10. polynomials

7/21/2019 10. Polynomials

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Z F A V H D M T C A H J S^ -E

IMTBFIMTCK[

=

ÂOSHAICMO[

CHTZADVKTCAHCh komss CR, bmvf studcfd tbf paoyhaicmos ch ahf vmrcmjof mhd tbfcr df`rffs. Pf bmvf mosa ofmrht mjaut tbf vmoufs t

zfras ae m paoyhaicmo. Ch tbf tbcs kbmptfr, wf wcoo dcskuss iarf mjaut tbf zfras ae m paoyhaicmo mhd tbf rfomtcahsb

jftwffh tbf zfras mhd tbf kafeeckcfhts ae m paoyhaicmo wctb pmrtckuomr rfefrfhkf ta qumdrmtck paoyhaicmos. Ch mddctcastmtfifht mhd scipof prajofis ah dcvcscah moàrctbi ear paoyhaicmos wctb rfmo kafeeckcfhts wcoo jf dcskussfd.

BC[TAZCKMO EMKT[Dftfrichch` tbf raats ae paoyhaicmos, ar saovch` mo fjrmck fqumtcahs, cs miah`

tbf aodfst prajofis ch imtbfimtcks. Bawfvfr, fof`mht mhd prmktckmo hatmtcah wf

usf tadmy ahoy dfvfoapfd jf`chhch` ch tbf =6tb kfhtury. Jfearf tbmt, fqumtcahs

wfrf wrcttfh aut ch wards. Ear fxmipof, mh mo`fjrm prajofi erai tbf Kbchfsf

Mrctbiftck ch Hchf [fktcahs, jf`chs Tbrff sbfmes ae àad krap, twa sbfmes ae

ifdcakrf krap, mhd ahf sbfme ae jmd krap mrf saod ear :3 dau. Pf wauod wrctf ;x

+ :y + z 9 :3.

Tbf fmrocfst ghawh usf ae tbf fqumo sc`h cs ch Zajfrt Zfkardfrs Tbf Pbftstahf ae Pcttf, =660. Tbf sc`hs + e

mddctcah, - ear sujtrmktcah, mhd tbf usf ae ofttfr ear mhd uhghawh mppfmr ch Ickbmfo [tcefos Mrctbiftckmo Chtf`=644. Zfhf Dfskmrtfs, ch Om `faiftrck, =5;0, chtradukfd tbf kahkfpt ae tbf rmpb ae paoyhaicmo fqumtcah. B

papuomrczfd tbf usf ae ofttfrs erai tbf jf`chhch` ae tbf mopbmjft ta dfhatf kahstmhts mhd ofttfrs erai tbf fhd ae t

mopbmjft ta dfhatf vmrcmjofs, ms kmh jf sffh ch tbf `fhfrmo eariuom ear m paoyhaicmo, wbfrf tbf ms dfhatf kahstmh

mhd x dfhatfs m vmrcmjof. Dfskmrtfs chtradukfd tbf usf ae supfrskrcpts ta dfhatf fxpahfhts ms wfoo.

ZFKMOO

(c) âoyhaicmos < Mh mo`fjrmck fxprfsscah ae tbf eari >=

=

:

:

=

=.................)( xmxmxmxmxmxp hh

h

h

h

h

wbfrf >hm mhd hmmmm .....,.........,, :=> mrf rfmo huijfrs mhd fmkb pawfr ae x cs m pasctcvf chtf`fr, cs kmoofd

paoyhaicmo.

Bfhkf, ,,, := hhh mmm mrf kafeeckcfhts ae>= ................, xxx hh mhd ...,.........,, ::

=

=

h

h

h

h

h

h xmxmxm mrf tfris ae t

paoyhaicmo . Bfrf tbf tfri

h

hxm cs kmoofd tbf ofmdch` tfri mhd cts kafeeckcfht hm , tbf ofmdch` kafeeckcfht, E

fxmipof < 4:;:

=)( :; uuuup cs m paoyhaicmo ch vmrcmjof u.

4,:,;,:

= :; uu mrf ghaw ms tfris ae paoyhaicmo mhd 4,:,;,:

= mrf tbfcr rfspfktcvf kafeeckcfhts.

:5 x Tbcs cs HAT m paoyhaicmo tfri Jfkmusf tbf vmrcmjof bms m hf`mtcvf fxpahfht

:

=

x

Tbcs cs HAT m paoyhaicmo tfri Jfkmusf tbf vmrcmjof cs ch tbf dfhaichmtar

sqrt (x) Tbcs cs HAT m paoyhaicmo tfri Jfkmusf tbf vmrcmjof cs chscdf m rmdckmo

:4x Tbcs C[ m paoyhaicmo tfri Jfkmusf ct ajfys moo tbf ruofs

(cc) Typfs ae âoyhaicmos


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IMTBFIMTCK[

:

âoyhaicmos komsscecfd jy huijfr ae dcstchkt vmrcmjofs

Huijfr ae dcstchkt vmrcmjofs Hmif Fxmipof

= Vhcvmrcmtf x + 3

: Jcvmrcmtf x + y + 3

; Trcvmrcmtf x + y + z + 3

@fhfrmooy, m paoyhaicmo ch iarf tbmh ahf vmrcmjof cs kmoofd m iuotcvmrcmtf paoyhaicmo. M sfkahd imlar wmy

komssceych` paoyhaicmos cs jy tbfcr df`rff. Zfkmoo tbmt tbf df`rff ae m tfri cs tbf sui ae tbf fxpahfhts ah vmrcmjof

mhd tbmt tbf df`rff ae m paoyhaicmo cs tbf omr`fst df`rff ae mhy ahf tfri.

âoyhaicmos komsscecfd jy df`rff

Df`rff Hmif Fxmipof

\fra >> (hah-zfra) kahstmht =

= Ochfmr x + =

: qumdrmtck x:+ =

; kujck x;+ :4 qumrtck (ar jcqumdrmtck) x

4+ ;

6 quchtck x6 + 4

5 sfxtck (ar bfxck) x5+ 6

0 sfptck (ar bfptck) x0+ 5

2 aktck x2+ 0

3 hahck x3+ 2

=> dfkck x=>+ 3

Vsumooy, m paoyhaicmo ae df`rff h, ear h `rmtfr tbmh ;, cs kmoofd m paoyhaicmo ae df`rff h, motbau`b tbf pbrmsfs qumrt

paoyhaicmo mhd quchtck paoyhaicmo mrf saiftcifs usfd.

Tbf paoyhaicmo >, wbckb imy jf kahscdfrfd ta bmvf ha tfris mt moo, cs kmoofd tbf zfra paoyhaicmo.Vhocgf atbkahstmht paoyhaicmos, cts df`rff cs hat zfra. Zmtbfr tbf df`rff ae tbf zfra paoyhaicmo cs fctbfr ofet fxpockctoy uhdfech

, ar dfechfd ta jf hf`mtcvf (fctbfr = ar )

âoyhaicmos komsscecfd jy huijfr ae hah-zfra tfris

Huijfr ae hah-

zfra tfris

Hmif Fxmipof

> zfra paoyhaicmo >

= iahaicmo x:

: jchaicmo x:+ =

; trchaicmo x:+ x + =

Ce m paoyhaicmo bms ahoy ahf vmrcmjof, tbfh tbf tfris mrf usumooy wrcttfh fctbfr erai bc`bfst df`rff ta oawfst df`r

(dfskfhdch` pawfrs) ar erai oawfst df`rff ta bc`bfst df`rff (mskfhdch` pawfrs ).

(ccc) Umouf ae m âoyhaicmo < Ce p(x) cs m paoyhaicmo ch vmrcmjof x mhd cs mhy rfmo huijfr, tbfh tbf vmouf ajtmchfd jrfpomkch` x jy ch p(x) cs kmoofd vmouf ae p(x) mt x 9 mhd cs dfhatfd jy p(x).

Ear fxmipof < Echd tbf vmouf ae p(x) 9 x; 5x:+ ==x 5mt 9 :

p(:) 9 (:); 5 (:):+ == (:) 5 9 2 :4 :: 5 p(:) 9 5>


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IMTBFIMTCK[

;

(cv) \fra ae m âoyhaicmo < M rfmo huijfr cs zfra ae tbf paoyhaicmo p(x) ce p( ) 9 >.

