NCERT पाठ्यपुस्तक पाठ 2 - बहुपद, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 2 - बहुपद, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


cgqin 23
2
2.1 Hkwfedk
d{kk IX esa] vkius ,d pj okys cgqinksa (polynomials) ,oa mudh ?kkrksa (degree) osQ ckjs
esa vè;;u fd;k gSA ;kn dhft, fd pj x osQ cgqin p(x) esa x dh mPpre ?kkr (power)
cgqin dh ?kkr (degree) dgykrh gSA mnkgj.k osQ fy,]  4x + 2 pj  x esa ?kkr 1 dk
cgqin gS] 2y
2
 – 3y + 4 pj  y esa ?kkr 2 dk cgqin gS] 5x
3
 – 4x
2
 + x – 
2
pj x esa ?kkr 3 d k
cgqin gS vkSj 7 u
6
 – 
4 2
3
4 8
2
u u u + + - pj u esa ?kkr 6 dk cgqin gSA O;atd 
1
1 x -
, 2 x + ,
2
1
2 3 x x + +
bR;kfn cgqin ugha gSaA
?kkr 1 osQ cgqin dks jSf[kd cgqin (linear polynomial) dgrs gSaA mnkgj.k osQ
fy,] 2x – 3, 3 5 , x + 2 y+ , 
2
11
x -
, 3z + 4, 
2
1
3
u +
, bR;kfn lHkh jSf[kd cgqin gSaA
tcfd 2x + 5 – x
2
, x
3
 + 1, vkfn izdkj osQ cgqin jSf[kd cgqin ugha gSaA
?kkr 2 osQ cgqin dks f}?kkr cgqin (quadratic polynomial) dgrs gSaA f}?kkr
(quadratic) 'kCn DokMªsV  (quadrate) 'kCn ls cuk gS] ftldk vFkZ gS ^oxZ*A 
2
2
,
2 3
5
x x + -
y
2
 – 2, 
2
2 3, x x - +
2 2 2
2 1
2 5 , 5 , 4
3 3 7
u
u v v z - + - + , f}?kkr cgqinksa osQ oqQN mnkgj.k
gSa (ftuosQ xq.kkad okLrfod la[;k,¡ gSa)A vf/d O;kid :i esa] x esa dksbZ f}?kkr cgqin
ax
2
 + bx + c, tgk¡ a, b, c okLrfod la[;k,¡ gSa vkSj a ? 0 gS] osQ izdkj dk gksrk gSA ?kkr
3 dk cgqin f=k?kkr cgqin (cubic polynomial) dgykrk gSA f=k?kkr cgqin osQ oqQN
mnkgj.k gSa%
2 – x
3
, x
3
, 
3
2 , x 3 – x
2
 + x
3
, 3x
3 
– 2x
2
 + x – 1
cgqin
2018-19
Page 2


cgqin 23
2
2.1 Hkwfedk
d{kk IX esa] vkius ,d pj okys cgqinksa (polynomials) ,oa mudh ?kkrksa (degree) osQ ckjs
esa vè;;u fd;k gSA ;kn dhft, fd pj x osQ cgqin p(x) esa x dh mPpre ?kkr (power)
cgqin dh ?kkr (degree) dgykrh gSA mnkgj.k osQ fy,]  4x + 2 pj  x esa ?kkr 1 dk
cgqin gS] 2y
2
 – 3y + 4 pj  y esa ?kkr 2 dk cgqin gS] 5x
3
 – 4x
2
 + x – 
2
pj x esa ?kkr 3 d k
cgqin gS vkSj 7 u
6
 – 
4 2
3
4 8
2
u u u + + - pj u esa ?kkr 6 dk cgqin gSA O;atd 
1
1 x -
, 2 x + ,
2
1
2 3 x x + +
bR;kfn cgqin ugha gSaA
?kkr 1 osQ cgqin dks jSf[kd cgqin (linear polynomial) dgrs gSaA mnkgj.k osQ
fy,] 2x – 3, 3 5 , x + 2 y+ , 
2
11
x -
, 3z + 4, 
2
1
3
u +
, bR;kfn lHkh jSf[kd cgqin gSaA
tcfd 2x + 5 – x
2
, x
3
 + 1, vkfn izdkj osQ cgqin jSf[kd cgqin ugha gSaA
?kkr 2 osQ cgqin dks f}?kkr cgqin (quadratic polynomial) dgrs gSaA f}?kkr
(quadratic) 'kCn DokMªsV  (quadrate) 'kCn ls cuk gS] ftldk vFkZ gS ^oxZ*A 
2
2
,
2 3
5
x x + -
y
2
 – 2, 
2
2 3, x x - +
2 2 2
2 1
2 5 , 5 , 4
3 3 7
u
u v v z - + - + , f}?kkr cgqinksa osQ oqQN mnkgj.k
gSa (ftuosQ xq.kkad okLrfod la[;k,¡ gSa)A vf/d O;kid :i esa] x esa dksbZ f}?kkr cgqin
ax
2
 + bx + c, tgk¡ a, b, c okLrfod la[;k,¡ gSa vkSj a ? 0 gS] osQ izdkj dk gksrk gSA ?kkr
