Polynomial Functions, Rational Functions, Partial Fraction ...

Polynomial Functions, Rational Functions, Partial Fraction, Remaining Theorem & Factor Theorem

1 Pure Mathematics

1) oS¾> fnoSfuka fnod fYaIh iy ,íêh fidhkak.

( )

( )

( )

( )

( )

2) my; nyq mo Y%s; ( ) ( ) ( ) ys nyq mo f,i m%ldY lrkak.

I. II .

III. IV .

V . VI .

( )

( ) nyq moh ( ) moh u.ska, f.ka iajdh;a; fYaIhla ,efnk ;=re fnÿ úg ,efnk

fYaIh ( ) fú.

idOkh

( ) nyq moh ( ) moh u.ska, f.ka iajdh;a; fYaIhla ,efnk ;=re fnÿ úg ,efnk

,íêh ( ) o fYaIh hehs o .ksuq

( ) ( )( )

( ) ( )( ) , ( )


2 Pure Mathematics

( )

I. ( ) nyq mofhys ( ) idOlhla fõ. Ok

ksÅ,hls' iy fidhkak.

II. nyq mofha ( ) hkak idOlhla nj fmkajd tys b;srs idOl

fidhkak.

III. ( ) – ( ) ys idOlhla ( ) jk fia ys w.h fidhkak.

ys tla tla w.hka yd iudk fiiq idOl fidhkak.

IV. hkq ( ) j,ska ksh;j fnosh yels jq;a ;d;aúl ixLHd w;r

iïnkaO;djhla ,nd .kak. fuu iïnkaOfhka jk úg ys w.h fidhkak. ys

fuu w.hg ys idOl fidhkak.

V. nyq mofha ( ) idOlhla fõ fidhkak.

VI. nyq mofha ( ) hkak idOlhla nj fmkajd tys b;srs idOl

fidhkak.

VII. hkak ( ) ys idOlhla kï fyda nj fmkajkak'

o idOlhla kï iy g ;sìh hq;= w.h ish,a,u fidhkak'

( ) nyq mofhys ( ) idOlhla ùug ;sìh hq;= wksjd¾h iy m%udKj;a wjYH;djh kï

( ) úh hq;=h.


3 Pure Mathematics

VIII. ( ) ( ) fõ ( ) j¾.c m%ldYkhl mq¾K j¾.hlao

( ) hkq ( ) ys idOlhla o kï yd fidhkak. tu w.h i|yd ( ) ys ish¿

idOl fidhkak.

IX. hkq T;af;a Ok ksÄ,hla jk úg hkq ys idOlhla nj

fmkajkak. tkhska ys idOl la fidhkak.

X. ( ) ( ) ( ) ( ) fú.

hkak ( ) ys idOlhla kï ys w.hhka fidhkak. ys tla tla w.hg wkqrEm

( ) ys b;srs idOl o fidhkak.

XI. ( ) hkq j¾.P nyq mohls. ( ) nyq moh ( ) ( ) ( ) u.ska fnÿ úg

,efnk fYaIhka

fõ. ( ) ( ) fõ' ( ) ( ) ( ) hkq ( ) ys

idOl nj fmkajkak. tkhska ( ) fidhkak.


4 Pure Mathematics

3) fYaI m%fïhh kej; kej; fhoSu.

I. nyqmo ms,sno fYaI m%fïhh m%ldY lr idOkh lrkak. ( ) jk ud;%fha nyqmohls.

( ) nyqmoh ( ) ka fn¥ úg fYaIh fú' ( ) nyqmoh ( ) ka fn¥ úg

fYaIh fú' fYaI m%fïhh kej; kej; fhosfuka ( ) nyqmoh ( )( ) ka

fn¥ úg fYaIh fidhkak'

II. ( ) nyqmoh Y%s;h ( )( )( ) u.ska fnÿ úg ,efnk fYaIh

( )( ) ( ) wdldrfhka jk nj fYaI m%fïhh kej; kej;

fhoSfuka fmkajkak. tkhska yd ks¾Kh lrkak.

III. fYaI m%fïhh kej; kej; fhoSfuka nyq moh

( ) u.ska fnÿ úg ,efnk fYaIh ( ) jk nj fmkajkak. tkhska

fYaIh fidhkak'

IV. ( ) nyqmoh Y%s;h ( )( ) u.ska fnÿ úg ,efnk fYaIh

( ) ( ) ( ) ( )

nj fmkajkak. tkhska nyq moh

( ) u.ska fnÿ úg ,efnk fYaIh fidhkak'

V. ( ) nyqmoh Y%s;h ( ) u.ska fnÿ úg ,efnk fYaIh ( ) nj fYaI

m%fïhh kej; kej; fhoSfuka fmkajkak. yd ks¾Kh lrkak.


