Welcome to Unified Communication classroom
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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=,