3 dk cgqin f=k?kkr cgqin (cubic polynomial) dgykrk gSA f=k?kkr cgqin osQ oqQN
mnkgj.k gSa%
2 – x
3
, x
3
, 
3
2 , x 3 – x
2
 + x
3
, 3x
3 
– 2x
2
 + x – 1
cgqin
2018-19
24 xf.kr
okLro esa] f=k?kkr cgqin dk lcls O;kid :i gS%
ax
3
 + bx
2
 + cx + d,
tgk¡ a, b, c, d okLrfod la[;k,¡ gSa vkSj a ? 0 gSA
vc cgqin p(x) = x
2
 – 3x – 4 ij fopkj dhft,A bl cgqin esa x = 2 j[kus ij ge
p(2) = 2
2
 – 3 × 2 – 4 = – 6 ikrs gSaA x
2
 – 3x – 4 esa] x dks 2 ls izfrLFkkfir djus ls izkIr
eku ^&6*] x
2
 – 3x – 4 dk x = 2 ij eku dgykrk gSA blh izdkj p(0), p(x) dk x = 0 ij
eku gS] tks – 4 gSA
;fn p(x), x esa dksbZ cgqin gS vkSj k dksbZ okLrfod la[;k gS] rks p(x) esa x dks k ls
izfrLFkkfir djus ij izkIr okLrfod la[;k p(x) dk x = k ij eku dgykrh gS vkSj bls
p(k) ls fu:fir djrs gSaA
p(x) = x
2
 –3x – 4 dk x = –1 ij D;k eku gS\ ge ikrs gSa %
p(–1) = (–1)
2 
–{3 × (–1)} – 4 = 0
lkFk gh] è;ku nhft, fd p(4) =4
2
 – (3 × 4) – 4 = 0 gSA
D;ksafd p(–1) = 0 vkSj p(4) = 0 gS] blfy, –1 vkSj 4 f}?kkr cgqin x
2
 – 3x – 4 osQ
'kwU;d (zeroes) dgykrs gSaA vf/d O;kid :i esa] ,d okLrfod la[;k k cgqin p(x)
dk 'kwU;d dgykrh gS] ;fn p(k) = 0 gSA
vki d{kk IX esa i<+ pqosQ gSa fd fdlh jSf[kd cgqin dk 'kwU;d oSQls Kkr
fd;k tkrk gSA mnkgj.k osQ fy,] ;fn p(x) = 2x + 3 dk 'kwU;d k gS] rks p(k) = 0 ls] gesa
2k + 3 = 0 vFkkZr~ k = 
3
2
-
izkIr gksrk gSA
O;kid :i esa] ;fn p(x) = ax + b dk ,d 'kwU;d k gS] rks p(k) = ak + b = 0, vFkkZr~
b
k
a
-
=
gksxkA vr%] jSf[kd cgqin ax + b dk 'kwU;d 
b
a x
- -
=
(vpj in)
dk xq.kkd a
 gSA
bl izdkj] jSf[kd cgqin dk 'kwU;d mlosQ xq.kkadksa ls lacaf/r gSA D;k ;g vU;
cgqinksa esa Hkh gksrk gS\ mnkgj.k osQ fy,] D;k f}?kkr cgqin osQ 'kwU;d Hkh mlosQ xq.kkadksa
ls lacaf/r gksrs gSa\
bl vè;k; esa] ge bu iz'uksa osQ mÙkj nsus dk iz;Ru djsaxsA ge cgqinksa osQ fy,
foHkktu dyu fof/ (division algorithm) dk Hkh vè;;u djsaxsA
2018-19
Page 3


cgqin 23
2
2.1 Hkwfedk
d{kk IX esa] vkius ,d pj okys cgqinksa (polynomials) ,oa mudh ?kkrksa (degree) osQ ckjs
esa vè;;u fd;k gSA ;kn dhft, fd pj x osQ cgqin p(x) esa x dh mPpre ?kkr (power)
cgqin dh ?kkr (degree) dgykrh gSA mnkgj.k osQ fy,]  4x + 2 pj  x esa ?kkr 1 dk
cgqin gS] 2y
2
 – 3y + 4 pj  y esa ?kkr 2 dk cgqin gS] 5x
3
 – 4x
2
 + x – 
2
pj x esa ?kkr 3 d k
cgqin gS vkSj 7 u
6
 – 
4 2
3
4 8
2
u u u + + - pj u esa ?kkr 6 dk cgqin gSA O;atd 
1
1 x -
, 2 x + ,
2
1
2 3 x x + +
bR;kfn cgqin ugha gSaA
?kkr 1 osQ cgqin dks jSf[kd cgqin (linear polynomial) dgrs gSaA mnkgj.k osQ
fy,] 2x – 3, 3 5 , x + 2 y+ , 
2
11
x -
, 3z + 4, 
2
1
3
u +
, bR;kfn lHkh jSf[kd cgqin gSaA
tcfd 2x + 5 – x
2
, x
3
 + 1, vkfn izdkj osQ cgqin jSf[kd cgqin ugha gSaA
?kkr 2 osQ cgqin dks f}?kkr cgqin (quadratic polynomial) dgrs gSaA f}?kkr
(quadratic) 'kCn DokMªsV  (quadrate) 'kCn ls cuk gS] ftldk vFkZ gS ^oxZ*A 
2
2
,
2 3
5
x x + -
y
2
 – 2, 
2
2 3, x x - +
2 2 2
2 1
2 5 , 5 , 4