Ear fxmipof < kahscdfr p(x) 9 x; 5x:+ == x 5

p(=) 9 (=); 5(=): + == (=) 5 9 = 5 + == 5 9 >

p(:) 9 (:); 5(:): + == (:) 5 9 2 :4 + :: 5 9 >

p(;) 9 (;); 5(;): + == (;) 5 9 :0 64 + ;; 5 9 >

Tbus, =, : mhd ; mrf kmoofd tbf zfra ae paoyhaicmo p(x).

@FAIFTZCKMO IFMHCH@ AE TBF \FZA[ AE M ÂOSHAICMO

@faiftrckmooy tbf zfras ae m paoyhaicmos e(x) mrf tbf x-ka-ardchmtfs ae tbf pachts wbfrf tbf `rmpb y 9 e(x) chtfrsfk

x-mxcs. Ta uhdfrstmhd ct, wf wcoo sff tbf `faiftrckmo rfprfsfhtmtcahs ae ochfmr mhd qumdrmtck paoyhaicmos.

@faiftrckmo Zfprfsfhtmtcah ae tbf zfra ae m Ochfmr âoyhaicmo

Kahscdfr m ochfmr paoyhaicmo, y 9 :y 6.

Tbf eaooawch` tmjof ocsts tbf vmoufs ae y karrfspahdch` ta dceefrfht vmoufs ae x.

x = 4

y - ; ;:

Ah poattch` tbf pachts M(=, -; ) mhd J(4, ;) mhd lachch` tbfi, m strmc`bt ochf cs ajtmchfd.

Erai, `rmpb wf ajsfrvfr tbmt tbf `rmpb ae y 9 :x 6 chtfrsfkts tbf x-mxcs mt

>,

:

6wbasf x-kaardchmtf cs ,

:

6Mos

zfra ae :x 6 cs:

6.

Tbfrfearf, wf kahkoudf tbmt tbf ochfmr paoyhaicmo ms + j bms ahf mhd ahoy ahf zfra, wbckb cs tbf x

kaardchmtf ae tbf pacht wbfrf tbf `rmpb ae y 9 mx + j chtfrsfkts tbf x-mxcs

@faiftrckmo Zfprfsfhtmtcah ae tbf zfra ae m qumdrmtck âoyhaicmo = : ; 4 6 5

y 9 x: :x 2 =5 0 > 6 2 3 2 6 > 0 =5

Ah poattch` tbf pachts (-4, =5), (-;, 0)(-:, >), (-=, -6), (>, -2), (=, -3), (:, -2), (;, -6), (4, >), (6, 0) mhd (5, =5)


4/27

IMHC[BGVIMZ


IMTBFIMTCK[

4

ah m `rmpb pmpfr mhd drmwch` m siaatb erff bmhd kurvf pmssch` tbrau`b tbfsf pachts, tbf kurvf tbu

ajtmchfd rfprfsfhts tbf `rmpb ae tbf paoyhaicmo y 9 x: :x 2. Tbcs cs kmoofd m pmrmjaom.

Ct cs kofmr erai tbf tmjof tbmt : mhd 4 mrf tbf zfras ae tbf qumdrmtck paoyhaicmo x: :x 2. Mosa, wf ajsfrvf tbmt

mhd 4 mrf tbf x-kaardchmtfs ae tbf pachts wbfrf tbf `rmpb ae y 9 x: :x 2 chtfrsfkts tbf x-mxcs.

Kahscdfr tbf eaooawch` kmsfs

Kmsf-C


5/27

IMHC[BGVIMZ


IMTBFIMTCK[

6

Kmsf-CCC < Bfrf, tbf `rmpb cs fctbfr kaipoftfoy mjavf tbf x-mxcs ar kaipoftfoy jfoaw tbf x-mxcs, [a, ct dafs hat kut t

x-mxcs mt mhy pacht.

(v) (vc)

[a, tbf qumdrmtck paoyhaicmo mx: + jx + k bms ha zfra ch tbcs kmsf.

[a, yau kmh sff `faiftrckmooy tbmt m qumdrmtck paoyhaicmo kmh bmvf fctbfr twa dcstchkt zfrafs ar ahf zfra, ar ha zfr

Tbcs mosa ifmhs tbmt m paoyhaicmo ae df`rff : bms mt iast twa zfrafs.

Zfimrg < Ch `fhfrmo `cvfh m paoyhaicmo p(x) ae df`rff h, tbf `rmpb ae y 9 p(x) chtfrsfkts tbf x-mxcs mt mt iast

pachts. Tbfrfearf, m paoyhaicmo p(x) ae df`rff h bms mt iast h zfras.

Zfomtcahsbcp Jftwffh Tbf \fras Mhd Kafeeckcfhts Ae M ^aoyhaicmoEar m ochfmr paoyhaicmo mx + j, (m >), wf bmvf,

zfra ae m ochfmr paoyhaicmo m

j

Ear m qumdrmtck paoyhaicmo mx: + j + k (m >), wctb mhd ms cts zfras, wf bmvf

[ui ae zfras m

j

^radukt ae zfras mk

Ce mhd mrf tbf zfras ae m qumdrmtck paoyhaicmo e(x). Tbfh paoyhaicmo e(x) cs `cvfh jy

e(x) 9 G{x:(+ )x+ } ar e(x) 9 G{x: (sui ae tbf zfras) x + pradukt ae tbf zfras}

wbfrf G cs m kahstmht .

KAI^FTCTCAH PCHDAPZFOMTCAH[BC^ JFTPFFH TBF \FZA[ MHD KAFEECKCFHT[ AE KVJCK ^AOSHAICMO

Ear m kujck paoyhaicmo mx;+ jx:+ kx + d (m >), wctb , mhd mt cts zfras, wf bmvf ), wctb ,, mhd ms cts zfras, wf bmvf

x:+ 0x + =: 9 >

(x + 4)(x + ;) 9 >

x + 4 9 > ar, x + ; 9 >

x 9 - 4 ar x 9 - ;

Tbus, tbf zfras ae e(x) 9 x: + 0x + =: mrf 4 mhd ;

Haw, sui ae tbf zfras 0);()4(

mhd - 0=

0

[ui ae tbf zfras 9 -

^radukt ae tbf zfras =:);()4(

mhd, =:=

=:

^radukt ae tbf zfras 9

Kafeeckcfht ae x

Kafeeckcfht ae x:

Kafeeckcfht ae x

Kafeeckcfht ae x:

Kahstmht tfri

Kafeeckcfht ae x:

Kahstmht tfri

Kafeeckcfht ae x:


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IMTBFIMTCK[

0

Fx.: Echd tbf zfras ae tbf qumdrmtck paoyhaicmo e(x) 9 mjx:+ (j:+ mk) x + jf mhd vfrcey tbf rfomtcahsbcp jftwffh tbzfras mhd cts kafeeckcfhts.

[ao. e(x) 9 mjx:+ (j:+ mk) x + jk 9 mjx: + j:x + mkx + jk.9 jx (mx + j) + k (mx + j) 9 (mx + j) (jx + k)

[a, tbf vmouf ae e(x) cs zfra wbfh mx + j 9 > ar jx + k 9 >, c.f.m

jx

ar

j

kx

Tbfrfearf,m

j mhdj

k mrf tbf zfras (ar raats) ae e(x).

Haw, sui ae zfras

mj

mkj

mj

mkj

j

k

m

j )( ::

^radukt ae zfras

mj

jk

j

k

m

j

[SIIFTZCK EVHKTCAH[ AE TBF \FZA[

Oft , jf tbf zfras ae m qumdrmtck paoyhaicmo, tbfh tbf fxprfsscah ae tbf eari 7)(7 :: mrf kfootbf euhktcahs ae tbf zfras. Jy syiiftrck euhktcah wf ifmh tbmt tbf euhktcah rfimch chvmrcmht (uhmotfrfd) ch vmou

wbfh tbf raats mrf kbmh`fd kykockmooy. Ch atbfr wards, mh fxprfsscah chvaovch` mhd wbckb rfimchs uhkbmh`jy chtfrkbmh`ch` mhd cs kmoofd syiiftrck euhktcah ae mhd .