5 Pure Mathematics

4) fYaIh os we;s úg ksh; mohkays w.h fiùu.

I. nyq moh fnÿ úg ,efnk fYaIh kï yd

w.h fidhkak' by; nyq mofha idOlhla kï yd w.h fidhkak'

II. ( ) hkak ( ) ( ) ( ) j,ska fnÿ úg ,efnk fYaIhka

ms<sfj,ska iy fú. fidhkak'

III. ka fnosfïos fYaIh ( ) kï ys w.h fidhkak.

( )

IV. nyq moh ( ) ka iy ( ) ka fnÿ úg ,efnk fYaIh

iudkh ys w.h fidhkak.

V. ( ) nyqmoh Y%s;h ( )( )( ) u.ska fnÿ úg ,efnk fYaIh

kï yd fidhkak.

VI. ( ) hkak ( ) ( ) u.ska fnosfïos fYaIh ms<sfj,ska

kï fidhkak.

VII. hkak ( ) ( ) ( ) u.ska fnosfïos fYaIh

ms<sfj,ska kï fidhkak.

VIII. ( ) m%ldYkh ( ) j,ska fnÿ úg fYaIh ls. ( ) Y%s;h

( ) j,ska fnÿ úg fYaIh fú. fidhkak.

IX. ( ) ka [ ( ) ] fnÿ úg ,efnk fYaIhka ( ) ka

( ) fnÿ úg ,efnk fYaIhka iudk kï ys w.h fidhkak.


6 Pure Mathematics

X. ( ) kï, ( ) Ys%;h ( ) j,ska fnÿ úg ,efnk fYaIh;a,

( ) Ys%;h ( ) j,ska fnÿ úg ,efnk fYaIh;a, tl iudk fõ. g ;sìh yels

w.h fol fidhkak.

5) úfYaI wdldrfha .egz ( )

1. ( ) hkak [ ( )] ( ) wdldrfhka m%ldY lrkak'

fuys ( ) hkq j¾.c Y%s;hla o hkq ksh;hla o fú' ta khska ( ) iólrKfha uQ,

fidhkak'

2. hkak wdldrfhka m%ldY lrkak' fuys ksh; fidhkak'

fuys fú'

3. jeks ud;%fha nyqmo Y%s;hla idOl j,ska fnÿ úg ,efnk fYaI ms<sfj,ska

fú. Y%s;h ( ) idOlfhkao ( ) idOlfhkao fnÿ úg ,efnk fYaI

ms<sfj,ska ( ) o ( ) fú. tu nyqmo Y%s;h fidhkak.

4. nyq mo ms,sno fYaI m%fïhh m%ldY lr Tmamq lrkak.

( ) hkak ys nyqmo Y%s;hlaj ( ) ( ) ( ) f,i;a ;sfí kï ( )

hkak ( ) ka fnosfïos fYaIh ( ) ( ) hehs olajkak. ( )

hkak ( ) ka fnosfïos fYaIh fidhkak

kï nyqmo Y%s;h ( ) ( ) ka fnosfïos fYaIh ms<sfj,ska

( ) ( ) kï fidhkak.


7 Pure Mathematics

5. ( ) hkak [ ( )] ( ) wdldrfhka m%ldY lrkak'

fuys ( ) fú' ;d;aúl úg ( ) nj fmkajkak.

6. ( ) ( ) ( ) fõ kï o ( ) ka ( ) ka fnÿ úg fYaIh o kï

( ) fidhkak.

7. ( ) ( ) m%ldYkfhys ( ) hkq idOlhls. ;jo

( ) m%ldYkh mrsmQ¾K j¾.hls. ys o yso w.h fidhkak. ( ) m%ldYkh lskï

m%ldYkfha mrsmQ¾K j¾.hla o@

8. hkak j¾.c m%ldYkhl mq¾K j¾.hla kï yd fidhkak.

9. ( ) hkq ud;%h jk nyqmo Y%s;hls th ( ) ka ksh;j fnfoa ( ) ka fnÿ

úg fYaIh o ( ) ka fnÿ úg fYaIh o fõ ( ) Y%s;h fidhkak.

10. ( ) ( ) fú.

I. ( ) yd ( ) hkq ( ) ys idOl nj fmkajkak.

II. ( ) Ys%;h ka fnÿ úg fYaIh fú. fidhkak.

III. yd ys tu w.hhka i|yd [ ( ) ( )] hkak ka fnÿ úg fYaIh

fidhkak.