3 3 7
u
u v v z - + - + , f}?kkr cgqinksa osQ oqQN mnkgj.k
gSa (ftuosQ xq.kkad okLrfod la[;k,¡ gSa)A vf/d O;kid :i esa] x esa dksbZ f}?kkr cgqin
ax
2
 + bx + c, tgk¡ a, b, c okLrfod la[;k,¡ gSa vkSj a ? 0 gS] osQ izdkj dk gksrk gSA ?kkr
3 dk cgqin f=k?kkr cgqin (cubic polynomial) dgykrk gSA f=k?kkr cgqin osQ oqQN
mnkgj.k gSa%
2 – x
3
, x
3
, 
3
2 , x 3 – x
2
 + x
3
, 3x
3 
– 2x
2
 + x – 1
cgqin
2018-19
24 xf.kr
okLro esa] f=k?kkr cgqin dk lcls O;kid :i gS%
ax
3
 + bx
2
 + cx + d,
tgk¡ a, b, c, d okLrfod la[;k,¡ gSa vkSj a ? 0 gSA
vc cgqin p(x) = x
2
 – 3x – 4 ij fopkj dhft,A bl cgqin esa x = 2 j[kus ij ge
p(2) = 2
2
 – 3 × 2 – 4 = – 6 ikrs gSaA x
2
 – 3x – 4 esa] x dks 2 ls izfrLFkkfir djus ls izkIr
eku ^&6*] x
2
 – 3x – 4 dk x = 2 ij eku dgykrk gSA blh izdkj p(0), p(x) dk x = 0 ij
eku gS] tks – 4 gSA
;fn p(x), x esa dksbZ cgqin gS vkSj k dksbZ okLrfod la[;k gS] rks p(x) esa x dks k ls
izfrLFkkfir djus ij izkIr okLrfod la[;k p(x) dk x = k ij eku dgykrh gS vkSj bls
p(k) ls fu:fir djrs gSaA
p(x) = x
2
 –3x – 4 dk x = –1 ij D;k eku gS\ ge ikrs gSa %
p(–1) = (–1)
2 
–{3 × (–1)} – 4 = 0
lkFk gh] è;ku nhft, fd p(4) =4
2
 – (3 × 4) – 4 = 0 gSA
D;ksafd p(–1) = 0 vkSj p(4) = 0 gS] blfy, –1 vkSj 4 f}?kkr cgqin x
2
 – 3x – 4 osQ
'kwU;d (zeroes) dgykrs gSaA vf/d O;kid :i esa] ,d okLrfod la[;k k cgqin p(x)
dk 'kwU;d dgykrh gS] ;fn p(k) = 0 gSA
vki d{kk IX esa i<+ pqosQ gSa fd fdlh jSf[kd cgqin dk 'kwU;d oSQls Kkr
fd;k tkrk gSA mnkgj.k osQ fy,] ;fn p(x) = 2x + 3 dk 'kwU;d k gS] rks p(k) = 0 ls] gesa
2k + 3 = 0 vFkkZr~ k = 
3
2
-
izkIr gksrk gSA
O;kid :i esa] ;fn p(x) = ax + b dk ,d 'kwU;d k gS] rks p(k) = ak + b = 0, vFkkZr~
b
k
a
-
=
gksxkA vr%] jSf[kd cgqin ax + b dk 'kwU;d 
b
a x
- -
=
(vpj in)
dk xq.kkd a
 gSA
bl izdkj] jSf[kd cgqin dk 'kwU;d mlosQ xq.kkadksa ls lacaf/r gSA D;k ;g vU;
cgqinksa esa Hkh gksrk gS\ mnkgj.k osQ fy,] D;k f}?kkr cgqin osQ 'kwU;d Hkh mlosQ xq.kkadksa
ls lacaf/r gksrs gSa\
bl vè;k; esa] ge bu iz'uksa osQ mÙkj nsus dk iz;Ru djsaxsA ge cgqinksa osQ fy,
foHkktu dyu fof/ (division algorithm) dk Hkh vè;;u djsaxsA
2018-19
cgqin 25
2.2 cgqin osQ 'kwU;dksa dk T;kferh; vFkZ
vki tkurs gSa fd ,d okLrfod la[;k k cgqin p(x) dk ,d 'kwU;d gS] ;fn p(k) = 0 gSA
ijarq fdlh cgqin osQ 'kwU;d brus vko';d D;ksa gSa\ bldk mÙkj nsus osQ fy,] loZizFke
ge jSf[kd vkSj f}?kkr cgqinksa osQ vkys[kh; fu:i.k ns[ksaxs vkSj fiQj muosQ 'kwU;dksa dk
T;kferh; vFkZ ns[ksaxsA
igys ,d jSf[kd cgqin ax + b, a ? 0 ij fopkj djrs gSaA vkius d{kk IX esa i<+k gS
fd y = ax + b dk xzkiQ (vkys[k) ,d ljy js[kk gSA mnkgj.k osQ fy,]  y = 2x + 3 dk
xzkiQ ¯cnqvksa (– 2, –1) rFkk (2, 7) ls tkus okyh ,d ljy js[kk gSA
x –2 2
y = 2x + 3 –1 7
vko`Qfr 2.1 ls vki ns[k ldrs
gSa fd y = 2x + 3 dk xzkiQ x–v{k dks
x = –1 rFkk x = – 2 osQ chpks chp]
vFkkZr~ ¯cnq 
3
,
0
2
? ?