[aif usfeuo rfomtcahs chvaovch` mhd mrf


8/27

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IMTBFIMTCK[

2

(cc) Pf bmvf,km

jmjk

m

k

m

j

m

k

m

j

:

::

;

;;;::;

;)(;)(

Fx.4 Ce mhd mrf tbf zfras ae tbf qumdrmtck paoyhaicmo p(s) 9 ;s: 5s + 4, echd tbf vmouf

;

==:

[ao. [chkf mhd mrf tbf zfras ae tbf paoyhaicmo p(s) 9 ;s:- 5s + 4.

:;

)5(

mhd

;

4

Pf bmvf

;:;

==:

::

2;4;

;

4::

;

4;

4:):(

;)(::)(

::

Fx.6 Ce mhd mrf tbf raats (zfras) ae tbf paoyhaicmo e(x) 9 x:- ;x + g sukb tbmt ,= echd tbf vmouf ae g.

[ao. [chkf mhd mrf tbf raats (zfras) ae tbf paoyhaicmo e(x) 9 x:- ;x + g.

;=

);(

mhd 9 g.

Pf bmvf =:)=()(= ::::

=:}:){(=:)( :::

=4);(=4)( :: g

3 4 g 9 = 4 g 9 2 g 9 :

Bfhkf, tbf vmouf ae g cs :.

Fx.5 Ce , mrf tbf zfras ae tbf paoyhaicmo e(x) 9 :x:+ 6x + g smtcseych` tbf rfomtcah4

:=:: , tb

echd tbf vmouf ae g ear tbcs ta jf passcjof .

[ao. [chkf mhdmrf tbf zfras ae tbf paoyhaicmo e(x) 9 :x:+ 6x + g

:

6 mhd

:

g

Haw,4

:=::

4

:=):( ::

4

:=)( :


9/27

IMHC[BGVIMZ


IMTBFIMTCK[

3

:

gmhd

:

6

4

:=

:4

:6

g

=:

g

g 9 :

Fx.0 Echd m qumdrmtck paoyhaicmo fmkb wctb tbf cvfh huijfrs ms tbf sui mhd pradukt ae cts zfras prfspfktcvfoy .

(c) =,4

= (cc)

;

=,:

[ao. Pf ghaw tbmt m qumdrmtck paoyhaicmo wbckb tbf sui mhd pradukt ae cts zfras mrf `cvfh c `cvfh jy

e(x) 9 g{x:- ([ui ae tbf zfras) x + ^radukt ae tbf zfras}, wbfrf g cs m kahstmht.

(c) Zfqucrfd qumdrmtck paoyhaicmo e(s) cs `cvfh jy

e(x)

=

4

=: xxg

(cc) Zfqucrfd qumdrmtck paoyhaicmo e(s) cs `cvfh jy

e(x)

;

=:: xxg

Fx.2 Ce , mrf tbf zfras ae tbf paoyhaicmo mx:+ jx + k, echd m paoyhaicmo wbasf zfras mrfjm

=mhd

jm

=

[ao. schkf mhdmrf tbf zfras ae tbf paoyhaicmo mx:+ jk + k.

m

j mhd

m

k

[chkfjm

=mhd

jm

=mrf tbf zfras ae tbf rfqucrf paoyhaicmo

sui ae tbf zfras))((

==

jmjm

jmjm

jmjm

9mk

j

jm

jmj

m

km

jm

jm

jmjm

jm

::::

:

)(

:)(

^radukt ae tbf zfras:: )(

===

jmjmjmjm

mkj

m

jmj

m

km

==

::

Bfhkf, tbf rfqucrfd paoyhaicmo 9 x: (sui ae zfras) x + pradukt ae zfrasmk

xmk

jx

=:


10/27

IMHC[BGVIMZ


IMTBFIMTCK[

=>

DCUC[CAH MO@AZCTBI EAZ ^AOSHAICMO[

Ce e(x) cs m paoyhaicmo mhd `(x) cs m hah-zfra paoyhaicmo, tbfh tbfrf fxcst twa paoyhaicmos q(x) mhd r(x) sukb tbmt e(

9 `(x) q(x) + r(x) , wbfrf r(x) 9 > ar df`rff r(x) ? df`rff `(x). Ch atbfr wards,

Dcvcdfhd 9 Dcvcsar ]uatcfht + Zfimchdfr

Zfimrg < Ce r(x) 9 >, tbfh paoyhaicmo (x) cs m emktar ae paoyhaicmo e(x).

Fx.3 dcvcdf tbf paoyhaicmo :x:+ ;x + = jy tbf paoyhaicmo x + : mhd vfrcey tbf dcvcscah mo`arctbi .

[ao. Pf bmvf

Kofmroy, quatcfht 9 :x -= mhd rfimchdfr 9 ;

Mosa, (x + :) (:x - =) + ; 9 :x:+ 4x x : + ; 9 :x :+ ;x + =

c.f., :x:+ ;x + = 9 (x + :)(:x - =) + ;. Tbus, Dcvcdfhd 9 Dcvcsar ]uatcfht + Zfimchdfr.

Fx.=> Kbfkg wbftbfr tbf paoyhaicmo t: ; cs m emktar ae tbf paoyhaicmo :t

4+ ;t

; :t

:9 3t =:, jy dcvcdch` t

sfkahd paoyhaicmo jy tbf ecrst paoyhaicmo.

[ao. Pf bmvf

[chkf tbf rfimchdfr cs zfra, tbfrfearf, tbf paoyhaicmo t: ; cs m emktar ae tbf paoyhaicmo :t4+ ;t; :t: 3t =: .

Fx.== Echd moo tbf zfras ae :x4 ;x; ;x:+ 5x :, ce yau ghaw tbmt twa ae cts zfras mrf : mhd : .

[ao. Oft p(x) :x4- ;x

;- ;x

:+ 5x - : jf tbf `cvfh paoyhaicmo. [chkf twa zfras mrf : mhd - : sa, (x - : ) mh

(x + : ) mrf jatb emktars ae tbf `cvfh paoyhaicmo p(x).

Mosa, (x - : ) 9 (x:

- :) cs m emktar ae tbf paoyhaicmo. Haw , wf dcvcdf tbf `cvfh paoyhaicmo jy x:- :.


11/27


12/27

IMHC[BGVIMZ


IMTBFIMTCK[

=:

=>. Ce ,, mrf tbf zfras ae m kujck paoyhaicmo mx;+ jx:+ kx + d, m > tbfh

[ui ae cts zfras m

j

[ui ae tbf pradukts ae zfras tmgfh twa mt m tcifm

k

^radukt ae cts zfras m

d

==. Tbf dcvcscah mo`arctbi stmtfs tbmt `cvfh mhy paoyhaicmo p(x) mhd mhy hah-zfra paoyhaicmo `(x) tbfh wf kmh ec

quatcfht paoyhaicmo q(x) mhd rfimchdfr paoyhaicmo r(x) sukb tbmt .

[AOUFD HKFZT FRFZKC[F

FRFZKC[F < : . =

=. Tbf `rmpb ae y p(x) mrf `cvfh ch ec` jfoaw, ear saif paoyhaicmos p(x). Echd tbf huijfr ae zfras ae p(x), ch

fmkb kmsf.

(c) (cc) (ccc)

(cv) (v) (vc)

[ao. (c) @rmpb ae y 9 p(x) dafs hat chtfrsfkt tbf x-mxcs. Bfhkf, paoyhaicmo p(x) bms ha zfra.

(cc) @rmpb ae y 9 p(x) chtfrsfkts tbf x-mxcs mt ahf mhd ahoy ahf pacht.

Bfhkf, paoyhaicmo p(x) bms ahf fhd ahoy ahfrfmo zfra.

WZfst Try SaursfoeY

FRFZKC[F < :.:

=. Echd tbf zfras ae tbf eaooawch` qumdrmtck paoyhaicmos mhd vfrcey tbf rfomtcahsbcp jftwffh tbf zfras mhd tbf

kafeeckcfhts.