11. ys nyq mohla jk ( ) hkak ( ) hkafkka fnÿ úg fYaIhla ,efnhs.

( ) nj fmkajkak. tkhska úg ( ) ( ) hkafkka ( ) fnÿ

úg ( )

( )( )

( )

( )( ) fYaIhla ,efnk nj fmkajkak.


8 Pure Mathematics

12. ( ) nyqmo Y%s;h ( ) ka fnÿ úg fYaIh ( ) kï

[ ( ) ( )] yd

[ ( ) ( )] nj fmkajkak.

13. ( ) fú. fuu nyq moh ka fnÿ úg fYaIh la kï

ka fnÿ úg fYaIh fidhkak. tu.ska ( ) ys idOlhla wfmdaykh lrkak. ( )

talP idOl j, .=Ks;hla f,i fidhkak.

14. ( ) nyqmoh ( ) ( ) ( ) ka fnosfïos fYaIh

( )( ) ( )( ) ( )( ) fú. ksh; mo

( ) ( ) ( ) ys fidhkak. ksh; mohla jk úg( )

( )( )( ) ka fnosfïos fYaIfha mo fkdue;s ùug ys w.h fidhkak.

15. ( ) ka fnosfïos fYaIh fidhkak.

16. nyqmoh ( ) ka fnÿ úg fYaIh l%u follska fidhkak.

17. ( ) yd ( ) hkq ys nyqmo fú. ( ) nyqmoh ka o, ( ) nyqmoh ka o

fnÿ úg fYaIhka ms<sfj,ska yd fú. ( ) ( ) nyqmofhaa talP idOlhla

fidhkak. tu talP idOlfhka ( ) ( ) fnÿ úg ,efnk fYaIh fidhkak.

18. ( ) ( ) ( ) m%ldYkfha idOl ish,a, fidhkak.


9 Pure Mathematics

19. ( ) ( ) ys idOl fidhkak. fuu m%ldYkhg;a

( ) ( ) m%ldYkhg;a fmdÿ idOlhla jk ys w.hhka folla

;sfí. ys tu w.hhka fol fidhkak.

20. ( )

hkqfjka oelafjk jeks ud; %fha nyq

mo Y%s;h ( ) j,ska fmkakqï lrhs. ( ) Y%s;h ( )( ) j,ska fnÿ úg fYaIh

( ) ( ) wdldrhg ,súh yels nj;a fuúg ( )

yd

( )

nj;a

fmkajkak.

21. hkq ( ) ( ) ys idOlhla nj fmkajd tys b;srs

idOl fidhkak.

22. ) ( ) nyq mohg ( ) yd ( ) hk idOl ;sfnhs.

I. yd nj fmkajkak.

II. ( ) ys idOl mq¾K f,i fidhkak'

III. ( ) Ok jk mrsos ys w.hka fidhkak'

)

( )( )( ) ys Nskak Nd. fidhkak'

23. ( ) ( ) nyq mo folg ( ) tla fmdÿ idOlhla kï ( ) hkq [ ( ) ( )] ys

idOlhla nj fmkajkak.

24. hk nyq mo folg tla fmdÿ idOlhla we;akï

fyda nj fmkajkak.


10 Pure Mathematics

25. ( ) ( ) kï ( ) hkak ( ) ys idOlhla nj fmkajkak. ;jo

( ) ( )( ) nj o fmkajkak. tu.ska

ys idOl fidhd wdldrfha mq¾K ixLHd ish,a, ka fnoshyels nj

fmkajkak. by; m%;sM, Ndú;d fldg T;af;a ixLHd i|yd jk jk mq¾K ixLHd

ish,a, ka fnoshyels nj fmkajkak.

26. idOl m%fïhh Ndú;fhka idOl fidhkak.

27. fYaI m%fïhh kej; kej; fhoSfuka ( ) nyq mofha

( ) idOlhla nj fmkajkak. ( ) ( ) ( ) jk mrsos ( ) fidhkak.

( ) [ ] jk whqrska yd fidhkak. tkhska ( ) ys m%ia;drh w¢kak

28. hkq ( ) iólrKfha uq,hla nj fmkajkak. yd ;d;aúl

o o kï by; iólrKhg ;d;aúl uq, tlla muKla we;s nj o fmkajkak.

g ;d;aúl uq, tlla muKla ;sfnk nj wfmdaykh lr th fidhkak.

29. ( ) hkq ;=kajk ud;%fha nyq mohla jk w;r yd os ( ) jk w;r

os ( ) fú. ;jo ( ) hkak ( ) hkafkka fnÿ úg fYaIh

ls.