-
? ?
? ?
ij izfrPNsn
djrk gSA vki ;g Hkh tkurs gSa fd
2x + 3 dk 'kwU;d 
3
2
-
gSA vr% cgqin
2x + 3 dk 'kwU;d ml ¯cnq dk
x-funsZ'kkad gS] tgk¡ y = 2x + 3 dk
xzkiQ x-v{k dks izfrPNsn djrk gSA
O;kid :i esa] ,d jSf[kd cgqin ax + b, a ? 0 osQ fy,] y = ax + b dk xzkiQ ,d
ljy js[kk gS] tks x-v{k dks Bhd ,d ¯cnq 
,
0
b
a
- ? ?
? ?
? ?
ij izfrPNsn djrh gSA vr%] jSf[kd
cgqin ax + b, a ? 0 dk osQoy ,d 'kwU;d gS] tks ml ¯cnq dk x–funsZ'kkad gS] tgk¡a
y = ax + b dk xzkiQ x–v{k dks izfrPNsn djrk gSA
vc vkb, ge f}?kkr cgqin osQ fdlh 'kwU;d dk T;kferh; vFkZ tkusA f}?kkr
cgqin x
2
 – 3x – 4 ij fopkj dhft,A vkb, ns[ksa fd y = x
2
 – 3x – 4 dk xzkiQ* fdl izdkj
* f}?kkr ;k f=k?kkr cgqinksa osQ xzkiQ [khapuk fo|k£Fk;ksa osQ fy, visf{kr ugha gS vkSj u gh budk
ewY;kadu ls laca/ gSA
vko`Qfr 2.1
2018-19
Page 4


cgqin 23
2
2.1 Hkwfedk
d{kk IX esa] vkius ,d pj okys cgqinksa (polynomials) ,oa mudh ?kkrksa (degree) osQ ckjs
esa vè;;u fd;k gSA ;kn dhft, fd pj x osQ cgqin p(x) esa x dh mPpre ?kkr (power)
cgqin dh ?kkr (degree) dgykrh gSA mnkgj.k osQ fy,]  4x + 2 pj  x esa ?kkr 1 dk
cgqin gS] 2y
2
 – 3y + 4 pj  y esa ?kkr 2 dk cgqin gS] 5x
3
 – 4x
2
 + x – 
2
pj x esa ?kkr 3 d k
cgqin gS vkSj 7 u
6
 – 
4 2
3
4 8
2
u u u + + - pj u esa ?kkr 6 dk cgqin gSA O;atd 
1
1 x -
, 2 x + ,
2
1
2 3 x x + +
bR;kfn cgqin ugha gSaA
?kkr 1 osQ cgqin dks jSf[kd cgqin (linear polynomial) dgrs gSaA mnkgj.k osQ
fy,] 2x – 3, 3 5 , x + 2 y+ , 
2
11
x -
, 3z + 4, 
2
1
3
u +
, bR;kfn lHkh jSf[kd cgqin gSaA
tcfd 2x + 5 – x
2
, x
3
 + 1, vkfn izdkj osQ cgqin jSf[kd cgqin ugha gSaA
?kkr 2 osQ cgqin dks f}?kkr cgqin (quadratic polynomial) dgrs gSaA f}?kkr
(quadratic) 'kCn DokMªsV  (quadrate) 'kCn ls cuk gS] ftldk vFkZ gS ^oxZ*A 
2
2
,
2 3
5
x x + -
y
2
 – 2, 
2
2 3, x x - +
2 2 2
2 1
2 5 , 5 , 4
3 3 7
u
u v v z - + - + , f}?kkr cgqinksa osQ oqQN mnkgj.k
gSa (ftuosQ xq.kkad okLrfod la[;k,¡ gSa)A vf/d O;kid :i esa] x esa dksbZ f}?kkr cgqin
ax
2
 + bx + c, tgk¡ a, b, c okLrfod la[;k,¡ gSa vkSj a ? 0 gS] osQ izdkj dk gksrk gSA ?kkr
3 dk cgqin f=k?kkr cgqin (cubic polynomial) dgykrk gSA f=k?kkr cgqin osQ oqQN
mnkgj.k gSa%
2 – x
3
, x
3
, 
3
2 , x 3 – x
2
 + x
3
, 3x
3 
– 2x
2
 + x – 1
cgqin
2018-19
24 xf.kr
okLro esa] f=k?kkr cgqin dk lcls O;kid :i gS%
ax
3
 + bx
2
 + cx + d,
tgk¡ a, b, c, d okLrfod la[;k,¡ gSa vkSj a ? 0 gSA
vc cgqin p(x) = x
2
 – 3x – 4 ij fopkj dhft,A bl cgqin esa x = 2 j[kus ij ge
p(2) = 2
2
 – 3 × 2 – 4 = – 6 ikrs gSaA x
2
