Gbetlrcsk

(b) ?x? + 5x + 2, (o) ?x? + 4x + ?, (e) 5x? ∟ 5x ∟ 7, (d) ;x? ∟ 4x ∟ 2, (k) 27x? ∟ 2,

(g) ∟x? + 2, (m) ∟?x? + x + 5.

5. Gbetlrcsk

(b) x? + 3x + 24, (o) x? + 22x + 2>, (e) x? + =x ∟ 2>, (d) x? + 4x ∟ ==,(k) x? + ?x, (g) 5x? + x, (m) 5x? + 4x + 2, (n) 7x? + ;x + 2,(c) 7x? + 52x + 5;, (i) 7x? + =x ∟ ;, (j) ∟5x? + ?x + ;, (h) x? ∟ 5x + ?.

4. Gbetlrcsk (b) z ? ∟ 244, (o) z ? ∟ 24

, (e) s? ∟ 23

B`swkrs

2. (b) (x + =)(x + 2), (o) (x + =)(x ∟ 2), (e) (x + ?)(x + ;), (d) (x ∟ 5)(x ∟ 5)?. (b) (?x + 2)(x + 2), (o) ?(x + 2)?, (e) 5(x + 2)(x ∟ ?), (d)(;x + 2)(x ∟ 2),

(k) (4x + 2)(4x ∟ 2), (g) (x + 2)(2 ∟ x), (m) (x + 2)(5 ∟ ?x)

5. Tnk gbetlrs brk9

(b) (= + x)(? + x), (o) (3 + x)(? + x), (e) (x + 3)(x ∟ ?), (d) (x + 22)(x ∟ =),(k) (x + ?)x, (g) (5x + 2)x, (m) (5x + 2)(x + 2), (n) (5x + 2)(?x + 2),(c) (5x + ;)(?x + =), (i) (5x + ;)(?x ∟ 2), (j) (; ∟ 5x)(x + 2), (h) (? ∟ x)(2 ∟ x).

4. (b) (z + 2?)(z ∟ 2?), (o) (z +

2

?)(z ∟

2

?), (e) (s +

2

5)(s ∟

2

5)

Simplification and Factorisation

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