Rational Number

pirmyX ivAMjk

iqAwr krqw:-jiqMdr kumwr mYQ mwstr trynr

srkwrI mwfl sInIAr sYkMMfrI skUl simtrI rof

luiDAwxw

Awau A`j pirmyX ivAMjkW Awau A`j pirmyX ivAMjkW bwry jwxkwrI leIey Aqy svwl bwry jwxkwrI leIey Aqy svwl

h`l kreIey[h`l kreIey[

1,2,3,4………………………. iks qrHW dI sMiKAwvW hn ?

0,1,2,3,4,5,………………….. iks qrHW dI sMiKAwvW hn ?

……-3,-2,-1,0,1,2,3,……… iks qrHW dI sMiKAwvW hn ?

2/3 , 4/5 , 6/5 ……………….iks qrHW dI sMiKAwvW hn

x nUM iks qrHW dI sMiKAwvW AwKdy hW ?

1+2x iv`c ikMny p`d hn?

1+2x+3x2+4x3 iv`c ikMny p`d hn?

a0+a1x+a2x2+a3x3+………+anxn iv`c ikMny p`d hn?

auprokq swrIAW sMiKAwvW nMU vI A`sI bhup`dI k`h s`kdy hW [

ik A`sI iehnW swrIAW sMiKAwvW dy hr iv`c 1 lgw s`kdy hW ?

x

1

1+2x

1

1+2x+3x2+4x3

1

a0+a1x+a2x2+a3x3+………+anxn

1

auprokq swrIAW sMiKAwvW hux A`sI AwKdy hW [

x2-1X2-5x+6

iks qrHW dI sMiKAwvW hY ?

P(x)q(x)

jykr p(x) Aqy q(x) do bhupdW hox qW nUM pirmyX ivMjAk khWgy[

ie`Qy q(x) dw ZYr is&r bhupd hoxw zrUrI hY[

p(x) nMU AMS Aqy q(x) nMU hr ikhw jWdw hY [

x2-1X2-5x+6 pirmyX ivMjAk hY [

P(x) 1

x2+3x+1 x2+1

3x-3+7 x2- 5x +6

vI pirmyX ivMjAk huMdw hY [kauik 1 iek bhup`d hY

pirmyX ivMjAk nhIN huMdy [jykr

1.

2.

c`l vwlI sMiKAwvW vrg mUl, Gx mUl Awid iv`c hox

c`l vwlI sMiKAwvW dI Gwq irxwqmk hovy

3x3/2+7 x2- 5x +6

3. c`l vwlI sMiKAwvW dI Gwq Anupwq iv`c hovy

3x3/2+7 0

4. hr 0 hovy [

3x+7 x2- 5x +6

ie`k pirmyX ivAMjk hY[ies dw AMS

ie`k ryKI bhupd hY[Aqy hr ie`k do GwqI bhupd hY

inaUnqm pdW iv`c hY kauNik X-5 Aqy X-4dy guxnKMf nhIN sMBv hn [

X-5 X-4

h`l:- id`qI hoeI pirmyX ivAMjk dy AMS Aqy hr AKMfnI

guxnKMf dy guxn&l rUp iv`c nhIN hn[AsIN pihlW iehnW nUM AKMfnI guxnKMfW dy guxnPl rUp iv`c bdlWgy[ mMn lau

p(x)=x2 -1 q(x) =(x2-6x+1x-6) =(x(x-6)-1(x-6)) =(x-1)(x-6), p(x) Aqy q(x) dw (x-1) sWJw guxnKMf hY[ ies leI

iehnW dw m.s.v (x-1) hY[ies leI

x2-1X2-5x+6

pVqwl kro ik ieh pirmyX ivAMjk Awpxy inaUnqm pdW iv`c hY jW nhIN[

p(x) =(x-1)(x+1)

(x -1)(x+1)(X-1)(x-6)

(x+1)(x-6)

=

hyT ilKy pirmyX ivAMjk nUM d`so ik ieh inaUnqm pdW iv`c hY jW nhIN[jy nhIN hY qW ies nUM inaUnqm pdW iv`c drswE[

id`qy hoey pirmyX ivAMjk dy AMS Aqy hr AKMfnI guxnKMfW dy guxnPl dy rUp iv`c nhIN hn[ieh inaUnqm pdW iv`c nhIN hY[

x3+8x2-4

h`l:-

x3+8x2-4 =

x3+23

x2-22 =(x+2)(x2-2x+22) (x-2)(x+2)

(x+2)(x2-2x+4) (x-2)(x+2)=

x2-2x+4 x-2=

3x+2X2-16

X-5 X-4pirmyX ivAMjkW Aqy dw joV kro[

h`l:-id`qIAW hoeIAW pirmyX ivAMjkW dy hr ie`ko ijhy nhIN hn[hrW dw l.s.v (x-4)(x+4) hovygw[

3x+2X2-16

X-5 X-4+

=

3x+2(x-4)(x+4) +

X-5 X-4

= (x-4)(x+4)

(3x+2).1+(x-5)(x-4) 3x+2+x2-5x-4x+20

=

X2-16X2-6x+22

X2-16=

X-1X-2

2x2+2x-7 X2+x-6iv`c ikhVw pirmyX ivAMjk joiVAw jwvy ik joVPl pRwpq ho jwvy[

pRSn:-

h`l:-mMn lauX-1X-2 iv`c

p(x)q(x)joVn qy swnUM

2x2+2x-7 X2+x-6pRwpq huMdw hY[

ies leIX-1X-2 +

p(x)q(x) =

2x2+2x-7 X2+x-6

p(x)q(x)

2x2+2x-7 X2+x-6=

X-1X-2- = 2x2+2x-7

X2+3x-2x-6 -X-1X-2

= 2x2+2x-7(x(x+3)-2(x+3))

X-1X-2- =

2x2+2x-7 (x+3)(x-2) -

X-1X-2

= (x+3)(x-2)

(2x2+2x-7) .1- (x-1)(x+3)= (x+3)(x-2)

2x2+2x-7 - (x2-x+3x-3)

= (x+3)(x-2)2x2+2x-7 - (x2+2x-3)

= (x+3)(x-2)

2x2+2x-7 - x2-2x+3

= x2-4(x+3)(x-2) =

x2-22

(x+3)(x-2) = (x-2)(x+2)(x+3)(x-2) =

(x+2)(x+3)

p(x)q(x)

x+2x+3=,