 – 3x – 4 esa] x dks 2 ls izfrLFkkfir djus ls izkIr
eku ^&6*] x
2
 – 3x – 4 dk x = 2 ij eku dgykrk gSA blh izdkj p(0), p(x) dk x = 0 ij
eku gS] tks – 4 gSA
;fn p(x), x esa dksbZ cgqin gS vkSj k dksbZ okLrfod la[;k gS] rks p(x) esa x dks k ls
izfrLFkkfir djus ij izkIr okLrfod la[;k p(x) dk x = k ij eku dgykrh gS vkSj bls
p(k) ls fu:fir djrs gSaA
p(x) = x
2
 –3x – 4 dk x = –1 ij D;k eku gS\ ge ikrs gSa %
p(–1) = (–1)
2 
–{3 × (–1)} – 4 = 0
lkFk gh] è;ku nhft, fd p(4) =4
2
 – (3 × 4) – 4 = 0 gSA
D;ksafd p(–1) = 0 vkSj p(4) = 0 gS] blfy, –1 vkSj 4 f}?kkr cgqin x
2
 – 3x – 4 osQ
'kwU;d (zeroes) dgykrs gSaA vf/d O;kid :i esa] ,d okLrfod la[;k k cgqin p(x)
dk 'kwU;d dgykrh gS] ;fn p(k) = 0 gSA
vki d{kk IX esa i<+ pqosQ gSa fd fdlh jSf[kd cgqin dk 'kwU;d oSQls Kkr
fd;k tkrk gSA mnkgj.k osQ fy,] ;fn p(x) = 2x + 3 dk 'kwU;d k gS] rks p(k) = 0 ls] gesa
2k + 3 = 0 vFkkZr~ k = 
3
2
-
izkIr gksrk gSA
O;kid :i esa] ;fn p(x) = ax + b dk ,d 'kwU;d k gS] rks p(k) = ak + b = 0, vFkkZr~
b
k
a
-
=
gksxkA vr%] jSf[kd cgqin ax + b dk 'kwU;d 
b
a x
- -
=
(vpj in)
dk xq.kkd a
 gSA
bl izdkj] jSf[kd cgqin dk 'kwU;d mlosQ xq.kkadksa ls lacaf/r gSA D;k ;g vU;
cgqinksa esa Hkh gksrk gS\ mnkgj.k osQ fy,] D;k f}?kkr cgqin osQ 'kwU;d Hkh mlosQ xq.kkadksa
ls lacaf/r gksrs gSa\
bl vè;k; esa] ge bu iz'uksa osQ mÙkj nsus dk iz;Ru djsaxsA ge cgqinksa osQ fy,
foHkktu dyu fof/ (division algorithm) dk Hkh vè;;u djsaxsA
2018-19
cgqin 25
2.2 cgqin osQ 'kwU;dksa dk T;kferh; vFkZ
vki tkurs gSa fd ,d okLrfod la[;k k cgqin p(x) dk ,d 'kwU;d gS] ;fn p(k) = 0 gSA
ijarq fdlh cgqin osQ 'kwU;d brus vko';d D;ksa gSa\ bldk mÙkj nsus osQ fy,] loZizFke
ge jSf[kd vkSj f}?kkr cgqinksa osQ vkys[kh; fu:i.k ns[ksaxs vkSj fiQj muosQ 'kwU;dksa dk
T;kferh; vFkZ ns[ksaxsA
igys ,d jSf[kd cgqin ax + b, a ? 0 ij fopkj djrs gSaA vkius d{kk IX esa i<+k gS
fd y = ax + b dk xzkiQ (vkys[k) ,d ljy js[kk gSA mnkgj.k osQ fy,]  y = 2x + 3 dk
xzkiQ ¯cnqvksa (– 2, –1) rFkk (2, 7) ls tkus okyh ,d ljy js[kk gSA
x –2 2
y = 2x + 3 –1 7
vko`Qfr 2.1 ls vki ns[k ldrs
gSa fd y = 2x + 3 dk xzkiQ x–v{k dks
x = –1 rFkk x = – 2 osQ chpks chp]
vFkkZr~ ¯cnq 
3
,
0
2
? ?
-
? ?
? ?
ij izfrPNsn
djrk gSA vki ;g Hkh tkurs gSa fd
2x + 3 dk 'kwU;d 
3
2
-
gSA vr% cgqin
2x + 3 dk 'kwU;d ml ¯cnq dk
x-funsZ'kkad gS] tgk¡ y = 2x + 3 dk
xzkiQ x-v{k dks izfrPNsn djrk gSA
O;kid :i esa] ,d jSf[kd cgqin ax + b, a ? 0 osQ fy,] y = ax + b dk xzkiQ ,d
ljy js[kk gS] tks x-v{k dks Bhd ,d ¯cnq 
,
0
b
a
- ? ?
? ?