( ) fidhkak.

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )


11 Pure Mathematics

6) fmdÿ idOl we;s nyqmo Y%s;

I. iy hk nyqmo folgu tla f¾Çh fmdÿ idOlhla

mj;S kï th ( ) ( ) yso idOlhla nj fmkajkak.

II. g wdldrfha idOlhla we;akï ( ) nj

fmkajkak. b;srs idOlho fidhkak. tkhska ys o idOlhla

nj fmkajkak.

III. yd Y%s; folgu ( ) idOlhls

Y%s;fha o th idOlhla nj fmkajkak.

7) ( ) m%ldYkh f¾Çh idOl foll .=Ks;hla f,i ,sùu

i|yd ys w.h fidhkak.

8) hkq ys idOlhla kï túg ( )( )

nj idOkh lrkak. fuu wjYH;djh imqrd,hs kï túg hkq

ys idOlhla nj fmkajkak.

9) ( ) ( ) ( ) ( ) fú.

hkak ( ) ys idOlhla kï túg ys w.hhka fidhkak. ys tla tla w.hg

wkqrEm ( ) ys b;srs idOl o fidhkak.

10) ( ) kï, ( ) iy f¾Åh m%ldYk foll

.=Ks;hla f,i m%ldY l, yels kï, ys w.h fidhkak. tu.ska by; Y%s;h f¾Åh m%ldYk

foll .=Ks;hla f,i m%ldY lrkak.


12 Pure Mathematics

11) ( ) ys idOlhla kï yd fidhkak. tkhska ( )

ys idOl ish,a, fidhkak. tkhska

I.

II.

III. ( ) ( ) ( ) hk nyq mo j, idOl wfmdaykh lrkak.

12) idOl m%fïhh Ndú;fhka

I. idOl ish,a, fidhkak.

II. ( ) ( ) ( ) idOl la fidhkak.


13 Pure Mathematics

mrsfïh Ys%; (Rational Functions)

( ) yd ( ) hkq nyq mo o ( ) o úg ( ) ( )

( ) f,i w¾: olajkq ,nk

( ) Ys%;hg mrsfïh Ys%;hla hehs lshkq ,efí.

( ) ys ud;%h ( ) ys ud;%h kï, ( ) Ys%;hg kshu mrsfïh Ys%;hla fyj;a

kshu Nd.hla hehs lshkq ,efí.

( ) ys ud;%h ( ) ys ud;%h kï, ( ) Ys%;hg úIu mrsfïh Ys%;hla fyj;a úIu

Nd.hla hehs lshkq ,efí.

Nskak Nd. (Partial Fractions)

Nskak Nd. lr olajkak (1- l%uh)

– 2.

3.

( )( )( )

4.

( )( ) 5.

( )( )( )

( )( )

7.

( )( )

( )( ) 9.

( )( )

( )( )

( )( )( ) 12.

( )( )( )( )

( )( )( )( )

( )( )( )( )

( )( )( )( )

( )( ) 15.

( )( )

( )( )( )


14 Pure Mathematics

Nskak Nd. lr olajkak (2 - l%uh)

1.

( )( ) 2.

( ) ( ) 3.

( )

( ) 5.

( )

( ) ( )

( )( ) 8.

( )

( )( )

( )( )( )

( ) ( ) 12.

( )


1.

( ) ( ) 2.

( ) ( ) 3.

( ) ( )

( ) ( )

( ) ( )

( )( )

( )( ) 8.

( )( ) 9.

( )( )

( )( )

( )( )

( )( )


( ) ( )

( ) 3.

( )

( )

( )( )

( )

( )

7.

( )

( )

( )

( )( )

( )

( ) 12.


15 Pure Mathematics

( )( ) Nskak Nd. fidhkak. tkhska kï

( )( )

( )( )

( )( ) nj fmkajkak.

;jo ( ) ( ) ( ) ys idOl o wfmdaykh lrkak.

( ) fõ' fuu m%ldYkhg wdldrfha idOlhla mj;S kï

nj fmkajkak. tkhska m%ldYkh idOl j,g fjka lrkak.

( ) Nskak Nd. fidhkak. tkhska

( ) Nskak Nd. wfmdaykh lrkak.

( ) hkq ;=kajk ud;%fha nyq mohls. ( ) ys my; i|yka ,laIK mj; S.

I. tys idOlhls.

II. ( ) u.ska b;srs ke;sj fnfoa.

III. ( ) úg [ ( ) ( )] nyq moh ( ) ys mqkrdj¾;k

idOlj,ska muKla m%ldY l, yelsh. ( ) fidhkak.