(c) x: :x 2 (cc) 4s: 4s + = (ccc) 5x: ; 0x (cv) 4u:+ 2u

(v) t: =6 (vc) ;x: x 4

Kafeeckcfht ae x:

Kafeeckcfht ae x;

Kahstmht tfri

Kafeeckcfht ae x;


13/27

IMHC[BGVIMZ


IMTBFIMTCK[

=;

[ao. (c) x: :x 2 9 x: 4x + :x 2 9 x (x 4) + :(x 4) 9 (x + :)(x 4)

\fras mrf : mhd 4.

[ui ae tbf zfras 9 (:) + (4) 9 : 9

=

):(

^radukt ae tbf zfras

=

)2(

)2()4)(:(

(cc) 4s: 4s + = 9 (:s =):

Tbf twa zfras mrf:

=,

:

=

[ui ae tbf twa zfras

4

)4(=

:

=

:

=

^radukt ae twa zfras

4

=

:

=

:

=

WZfst Try SaursfoeY

:. Echd m qumdrmtck paoyhaicmo fmkb wctb tbf `cvfh huijfrs ms tbf sui mhd pradukt ae cts zfras rfspfktcvfoy.

(c) =,4

= (cc)

;

=,: (ccc) 6,> (cv) =, = (v)

4

=,

4

= (cv) 4, =

[ao. (c) Oft tbf qumdrmtck paoyhaicmo jf mx:+ jx + k

Tbfh4

=

m

jmhd =

m

k

c.f.,4

=

m

jmhd

=

=

m

k

Pf sfofkt m 9 OKI (4, =) 9 4

Tbfh4

=

4

jmhd ==

4 j

kmhd .4k

[ujstctutch` 4,=,4 kjm ch ,: kjxmx wf `ft tbf rfqucrfd paoyhaicmo 44 : xx

(cc);

=,:

m

k

m

j

;

=,

=

:

m

k

m

j

[fofkt m 9 OKI (=, ;) 9 ;.

Tbfh :;

j mhd :;;= j

mk mhd k 9 =.

[ujstctutch` :;,; jm mhd k 9 = ch mx:+ jx + k, wf `ft tbf rfqucrfd paoyhaicmo =:;; : xx

WZfst Try SaursfoeY

-Kafeeckcfht ae x

Kafeeckcfht ae x:

Kahstmht tfri

Kafeeckcfht ae x:

-Kafeeckcfht ae x

Kafeeckcfht ae x:

Kahstmht tfri

Kafeeckcfht ae x:


14/27

IMHC[BGVIMZ


IMTBFIMTCK[

=4

FRFZKC[F < :.;

=. Dcvcdf tbf paoyhaicmo p(x) jy tbf paoyhaicmo `(x) mhd echd tbf quatcfht mhd rfimchdfr ch fmkb ae tbf eaooawch

x 6, Ce twa ae cts zfras mrf

;

6mhd

;

6 .

[ao. Twa ae tbf zfras ae ;x4 + 5x; :x: =>x 6, mrf;

6mhd

;

6 .

;

6

;

6xx cs m emktar ae tbf paoyhaicmo .


15/27

IMHC[BGVIMZ


IMTBFIMTCK[

=6

c.f.,;

6: x cs m emktar.

c.f., (;x: 6) cs m emktar ae tbf paoyhaicmo. Tbfh wf mppoy tbf dcvcscah mo`arctbi ms jfoaw ,

[ao. (c) p(x) 9 :x:+ :x + 2, `(x) 9 :x>9 : 7 q(x) 9 x:+ x + 4 7 r(x) 9 >

(cc) p(x) 9 :x:+ :x + 2, `(x) 9 x:+ x + 3 7 q(x) 9 : 7 r(x) 9 - =>

(ccc) p(x) 9 x;+ x + 6 7 `(x) 9 x:+ = 7 `(x) 9 x 7 r(x) 9 6.


16/27

IMHC[BGVIMZ


IMTBFIMTCK[

=5

FRFZKC[F = (EAZ [KBAAO / JAMZD FRMI[

AJLFKTCUF TS^F ]VF[TCAH[

Kbaasf Tbf Karrfkt Ahf

=. ]umdrmtck paoyhaicmo bmvch` zfras = mhd : cs -

(M) x: x + : (J) x: x :

(K) x:+ x : (D) Hahf ae tbfsf

:. Ce (x =) cs m emktar ae g:x; 4gx = , tbfh tbf vmouf ae g cs

(M) = (J) =

(K) : (D) :

;. Ear wbmt vmouf ae m cs tbf paoyhaicmo :x4 mx;9 4x: + :x + = dcvcscjof jy = :x 1

(M) m 9 :6 (J) m 9 :4 (K) m 9 :; (D) m 9 ::

4. Ce ahf ae tbf emktars ae x:+ x :> cs (x + 6), tbfh atbfr emktar cs -

(M) (x 4) (J) (x 6) (K) (x 5) (D) (x 0)

6. Ce , jf tbf zfras ae tbf qumdrmtck paoyhaicmo :x:+ 6x + =, tbfh vmouf ae

(M) : (J) = (K) = (D) Hahf ae tbfsf

5. Ce , jf tbf zfras ae tbf qumdrmtck paoyhaicmo : ;x x:, tbfh

(M) : (J) ; (K) = (D) Hahf ae tbfsf

0. ]umdrmtck paoyhaicmo bmvch` sui ae cts zfras 6 mhd pradukt ae cts zfras =4 cs -

(M) x: 6x =4 (J) x: =>x =4 (K) x: 6x + =4 (D) Hahf ae tbfsf

2. Ce x 9 : mhd x 9 ; mrf zfras ae tbf qumdrmtck paoyhaicmo x:+ mx + j, tbf vmoufs ae m mhd j rfspfktcvfoy mrf . Tbf sui mhd pradukt ae zfras ae tbf qumdrmtck paoyhaicmo mrf 6 mhd ; rfspfktcvfoy tbf qumdrmtck paoyhaicmo cs fqu

ta -

(M) x:+ :x + ; (J) x: 6x + ; (K) x:+ 6x + ; (D) x:+ ;x 6

==. Ah dcvcdch` x; ;x:+ x + : jy paoyhaicmo (x), tbf quatcfht mhd rfimchdfr wfrf x : mhd 4 :x rfspfktcvfoy tbfh

`(x) . Ce mhd mrf tbf zfras ae tbf paoyhaicmo e(x) 9 =6x: 6x + 5 tbfh

==

== cs fqumo ta

(M);

=; (J)

:

=; (K)

;

=5 (D)

:

=6

AJLFKTCUF MH[PFZ GFS FRFZKC[F -

]uf. = : ; 4 6 5 0 2 3 => == =: =; =4 =

Mhs. K M M M M D M K J K D D M K J

]uf. =5 =0 =2 =3 :>

Mhs. J K D D M


18/27

IMHC[BGVIMZ


IMTBFIMTCK[

=2

FRFZKC[F : (EAZ [KBAAO / JAMZD FRMI[

[VJLFKTCUF TS^F ]VF[TCAH[

Ufry [bart Mhswfr Typf ]ufstcahs

=. Oaag mt tbf `rmpb ch ec` `cvfh jfoaw. Fmkb cs tbf `rmpb ae y 9 p(x) , wbfrf p(x) cs m paoyhaicmo. Ear fmkb ae tbf `rmp

echd tbf huijfr ae zfras ae p(x).

(c) (cc) (ccc)

(cv) (v) (vc)

:. Kahscdfr tbf kujck paoyhaicmo e(x) 9 x; 4x. Echd erai tbf ec`, tbf huijfr ae zfras ae tbf mjavf stmtfd paoyhaicmo


19/27

IMHC[BGVIMZ


IMTBFIMTCK[

=3

;. Oft e(x) 9 x;

Tbf `rmpb ae tbf paoyhaicmo cs sbawh ch ec`.

(c) Echd tbf huijfr ae zfras ae paoyhaicmo e(x).