? ?
ij izfrPNsn djrh gSA vr%] jSf[kd
cgqin ax + b, a ? 0 dk osQoy ,d 'kwU;d gS] tks ml ¯cnq dk x–funsZ'kkad gS] tgk¡a
y = ax + b dk xzkiQ x–v{k dks izfrPNsn djrk gSA
vc vkb, ge f}?kkr cgqin osQ fdlh 'kwU;d dk T;kferh; vFkZ tkusA f}?kkr
cgqin x
2
 – 3x – 4 ij fopkj dhft,A vkb, ns[ksa fd y = x
2
 – 3x – 4 dk xzkiQ* fdl izdkj
* f}?kkr ;k f=k?kkr cgqinksa osQ xzkiQ [khapuk fo|k£Fk;ksa osQ fy, visf{kr ugha gS vkSj u gh budk
ewY;kadu ls laca/ gSA
vko`Qfr 2.1
2018-19
26 xf.kr
dk fn[krk gSA ge x osQ oqQN ekuksa osQ laxr  y = x
2
 – 3x – 4 osQ oqQN ekuksa dks ysrs gSa] tSls
lkj.kh 2.1 esa fn, gSaA
lkj.kh 2.1
x – 2 –1 0 1 2 3 4 5
y = x
2
 – 3x – 4 6 0 – 4 – 6 – 6 – 4 0 6
;fn ge mi;ZqDr ¯cnqvksa dks
,d xzkiQ issij ij vafdr djsa vkSj
xzkiQ [khapsa] rks ;g vko`Qfr 2.2 esa
fn, x, tSlk fn[ksxkA
okLro esa fdlh f}?kkr cgqin
ax
2
 + bx + c, a ? 0 osQ fy, laxr
lehdj.k  y = ax
2
 + bx + c osQ xzkiQ
dk vkdkj ;k rks Åij dh vksj
[kqyk  dh rjg vFkok uhps dh
vksj [kqyk  dh rjg dk gksxk]
tks bl ij fuHkZj djsxk fd a > 0 gS
;k a < 0 gS (bu oØksa dks ijoy;
(parabola) dgrs gSa)A
lkj.kh 2.1 ls vki ns[k ldrs
gSa fd f}?kkr cgqin osQ 'kwU;d –1
rFkk  4 gSaA bl ij Hkh è;ku nhft,
fd –1 rFkk 4 mu ¯cnqvksa osQ
x–funsZ'kkad gSa] tgk¡ y = x
2
 – 3x – 4
dk xzkiQ x–v{k dks izfrPNsn djrk
gSA bl izdkj] f}?kkr cgqin
x
2
 – 3x – 4 osQ 'kwU;d mu ¯cnqvksa osQ
x–funZs'kkad gSa] tgk¡ y = x
2
 – 3x – 4 dk xzkiQ x–v{k dks izfrPNsn djrk gSA
;g rF; lHkh f}?kkr cgqinksa osQ fy, lR; gS] vFkkZr~ f}?kkr cgqin  ax
2
 + bx + c,
a ? 0 osQ 'kwU;d mu ¯cnqvksa osQ x–funsZ'kkad gSa] tgk¡  y = ax
2
 + bx + c dks fu:fir djus
okyk ijoy; x–v{k dks izfrPNsn djrk gSA
vko`Qfr 2.2
2018-19
Page 5


cgqin 23
2
2.1 Hkwfedk
d{kk IX esa] vkius ,d pj okys cgqinksa (polynomials) ,oa mudh ?kkrksa (degree) osQ ckjs
esa vè;;u fd;k gSA ;kn dhft, fd pj x osQ cgqin p(x) esa x dh mPpre ?kkr (power)
cgqin dh ?kkr (degree) dgykrh gSA mnkgj.k osQ fy,]  4x + 2 pj  x esa ?kkr 1 dk
cgqin gS] 2y
2
 – 3y + 4 pj  y esa ?kkr 2 dk cgqin gS] 5x
3
 – 4x
2
 + x – 
2
pj x esa ?kkr 3 d k
cgqin gS vkSj 7 u
6
 – 
4 2
3
4 8
2
u u u + + - pj u esa ?kkr 6 dk cgqin gSA O;atd 
1
1 x -
, 2 x + ,
2
1
2 3 x x + +
bR;kfn cgqin ugha gSaA
?kkr 1 osQ cgqin dks jSf[kd cgqin (linear polynomial) dgrs gSaA mnkgj.k osQ
fy,] 2x – 3, 3 5 , x + 2 y+ , 
2
11
x -
, 3z + 4, 
2
1
3
u +
, bR;kfn lHkh jSf[kd cgqin gSaA
tcfd 2x + 5 – x
2
, x
3
 + 1, vkfn izdkj osQ cgqin jSf[kd cgqin ugha gSaA
?kkr 2 osQ cgqin dks f}?kkr cgqin (quadratic polynomial) dgrs gSaA f}?kkr
(quadratic) 'kCn DokMªsV  (quadrate) 'kCn ls cuk gS] ftldk vFkZ gS ^oxZ*A 
2
2
,
2 3
5
x x + -
y
2
 – 2, 
2
2 3, x x - +
2 2 2
2 1
2 5 , 5 , 4
3 3 7
u
u v v z - + - + , f}?kkr cgqinksa osQ oqQN mnkgj.k