***


16 Pure Mathematics

úNd. .egz (Exam Problems)

1. ( ) ( ) hehs .ksuq. fuys yd hkq ;d;aúl ixLHd

fú. ( ) hkq ( ) ys idOlhla nj fmkajkak. ( ) ys wfkla idOlh j¾.c

wdldrfhka fidhkak.ta khska fyda fjk;a wdldrhlska fyda yd m%Nskak kï

( ) g ;d;aúl uq, tlla muKla ;sfnk nj fmkajkak. g

;d;aúl uq, tlla muKla ;sfnk nj wfmdaykh lr th

fidhkak. ( )

2. yd ( ) hehs .ksuq.( ) yd ( ) hkq

( ) ys idOl kï yd ys w.hhka fidhkak.ta khska fyda fjk;a wdldrhlska fyda

( ) ys wfkla j¾.c idOlh fidhkak. ( )

3. ( ) hehs .ksuq. fYaI m%fïhh kej; kej; fhoSfuka

( ) hkak ( ) ys idOlhla nj fmkajkak. ( ) hkak ( ) ( )

wdldrfhka m%ldY lrkak fuys yd hkq ks¾Kh l< hq;= ksh; fjhs ys ish,q

;d;a;aúl w.hka i|yd ( ) nj wfmdaykh lrkak ( )

4. ( ) hehs .ksuq. fuys fú.

( ) hkak ( ) ys idOlhla fú kï yd ( ) hkak ( ) ka fnÿ úg ,efnk

fYaIh kï yd j, w.hka fidhkak. ( ) ys wfkla talP idOl fol;a

fidhkak. ( )


17 Pure Mathematics

5. hehs o ( ) hehs o .ksuq.( )hkak ( ) ys idOlhla

nj oS we;. ys w.h fidhkak. ( ) hkak ( )( ) wdldrhg m%ldY lrkak.

fuys hkq kssh;hls. by; m%ldYkfhys ( ) hkak yd ksh; jk

( ) wdldrhg ,sùfuka, ( ) hkak ( ) fnÿ úg fYaIh fidhkak.( )

6. ( ) hehs .ksuq. fuys 𝜸 yd ;d;a;aúl ksh; fú.

(

) yd ( ) nj oS we;s úg ( ) ys ;d;a;aúl talP idOl fol fidhkak.

2. ish,q u ;d;a;aúl i|yd ( ) ( ) ( ) ( ) iólrKh

imqrd,k ( ) yd ( ) talP idOl fol fidhkak. ( )

7. ( ) yd ( ) hehs .ksuq. fYaI m%fïhh

Ndú;fhka ( ) hkak ( ) u.ska fnÿ úg fYaIh;A, ( ) hkak ( ) u.ska fnÿ

úg fYaIh;A fidhkak.

( ) ( ) ( ) nj i;Hdmkh lr ( ) hkak ( )( ) u.ska fnÿ úg

fYaIh fidhkak. ( )


18 Pure Mathematics

8. ( ) hehs .ksuq. fuys fú. ( )( ) u.ska ( )

fnÿ úg fYaIh fú. ( ) u.ska ( ) fnÿ úg fYaIh nj yd ( )

u.ska ( ) fnÿ úg fYaIh nj fmkajkak. yd ys w.hka fidhd ( )

hkak ( ) ys idOlhla nj fmkajkak. ( )

9. ( ) yd ;d;a;aúl ixLHd hehs o ( ) hehs o .ksuq.

( ) u.ska ( ) fnÿ úg fYaIh fú. ;jo ( ) hkak ( ) ys idOlhla

fú. yd nj fmkajkak. yd ys fuu w.hka i|yd ( ) u.ska

( ) fnÿ úg fYaIh fidhkak. ( )

10. yd hkq ksYaY=kH ;d;a;aúl ixLHd folla hehs o ( ) hehs

o .ksuq. ( ) hkak ( ) ys idOlhla nj;a ( ) u.ska ( ) fnÿ úg fYaIh

fú. yd w.hka fidhkak. yd ys fuu w.hka i|yd ( ) u.ska ( )

fnÿ úg fYaIh fidhkak. ( )


19 Pure Mathematics

11. ( ) hehs o .ksuq. fuys fú. hkak ( ) ys

idOlhla nj oS we;. nj fmkajkak.

( ) hkak u.ska fnÿ úg fYaIh nj o oS we;. fuys fú. ys

w.h fidhd ( ) hkak ( ) ( ) wdldrhfhka ,súh yels nj fmkajkak.

fú. ( )

***

Polynomial Functions, Rational Functions, Partial Fraction ...

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