(cc) Dftfrichf tbf ka-ardchmtfs ae tbf pachts, mt wbckb tbf `rmpb chtfrsfkts tbf x-mxcs

[bart mhswfr Typf ]ufstcahs

=. Echd tbf zfras ae tbf eaooawch` qumdrmtck paoyhaicmo mhd vfrcey tbf rfomtcahsbcp jftwffh tbf zfras mhd tbf

kafeeckcfhts.

(c) 5x: x = (cc) :6x (x + =) + 4 (ccc) 4x:+ 4x + = (cv) 42y: =;y = (v) 5; :x x:

(vc) :x: 6x (vcc) 43x: 2= (vccc) 4x: 4x ;

:. Echd m qumdrmtck paoyhaicmo fmkb wctb tbf `cvfh huijfrs ms tbf zfras ae tbf paoyhaicmo .

(c) 0;,0; (cc) ;:,;: (ccc);

:,

0

; (cv) ;;,; (v) :;:,:;: (vc)

:

6,

;

2

;. Echd m qumdrmtck paoyhaicmo fmkb wctb tbf `cvfh huijfrs ms tbf sui mhd pradukt ae cts zfras rfspfktcvfoy .

(c) 3,;4 (cc) ;;,=;: (ccc)4

=,> (cv) 0,

;

=> (v)

3

:6,

5

6(vc)

;

6,

;

6:

(vcc)

4

=,; (vccc)

:6

3,

6

5

(cx) =:,:

4. Ce mhd mrf tbf zfras ae tbf paoyhaicmo e(x) 6x:+ 4x 3 tbfh fvmoumtf tbf eaooawch` . Ce mhd jf twa zfras ae tbf qumdrmtck paoyhaicmo mx:+ jx + k, tbfh vmoumtf

==. Echd tbf vmouf ae g ::

(ccc) Ce mhd mrf tbf zfras ae tbfpaoyhaicmo x: 5x + g sukb tbmt .:>:; =:. Ce : mhd ; mrf zfras ae paoyhaicmo ;x: :gx + :i, echd tbf vmoufs ae g mhd i.

=;. Ce ahf zfras ae paoyhaicmo ;x:9 2x + :x + = cs sfvfh tcifs tbf atbfr, tbfh echd tbf zfras mhd tbf vmouf ae g.=4. Cemhd mrf tbf zfras ae tbf paoyhaicmo :x: 4x + 6. Eari tbf paoyhaicmo wbfrf zfras mrf x + 5::. Kbfkg wbftbfr `(y) cs m emktar ae e(y) jy mppoych` tbf dcvcscah mo`arctbi x:+ mx + j, tbfh echd tbf vmoufs ae m mhd j.

(k) Echd p mhd q sukb tbmt ; mhd = mrf tbf zfras ae e(x) 9 x4 + px;+ qx:+ =:x 3 .

(d) Ce ; cs tbf zfra ae e(x) 9 x4 x; 2 x: + gx + =:, tbfh echd tbf vmouf ae g.

:4. (m) Echd moo tbf zfras ae ;x;+ =5x:+ :;x + 5 ce twa cts zfras mrf ; mhd : .

(j) Dftfrichf moo tbf zfras ae 4x; + =:x: x ; ce twa ae cts zfras mrf:

= mhd .

:

=

(k) Dftfrichf moo tbf zfras ae x

;

+ 6x

:

:x => ce twa ae cts zfras mrf : mhd :

(d) Dftfrichf moo tbf zfras ae 4x;+ =:x: x ; ce ahf ae cts zfras cs

:

6

(f) Dftfrichf moo tbf zfras ae 4x;+ 6x: =2>x ::6 ce ahf ae cts zfras cs .4

6

:6. (m) Echd moo tbf zfras ae ;x4 =>x;+ 6x:+ =>x 2 ce tbrff ae cts zfras mrf =, : mhd = .(j) Ajtmch moo tbf zfras ae :x4+ 6x; 2x: =0x 5 ce tbrff ae cts zfras mrf =, ;, :.

(k) Dftfrichf moo tbf zfras ae x4 x; 2x:+ :x + =: ce twa ae cts zfras mrf : mhd : .


21/27

IMHC[BGVIMZ


IMTBFIMTCK[

:=

:5. (m) Ajtmch moo atbfr zfras ae tbf paoyhaicmo :x; 4x x:+ : ce twa ae cts zfras mrf : mhd : .

(j) Echd moo tbf zfras ae :x4 3x;+ 6x:+ ;x = , ce twa ae cts zfras mrf ;: mhd ;: .(k) Echd moo tbf zfras ae tbf paoyhaicmo x4+ x; ;4x: 4x + =:>, ce twa ae cts zfras mrf : mhd : .

(d) Echd moo tbf zfras ae tbf paoyhaicmo :x4 +0x; =3x: =4x + ;>, ce twa ae cts zfras mrf : mhd : .:0. (m) Ah dcvcdch` e(x) 9 ;x;+ x:+ :x + 6 jy m paoyhaicmo `(x) 9 x:+ :x + =, tbf rfimchdfr r(x) 9 3x + =>. Echd tbf

quatcfht paoyhaicmo q(x).(j) Ah dcvcdch` e(x) jf m paoyhaicmo x = x:, tbf quatcfht q(x) mhd rfimchdfr r(x) mrf (x :) mhd ; rfspfktcvfoy .Echd e(x).

(k) Ah dcvcdch` x6 4x;+ x:+ ;x + = jy paoyhaicmo `(x), tbf quatcfht mhd rfimchdfr mrf (x: =) mhd : rfspfktcvfoyEchd (x).

(d) Ah dcvcdch` e(x) 9 :x6+ ;x4+ 4x;+ 4x:+ ;x + : jy m paoyhaicmo `(x), wbfrf `(x) 9 x; + x:+ x + =, tbf quatcfhtajtmchfd ms :x:+ x + =. Echd tbf rfimchdfr r(x).

^AOSHAICMO[ MH[PFZ GFS FRFZKC[F : (R) KJ[

Ufry [bart Mhswfr Typf ]ufstcahs=. (c) Ahf zfra, (cc) Twa zfra, (ccc) Ahf zfra, (v) Ahf zfra, (vc) Eaur zfras :. Tbrff zfras ;. (c)Ahf zfra, (cc) (>, >)

[bart Mhswfr Typf ]ufstcahs

=. (c)

:

=,

;

= (cc) ,

6

4,

6

= (ccc) ,

:

=,

:

= (cv) ,

=5

=,

;

= (v) 0, -3 , (vc)

:

6,> (vcc) ,

0

3,

0

3 (vccc)

:

=,

:

;

:. (c) x: 5x + :, (cc) x: =:, (ccc) :=x:+ ;;x + 5, (cv) ,3;4: xx (v) x: 4x =4, (vc) 5x: ;=x + 4>

;. (c) ,3;4: xx (cc) ),;;()=;:(: xx (ccc) 4x: =, (cv) :=;=>; : xx (v) =2x: =6x + 6>

(vc) ,66:; : xx (vcc) =;44 : xx (vccc) :6x:+ ;>x + 3, (cx) =::: xx

4. (c) ,6

=4 (cc) ,

:6

=>5 (ccc) ,

:6

65 (cv)

=:6

5>4 (v)

=:6

264(vc)

=:6

63;5 6.p 9 - 5 , atbfr zfra 9 = 5.m 9 ; 0.

:

;m

2.: mhd6

:3.Sfs =>.(c)

:

: :

m

mkj (cc)

;

;;

m

jmjk (ccc)

;

;;

k

jmjk(cv)

km

jmjk:

;; ==.(c) 5 (cc) =: (ccc) =5

=:. 3,:

=6 ig =;.