gSa (ftuosQ xq.kkad okLrfod la[;k,¡ gSa)A vf/d O;kid :i esa] x esa dksbZ f}?kkr cgqin
ax
2
 + bx + c, tgk¡ a, b, c okLrfod la[;k,¡ gSa vkSj a ? 0 gS] osQ izdkj dk gksrk gSA ?kkr
3 dk cgqin f=k?kkr cgqin (cubic polynomial) dgykrk gSA f=k?kkr cgqin osQ oqQN
mnkgj.k gSa%
2 – x
3
, x
3
, 
3
2 , x 3 – x
2
 + x
3
, 3x
3 
– 2x
2
 + x – 1
cgqin
2018-19
24 xf.kr
okLro esa] f=k?kkr cgqin dk lcls O;kid :i gS%
ax
3
 + bx
2
 + cx + d,
tgk¡ a, b, c, d okLrfod la[;k,¡ gSa vkSj a ? 0 gSA
vc cgqin p(x) = x
2
 – 3x – 4 ij fopkj dhft,A bl cgqin esa x = 2 j[kus ij ge
p(2) = 2
2
 – 3 × 2 – 4 = – 6 ikrs gSaA x
2
 – 3x – 4 esa] x dks 2 ls izfrLFkkfir djus ls izkIr
eku ^&6*] x
2
 – 3x – 4 dk x = 2 ij eku dgykrk gSA blh izdkj p(0), p(x) dk x = 0 ij
eku gS] tks – 4 gSA
;fn p(x), x esa dksbZ cgqin gS vkSj k dksbZ okLrfod la[;k gS] rks p(x) esa x dks k ls
izfrLFkkfir djus ij izkIr okLrfod la[;k p(x) dk x = k ij eku dgykrh gS vkSj bls
p(k) ls fu:fir djrs gSaA
p(x) = x
2
 –3x – 4 dk x = –1 ij D;k eku gS\ ge ikrs gSa %
p(–1) = (–1)
2 
–{3 × (–1)} – 4 = 0
lkFk gh] è;ku nhft, fd p(4) =4
2
 – (3 × 4) – 4 = 0 gSA
D;ksafd p(–1) = 0 vkSj p(4) = 0 gS] blfy, –1 vkSj 4 f}?kkr cgqin x
2
 – 3x – 4 osQ
'kwU;d (zeroes) dgykrs gSaA vf/d O;kid :i esa] ,d okLrfod la[;k k cgqin p(x)
dk 'kwU;d dgykrh gS] ;fn p(k) = 0 gSA
vki d{kk IX esa i<+ pqosQ gSa fd fdlh jSf[kd cgqin dk 'kwU;d oSQls Kkr
fd;k tkrk gSA mnkgj.k osQ fy,] ;fn p(x) = 2x + 3 dk 'kwU;d k gS] rks p(k) = 0 ls] gesa
2k + 3 = 0 vFkkZr~ k = 
3
2
-
izkIr gksrk gSA
O;kid :i esa] ;fn p(x) = ax + b dk ,d 'kwU;d k gS] rks p(k) = ak + b = 0, vFkkZr~
b
k
a
-
=
gksxkA vr%] jSf[kd cgqin ax + b dk 'kwU;d 
b
a x
- -
=
(vpj in)
dk xq.kkd a
 gSA
bl izdkj] jSf[kd cgqin dk 'kwU;d mlosQ xq.kkadksa ls lacaf/r gSA D;k ;g vU;
cgqinksa esa Hkh gksrk gS\ mnkgj.k osQ fy,] D;k f}?kkr cgqin osQ 'kwU;d Hkh mlosQ xq.kkadksa
ls lacaf/r gksrs gSa\
bl vè;k; esa] ge bu iz'uksa osQ mÙkj nsus dk iz;Ru djsaxsA ge cgqinksa osQ fy,
foHkktu dyu fof/ (division algorithm) dk Hkh vè;;u djsaxsA
2018-19
cgqin 25
2.2 cgqin osQ 'kwU;dksa dk T;kferh; vFkZ
vki tkurs gSa fd ,d okLrfod la[;k k cgqin p(x) dk ,d 'kwU;d gS] ;fn p(k) = 0 gSA
ijarq fdlh cgqin osQ 'kwU;d brus vko';d D;ksa gSa\ bldk mÙkj nsus osQ fy,] loZizFke
ge jSf[kd vkSj f}?kkr cgqinksa osQ vkys[kh; fu:i.k ns[ksaxs vkSj fiQj muosQ 'kwU;dksa dk
T;kferh; vFkZ ns[ksaxsA
igys ,d jSf[kd cgqin ax + b, a ? 0 ij fopkj djrs gSaA vkius d{kk IX esa i<+k gS
fd y = ax + b dk xzkiQ (vkys[k) ,d ljy js[kk gSA mnkgj.k osQ fy,]  y = 2x + 3 dk
xzkiQ ¯cnqvksa (– 2, –1) rFkk (2, 7) ls tkus okyh ,d ljy js[kk gSA
x –2 2
y = 2x + 3 –1 7
vko`Qfr 2.1 ls vki ns[k ldrs
gSa fd y = 2x + 3 dk xzkiQ x–v{k dks
x = –1 rFkk x = – 2 osQ chpks chp]
vFkkZr~ ¯cnq 
3
,
0
2
? ?
-
? ?