;

6,

;

0,

;

= g =4. (c) ):46(

6

= : xx (cc) )44:6(:6

= : xx (ccc) )226(6

= : xx

=6. (c) :>x: 3x + = (cc) ;x: x =5.: =0.;

= =2.6x: =:x + 4

:>. (c) q(x) 9 ;, r 9 :x: 2x 6 , (cc) q(x) 9 x;+ x:+ x + =, r(x) 9 >, (ccc) q(x) 9 :x:+ x + =, (x) 9 x + =,

(cv) q(x) 9 x + =, r(x) 9 >

:=. (c) q(x) 9 x4 :x: + 6x + 4, r(x) 9 - (;x + 6), (cc) q(x) 9 x, r(x) 9 :x: x + =,

(ccc) q(x) 9 :x; x: ;x +:

==, r(x) 9 ,

:

=; (cv) q(x) 9 6x :>, r(x) 9 =:0x ::

::. (c) `(y) cs m emktar ae e(y), (cc) `(x) cs m emktar ae e(x), (ccc) `(t) cs hat m emktar ae e(t)

:;. (m) g 9 =, (j) m 9 0, j 9 -=2, (k) p 9 - 2, q 9 =:, (d) g 9 :

:4. (m) ,

;

=,;,: (j) ,;,

:

=,

:

= (k) ,6,:,: (d) ,

:

=,

:

;,

:

6(f) 6;,6;,

4

6

:6. (m) =, :,;

4,= (j) ,

:

=,:,;,= (k) :,;,:,:

:5. (m) ,:

=(j) : ,

:

=,=,; (k) :, : , 6 mhd 5, (d)

:

;,: mhd 6

:0. (m) q(x) 9 ;x 6 , (j) e(x) 9 x;+ ;x: ;x + 6, (k) `(x) 9 x; ;x + =, (d) r(x) 9 x + =,


22/27

IMHC[BGVIMZ


IMTBFIMTCK[

::

FRFZKC[F ; (EAZ [KBAAO / JAMZD FRMI[

^ZFUCAV[ SFMZ[ JAMZD (KJ[F) ]VF[TCAH[

]ufstcahs Kmrrych` = Imrg

=. Prctf tbf zfras ae tbf paoyhaicmo x:

+ :x + =. WDfobc :>>:. Prctf tbf zfras ae tbf paoyhaicmo x: :x 5. WDfobc :>>

;. Prctf m qumdrmtck paoyhaicmo , tbf sui mhd pradukt ae wbasf zfras mrf ; mrf : rfspfktcvfoy. WDfobc :>>

4. Prctf tbf huijfr ae zfras ae tbf paoyhaicmo y 9 e(x) wbasf `rmpb cs `cvfh ch ec`urf. WMC :>>

6. Ce (x + m) cs m emktar ae :x:+ :mx + 6x + =>, echd m WEarfc`h :>>5. Ear wbmt vmouf ae g, (- 4) cs m zfra ae tbf paoyhaicmo x: x (:x + :) 1 WDfobc :>>30. Ear wbmt vmouf ae p, (-4) cs m zfra ae tbf paoyhaicmo x: :x (0p + ;) 1 WDfobc :>>3

2. Ce = cs m zfra ae tbf paoyhaicmo p(x) 9 mx

:

; (m =) x =, tbfh echd tbf vmouf ae m. WMC :>>33. Prctf tbf paoyhaicmo , tbf pradukt mhd sui ae wbasf zfras

:

3 mhd

:

; rfspfktcvfoy WEarfc`h :>>3

=>. Prctf tbf paoyhaicmo , tbf pradukt mhd sui ae wbasf zfras mrf6

=; mhd

6

; rfspfktcvfoy WEarfc`h :>>3

]ufstcahs Kmrrych` : Imrgs==. Echd tbf zfras ae tbf qumdrmtck paoyhaicmo 5x: ; 0x mhd vfrcey tbf rfomtcahsbcp jftwffh tbf zfras mhd tbf k

feeckcfht ae tbf paoyhaicmo . WDfobc :>>2Y=:. Echd tbf zfras ae tbf qumdrmtck paoyhaicmo 6x: 4 2x mhd vfrcey tbf rfomtcahsbcp jftwffh tbf zfras mhd tb

kafeeckcfhts ae tbf qumdrmtck paoyhaicmo WDfobc :>>2Y=;. Echd tbf qumdrmtck paoyhaicmo sui ae wbasf zfras cs 2 mhd tbfcr pradukt cs =:. Bfhkf, echd tbf zfras ae tb

paoyhaicmo . WMC :>>2Y=4. Ce ahf zfra ae tbf paoyhaicmo (m: + 3) x:+ =;x + 5m cs rfkcprakmo ae tbf atbfr. Echd tbf vmouf ae m WMC :>>2Y=6. Ce tbf pradukt ae zfras ae tbf paoyhaicmo mx: 5x 5 cs 4, echd tbf vmouf ae m WMC :>>2Y=5. Echd moo tbf zfras ae tbf paoyhaicmo x4+ x; ;4x: 4x + =:>, ce twa ae cts zfras mrf : mhd : . WEarfc`h :>>2Y

=0. Echd moo tbf zfras ae tbf paoyhaicmo :x4+ 0x; =3x: =4x + ;>, ce twa ae cts zfras mrf : mhd : WEarfc`h :>>

=2. Ce tbf paoyhaicmo 5x4+ 2x;+ =0x:+ :=x + 0 cs dcvcdfd jy mhatbfr paoyhaicmo ;x:+ 4x + =, tbf rfimchdfr kaifs auae jf (mx +j ) , echd m mhd j. WDfobc :>>3Y

=3. Ce tbf paoyhaicmo x4 + :x; + 2x:+ =:x + =2 cs dcvcdfd jy mhatbfr paoyhaicmo x:+ 6, tbf rfimchdfr kaifs aut ta jfpx + q. Echd tbf vmoufs ae p mhd q. WDfobc :>>3Y

:>. Echd moo tbf zfras ae tbf paoyhaicmo x;+ ;x: :x 5, ce twa ae cts zfras mrf : mhd : WMC :>>3Y

:=. Echd moo tbf zfras ae tbf paoyhaicmo :x;+ x: 5x ; , ce twa ae cts zfras mrf ; mhd .; WMC :>>3Y::. Ce tbf paoyhaicmo 5x4 + 2x; 6x:+ mx + j cs fxmktoy dcvcscjof jy paoyhaicmo :x: 6, tbfh echd tbf vmouf ae tbf m mhd

WEarfc`h :>>

^AOSHAICMO[ MH[PFZ GFS FRFZKC[F ; (R)- KJ[

=. x 9 - = :.;, -: ;. x: ;x : 4. ; 6. : 5. 3 0. ; 2. m 9 = 3. :x:+ ;x 3 =>.6x:+ ;x =;

==.

:

;,

;

= =:.

:,

:

: =;. x: 2x + =: 7 (5, :) =4. ; =6.

:

; =5. :, -: , -5 mhd 6 =0. 6:,: mhd

:

;

=2. m 9 =, j 9 : =3.p 9 :, q 9 ; :>. :,: mhd ; :=. ;,; mhd:

= ::. m 9 - :>, j 9 - :6


23/27

IMHC[BGVIMZ


IMTBFIMTCK[

:;

FRFZKC[F 4 (EAZ AOSI^CMD[

Kbaasf Tbf Karrfkt Ahf

=. Ce , mhd mrf tbf zfras ae tbf paoyhaicmo :x; 5x: 4x + ;>. tbfh tbf vmouf ae ( ) cs

(M) : (J) : (K) 6 (D) ;>

:. Ce , mhd mrf tbf zfras ae tbf paoyhaicmo e(x) 9 mx;+ jx:+ kx + d, tbfh

=== 9

(M)m

j (J)

d

k (K)

d

k (D)

m

k

;. Ce , mhd mrf tbf zfras ae tbf paoyhaicmo e(x) 9 mx; jx:+ kx d, tbfh :::

(M):

:

m

mkj (J)

:

: :

j

mkj (K)

m

mkj :: (D)

:

: :

m

mkj

4. Ce , mhd mrf tbf zfras ae tbf paoyhaicmo e(x) 9 x;+ px: pqrx + r, tbfh

===

(M)p

r (J)

r

p (K)

r

p (D)

p

r

6. Ce tbf pmrmjaom e(x) 9 mx:+ jx + k pmssfs tbrau`b tbf pachts ( -=, =:), (>, 6) mhd (:, -;) , tbf vmouf ae m + j + k cs

(M) - 4 (J) -: (K) \fra (D) =

5. Ce m, j mrf tbf zfras ae e(x) 9 x:+ px + = mhd k, d mrf tbf zfras ae e(x) 9 x:+ qx + = tbf vmouf ae

F 9 (m k) (j k) (m + j) (j + d) cs

(M) p: q: (J) q: p: (K) q:+ p: (D) Hahf ae tbfsf

0. Ce , mrf zfras ae mx:+ jx + k tbfh zfras ae m;x:+ mjkx + k;mrf -

(M) , (J) :: , (K) ::, (D) ;; ,

2. Oft , jf tbf zfras ae tbf paoyhaicmo x: px + r mhd

:,

:

jf tbf zfras ae x: qx + r, Tbfh tbf vmouf ae r cs

(M) ):)((3

:pqqp (J) ):)((

3

:qppq (K) ):)(:(

3

:pqq (D) ):)(:(

3

:pqqp

3. Pbfh x:>>+ = cs dcvcdfd jy x:+ =, tbf rfimchdfr cs fqumo ta

(M) x + : (J) :x = (K) : (D) - =

=>. Ce m (p+q):+ :jpq +k 9 > mhd mosa m(q + r): + :jqr + k 9 > tbfh pr cs fqumo ta

(M)k

mp : (J)

m

kq : (K)

j

mp : (D)

k

mq :

==. Ce m, j mhd k mrf hat moo fqumo mhd mhd jf tbf zfras ae tbf paoyhaicmo mx:+ jx + k, tbfh vmouf ae (=+

(:= ) cs (J) pasctcvf (K) hf`mtcvf (D) hah-hf`mtcvf=:. Twa kaipofx huijfr mhd mrf sukb tbmt : mhd ,:0:44 tbfh tbf paoyhaicmo wbasf zfras m

mhd cs

(M) x: :x =5 9 > (J) x: :x + =: 9 > (K) x: :x 2 9 > (D) Hahf ae tbfsfs

=;. Ce : mhd ; mrf tbf zfras ae e(x) 9 :x;+ ix: =;x + h, tbfh tbf vmoufs ae i mhd h mrf rfspfktcvfoy

(M) -6 , - ;> (J) -6, ;> (K) 6, ;> (D) 6, -;>


24/27

IMHC[BGVIMZ


IMTBFIMTCK[

:4

=4. Ce , mrf tbf zfras ae tbf paoyhaicmo 5x: + 5px + p:, tbfh tbf paoyhaicmo wbasf zfras mrf :)( m:)( cs

(M) ;x:+ 4p:x + p4 (J) ;x:+ 4p:x p4(K) ;x: 4p:x + p4 (D) Hahf ae tbfsfs

=6. Ce k, d mrf zfras ae x: =>mx ==j mhd m, j mrf zfras ae x: =>kx ==d, tbfh vmouf ae m + j + k + d cs

(M) =:=> (J) -= (K) :6;> (D) -==

=5. Ce tbf rmtca ae tbf raats ae paoyhaicmo x:+ jx + k cs tbf smif ms tbmt ae tbf rmtca ae tbf raats ae x:+ qx + r, tbfh

(M) jr:9 qk

:(J) kq

:9 rj

:(K) q

:k

:9 j

:r

:(D) jq 9 rk

=0. Tbf vmouf ae p ear wbckb tbf sui ae tbf squmrfs ae tbf raats ae tbf paoyhaicmo x: (p :) x p = mssuif tbf of

vmouf cs -

(M) -= (J) = (K) > (D) :

=2. Ce tbf raats ae tbf paoyhaicmo mx:+ jx + k mrf ae tbf eari=

mhd

=tbfh tbf vmouf ae (m + j + k):cs-

(M) j: :mk (J) j: 4mk (K) :j: mk (D) 4j: :mk

=3. Ce , mhd mrf tbf zfras ae tbf paoyhaicmo x;+ m>x: + m=x + m:, tbfh ( := ) (:= ) ( = ) cs

(M) (= m=):+ (m> m:)

: (J) (= + m=): (m> + m:)

: (K) (= + m=):+ (m>+ m:)

: (D) Hahf ae tbfsf

:>. Ce ,, mrf tbf zfras ae tbf paoyhaicmo x; ;x + ==, tbfh tbf paoyhaicmo wbasf zfras mrf ))(( m

)( cs

(M) x;+ ;x + == (J) x; ;x + == (K) x;+;x == (D) x; ;x ==

:=. Ce ,, mrf sukb tbmt ,2,5,: ;;;:: tbfh 444 cs fqumo ta

(M) => (J) =: (K) =2 (D) Hahf ae tbfsf

::. Ce , mrf tbf raats ae mx:+ jx + k mhd gg , mrf tbf raats ae px:+ qx + r, tbfh g 9

(M)

q

p

j

m

:

= (J)

q

p

j

m (K)

p

q

m

j

:

= (D) (mj pq)

:;. Ce , mrf tbf raats ae tbf paoyhaicmo x: px + q, tbfh tbf qumdrmtck paoyhaicmo, tbf raats ae wbckb m

))(( ;;:: mhd ;::; (J) x: (p6 6p;q + 6pq:) x +(p5q: 6p4q; + 4p:q4) 9 >

(K) x: (p;q 6p6+ p4q) (p5q: 6p:q5) 9 > (D) Moo ae tbf mjavf

:4. Tbf kahdctcah tbmt x; mx:+ jx k 9 > imy bmvf twa ae tbf raats fqumo ta fmkb atbfr jut ae appasctf sc`hs cs )

(M) m mhd j mrf ae appasctf sc`hs. (J) m mhd k mrf ae appasctf sc`hs.

(K) j mhd k mrf ae appasctf sc`hs. (D) m,j,k mrf moo ae tbf smif sc`h.


25/27

IMHC[BGVIMZ


IMTBFIMTCK[

:6

:0. Ce , mrf tbf zfras ae tbf paoyhaicmo x: px + q. tbfh:

:

:

:

cs fqumo ta -

(M)q

p

q

p :

:

4 4: (J)

q

p

q

p :

:

4 4: (K)

q

pq

q

p :

:

4 4: (D) Hahf ae tbfsf

:2. Ce , mrf tbf zfras ae tbf paoyhaicmo x: px + ;5 mhd ,3::

tbfh p 9(M) 5 (J) ; (K) 2 (D) 3

:3. Ce , mrf zfras ae mx:+ jx + k, mk >, tbfh zfras ae kx:+ jx + m mrf

(M) , (J)

=

, (K)

=

, (D)

=,

=

;>. M rfmo huijfr cs smcd ta jf mo`fjrmck ce ct smtcsecfs m paoyhaicmo fqumtcah wctb chtf`rmo kafeeckcfhts. Pbckb ae t

eaooawch` huijfrs cs hat mo`fjrmck (D)

;=. Tbf jc-qumdrmtck paoyhaicmo wbasf zfras mrf =,

;

4,:,= cs x;+ 6x:+=>x 2 (J) ;x4+ =>x; 6x:+=>x 2

(K) ;x4+ =>x; + 6x: =>x 2 (D) ;x4 =>x; 6x:+ =>x 2

;:. Tbf kujck paoyhaicmos wbasf zfras mrf:

;,4 mhd -: cs x :4 (J) :x;+ 0x: =>x :4

(K) :x; 0x: =>x + :4 (D) Hahf ae tbfsf

;;. Ce tbf sui ae zfras ae tbf paoyhaicmo p(x) 9 gx; 6x: ==x ; cs :, tbfh g cs fqumo ta (D)

:

6g

;4. Ce e(x) 9 4x; 5x: + 6x = mhd , mhd mrf cts zfras , tbfh

(M):; (J)

46 (K)

:; (D)

4=

;6. Kahscdfr e(x) 9 2x4 :x:+ 5x 6 mhd .,, mrf cts zfras tbfh

(M)4

= (J)

4

= (K)

:

; (D) Hahf ae tbfsf

;5. Ce x: mx + j 9 > mhd x:9 px + q 9 > bmvf m raat ch kaiiah mhd tbf sfkahd fqumtcah bms fqumo raats, tbfh