? ?
ij izfrPNsn
djrk gSA vki ;g Hkh tkurs gSa fd
2x + 3 dk 'kwU;d 
3
2
-
gSA vr% cgqin
2x + 3 dk 'kwU;d ml ¯cnq dk
x-funsZ'kkad gS] tgk¡ y = 2x + 3 dk
xzkiQ x-v{k dks izfrPNsn djrk gSA
O;kid :i esa] ,d jSf[kd cgqin ax + b, a ? 0 osQ fy,] y = ax + b dk xzkiQ ,d
ljy js[kk gS] tks x-v{k dks Bhd ,d ¯cnq 
,
0
b
a
- ? ?
? ?
? ?
ij izfrPNsn djrh gSA vr%] jSf[kd
cgqin ax + b, a ? 0 dk osQoy ,d 'kwU;d gS] tks ml ¯cnq dk x–funsZ'kkad gS] tgk¡a
y = ax + b dk xzkiQ x–v{k dks izfrPNsn djrk gSA
vc vkb, ge f}?kkr cgqin osQ fdlh 'kwU;d dk T;kferh; vFkZ tkusA f}?kkr
cgqin x
2
 – 3x – 4 ij fopkj dhft,A vkb, ns[ksa fd y = x
2
 – 3x – 4 dk xzkiQ* fdl izdkj
* f}?kkr ;k f=k?kkr cgqinksa osQ xzkiQ [khapuk fo|k£Fk;ksa osQ fy, visf{kr ugha gS vkSj u gh budk
ewY;kadu ls laca/ gSA
vko`Qfr 2.1
2018-19
26 xf.kr
dk fn[krk gSA ge x osQ oqQN ekuksa osQ laxr  y = x
2
 – 3x – 4 osQ oqQN ekuksa dks ysrs gSa] tSls
lkj.kh 2.1 esa fn, gSaA
lkj.kh 2.1
x – 2 –1 0 1 2 3 4 5
y = x
2
 – 3x – 4 6 0 – 4 – 6 – 6 – 4 0 6
;fn ge mi;ZqDr ¯cnqvksa dks
,d xzkiQ issij ij vafdr djsa vkSj
xzkiQ [khapsa] rks ;g vko`Qfr 2.2 esa
fn, x, tSlk fn[ksxkA
okLro esa fdlh f}?kkr cgqin
ax
2
 + bx + c, a ? 0 osQ fy, laxr
lehdj.k  y = ax
2
 + bx + c osQ xzkiQ
dk vkdkj ;k rks Åij dh vksj
[kqyk  dh rjg vFkok uhps dh
vksj [kqyk  dh rjg dk gksxk]
tks bl ij fuHkZj djsxk fd a > 0 gS
;k a < 0 gS (bu oØksa dks ijoy;
(parabola) dgrs gSa)A
lkj.kh 2.1 ls vki ns[k ldrs
gSa fd f}?kkr cgqin osQ 'kwU;d –1
rFkk  4 gSaA bl ij Hkh è;ku nhft,
fd –1 rFkk 4 mu ¯cnqvksa osQ
x–funsZ'kkad gSa] tgk¡ y = x
2
 – 3x – 4
dk xzkiQ x–v{k dks izfrPNsn djrk
gSA bl izdkj] f}?kkr cgqin
x
2
 – 3x – 4 osQ 'kwU;d mu ¯cnqvksa osQ
x–funZs'kkad gSa] tgk¡ y = x
2
 – 3x – 4 dk xzkiQ x–v{k dks izfrPNsn djrk gSA
;g rF; lHkh f}?kkr cgqinksa osQ fy, lR; gS] vFkkZr~ f}?kkr cgqin  ax
2
 + bx + c,
a ? 0 osQ 'kwU;d mu ¯cnqvksa osQ x–funsZ'kkad gSa] tgk¡  y = ax
2
 + bx + c dks fu:fir djus
okyk ijoy; x–v{k dks izfrPNsn djrk gSA
vko`Qfr 2.2
2018-19
cgqin 27
y = ax
2
 + bx + c osQ xzkiQ osQ vkdkj dk izs{k.k djus ls rhu fuEufyf[kr fLFkfr;k¡
laHkkfor gSaA
fLFkfr (i) : ;gk¡ xzkiQ x-v{k dks nks fHkUu ¯cnqvksa A vkSj A' ij dkVrk gSA
bl fLFkfr esa] A vkSj A' osQ x–funsZ'kkad f}?kkr cgqin ax
2
 + bx + c osQ nks 'kwU;d gSa
(nsf[k, vko`Qfr 2-3)A
vko`Qfr 2.3
fLFkfr (ii) : ;gk¡ xzkiQ x-v{k dks osQoy ,d ¯cnq ij] vFkkZr~ nks laikrh ¯cnqvksa ij dkVrk
gSA blfy,] fLFkfr (i) osQ nks ¯cnq A vkSj A' ;gk¡ ij laikrh gksdj ,d ¯cnq  A gks tkrs gSa
(nsf[k, vko`Qfr 2-4)A
vko`Qfr 2.4
bl fLFkfr esa] A dk x–funsZ'kkad f}?kkr cgqin ax
2
 + bx + c dk osQoy ,d 'kwU;d gSA
2018-19
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