(M) j + q 9 :mp (J):

mpqj (K) j + q 9 mp (D) Hahf ae tbfsf

AJLFKTCUF MH[PFZ GFS FRFZKC[F -]uf. = : ; 4 6 5 0 2 3 => == =: =; =4 =

Mhs. M K D J K J J D K J D K J K M

]uf. =5 =0 =2 =3 :> := :: :; :4 :6 :5 :0 :2 :3 ;

Mhs. J J J J D K K J M K J M D D D

]uf. ;= ;: ;; ;4 ;6 ;5

Mhs. M K D D D J


26/27

IMHC[BGVIMZ


IMTBFIMTCK[

:5

FRFZKC[F - 6 (EAZ CCT-LFF/MCFFF

Kbaasf Tbf Karrfkt Ahf=. Ce tbf sui ae tbf twa zfras ae x;+ px:+ qx + r cs zfra, tbfh pq 9 WFMIKFT - :>>;

(M) r (J) r (K) :r (D) :r

:. Oft m > mhd p(x) jf m paoyhaicmo ae df`rff `rmtfr tbmh :. Ce p(x) ofmvfs rfimchdfrs m mhd m wbfh dcvcdrfspfktcvfoy jy x + m mhd x m , tbf rfimchdfr wbfh p(x) cs dcvcdfd jy x: m:cs WFMIKFT - :>>;

(M) :x (J) :x (K) x (D) x

;. Ce ahf raat ae tbf paoyhaicmo x:+ px + q cs squmrf ae tbf atbfr raat, tbfh WCCT-[krffhch` - :>>

(M) p; q (;p =) + q:9 > (J) p; q (;p + =) + q:9 >

(K) p; + q (;p =) q:9 > (D) p;+ q (;p + =) q:9 >

4. Ce , mrf tbf zfras ae x: + px + = mhd , jf tbasf ae x: + qx + =, tbfh tbf vmouf

))(( ))(( 9 WDKF-:>>>

(M) p: q: (J) q: p: (K) p: (D) q:

6. Tbf qumdrmtck paoyhaicmo wbasf zfras mrf twckf tbf zfras ae :x: 6x + : 9 > cs WGfrmom Fh`chffrch` -:>>;

(M) 2x

:

=>x + : (J) x

:

6x + 4 (K) :x

:

6x + : (D) x

:

=>x + 55. Tbf kafeeckcfht ae x ch x:+ px + q wms tmgfh ms =0 ch pomkf ae =; mhd cts zfras wfrf eauhd ta jf : mhd =6. T

zfras ae tbf arc`chmo paoyhaicmo mrf - WGfrmom Fh`chffrch` -:>>;Y

(M) ;, 0 (J) ; , 0 (K) ; , 0 (D) ;, =>

0. Ce 4 mhd ,44:: tbfh , mrf tbf zfras ae tbf paoyhaicmo . WGfrmom Fh`chffrch` -:>>;Y

(M) :x: 0x + 5 (J) ;x:+ 3x + == (K) 3x: :0x + :> (D) ;x: =:x + 6

2. Ce ,, mrf tbf zfras ae tbf paoyhaicmo x;+ 4x + =, tbfh === )()()(

(M) : (J) ; (K) 4 (D) 6 WFMIKFT-:>>;Y

3. Ce , mrf tbf zfras ae tbf qumdrmtck paoyhaicmo 4x: 4x + = , tbfh ;; cs

(M)

4

= (J)

2

= (K) =5 (D) ;:

=>. Tbf vmouf ae m, ear wbckb ahf raat ae tbf qumdrmtck paoyhaicmo (m: 6m + ;) x:+ (;m =) x + : cs twckf ms omr`f

tbf atbfr, cs - WMCFFF -:>>;Y

(M);

= (J)

;

: (K)

;

: (D)

;

=

==. Oft , jf tbf zfras ae x:+ (: ) x . Tbf vmoufs ae ear wbckb :: cs ichciui cs

(M) > (J) = (K) : (D) ; WMIV-:>>:Y

=:. Ce = + :c cs m zfra ae tbf paoyhaicmo x:+ jx + k, j, k Z , tbfh (j, k) cs `cvfh jy (M) (:. 6 ) (J) (- ;, =) (K) (-:, 6) (D) (;, =)

=;. Ce : + c cs m zfra ae tbf paoyhaicmo x; 6x:+ 3x 6, tbf atbfr zfras mrf

(M) = mhd : c (J) = mhd ; + c (K) > mhd = (D) Hahf ae tbfsf=4. Tbf vmouf ae ear wbckb ahf zfra ae ;x: (= + 4) x +

:+ : imy jf ahf-tbcrd ae tbf atbfr cs

(M) 4 (J)2

;; (K)

4

=0 (D)

2

;=

=6. Ce = c cs m zfra ae tbf paoyhaicmo x:+ mx + j, tbfh tbf vmoufs ae m mhd j mrf rfspfktcvfoy .

(M) :, = (J) : , : (K) :, : (D) :, - : WTmico Hmdu Fh`chffrch` :>>:Y

=5. Ce tbf sui ae tbf zfras ae tbf paoyhaicmo x:+ px + q cs fqumo ta tbf sui ae tbfcr squmrfs, tbfh

(M) ^: q:9 > (J) p:+ q:9 > (K) p:+ p 9 :q (D) Hahf ae tbfsf


27/27

IMHC[BGVIMZ


IMTBFIMTCK[

=0. Oft , jf tbf zfras ae tbf paoyhaicmo (x m) (x j) k wctb k >. tbfh tbf zfras ae tbf paoyhai

( x )( x ) + k mrf < WCCT-=33:, MCFFF - :>>

(M) m, k (J) j, k (K) m, j (D) m + k, j + k

=2. Ce p, q mrf zfras ae x:+ px + q. tbfh WMCFFF - :>>

(M) p 9 = (J) p 9 = ar > (K) p 9 - : (D) p 9 - : ar >

=3. Ce mhd ,;6,;6 :: tbfh tbf paoyhaicmo wbasf zfras mrf

mhd

cs < WMCFFF - :>>:

(M) ;x: :6x + ; (J) x: 6x + ; (K) x:+6x ; (D) ;x: =3x + ;

:>. Ce mhd tbf dceefrfhkf jftwffh tbf raats ae tbf paoyhaicmos x:+ mx + j mhd x:+ jx + m cs tbf smif, tbfh

WMCFFF - :>>

(M) m + j + 4 9 > (J) m + j 4 9 > (K) m j + 4 9 > (D) m j 4 9 >

:=. Ce tbf zfras ae tbf paoyhaicmo mx:+ jx + k jf ch tbf rmtca i < h, tbfh

(M) j:ih 9 (i:+ h:) mk (J) (i + h):mk 9 j:ih

(K) j:(i: + h:) 9 ihmk (D) Hahf ae tbfsf

KAI^ZFBFH[CAH JM[FD ]VF[TCAH[

Imxciui mhd Ichciui vmouf ae m qumdrmtck fxprfsscah , tbf fxprfsscah mx:+ jx + k `cvfs ichciui vmoufm

jmk

4

4 :

(cc) Pbfh m ? >, tbf fxprfsscah mx: + jx + k `cvfs imxciui vmoufm

jmk

4

4 :

Jmsfd ah mjavf chearimtcah, da tbf eaooawch` qufstcahs x 6x:cs

(M) =: (J) =6 (K) =5 (D) =2

:4. Ce p mhd q )>( mrf tbf zfras ae tbf paoyhaicmo x:+ px + q, tbfh tbf ofmst vmouf ae x:+ px + q )( Zx cs

(M)4

= (J)

4

= (K)

4

3 (D)

4

3

:6. Ce x cs rfmo, tbf ichciui vmouf ae x: 2x + =0 cs

(M) = (J) > (K) = (D) :

AJLFKTCUF MH[PFZ GFS FRFZKC[F -]uf. = : ; 4 6 5 0 2 3 => == =: =; =4 =

Mhs. J D M J J D D K M J J K M D J

]uf. =5 =0 =2 =3 :> := :: :; :4 :6

Mhs. K K J D M J K M K K

10. polynomials

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