Unit 8 edited
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1. The document discusses factorizing algebraic expressions. It explains how to factorize expressions of different types like a2 - b2, a3 + b3, a3 - b3, and ax2 + bx + c through practical activities using cutouts and diagrams. 2. Students are instructed to factorize expressions by cutting shapes from a grid, rearranging the pieces, and determining the resulting factors. This helps visualize the factoring process. 3. Examples from the textbook are also discussed in groups and as exercises to solidify understanding of factorizing different expressions.
![8 aLhul)ftLo cleJo~hs
(Algebraic Expression)
kf7306L M 6
!= kl/ro
aLhul0ftLo cleJo~hsx¿nfO{ b'O{ jf ;f] eGbf a9L cleJo~hs jf kbx¿sf] u'0fgsf] ¿kdf JoQm
ug]{ k|lqmofnfO{ v08Ls/0f elgG5 . o; PsfOdf ljleGg :j¿ksf aLhul0ftLo cleJo~hsx¿sf]
v08Ls/0f ug]{ ljlwx¿ l;sfpg] k|of; ul/Psf] 5 .
@= p2]Zo
of] PsfOsf] cGTodf ljBfyL{x¿ lgDglnlvt s'/fx¿df ;Ifd x'g]5g M
1. a2 – b2 :j¿ksf aLhLo cleJo~hsx¿sf] v08Ls/0f ug{ .
2. a3 + b3 tyf a3 - b3 :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ .
3. ax2 + bx + c :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ .
4. a4 + a2b2 + b4 :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ .
#= z}lIfs ;fdu|L
sf8{af]8{ k]k/, s}“rL, HofldtL afs;, a3 + b3 / a3 - b3 sf] df]8n, nfg]n af]8{, vfU;L, wfuf], ud,
cleJo~hs n]lvPsf cfotfsf/ sf8{x¿
x2 + 5x + 6 x2 - 5x + 6 x2 + x - 6 x2 - x - 6
2x2 - 5x + 6 2x2 - 7x + 6
$= lzIf0fl;sfO lqmofsnfk
1. a2 – b2 :j¿ksf] cleJo~hssf] v08Ls/0f lgDgfg';f/ l;sfpg'xf]; M
(i) df]6f] sf8{af]8{ k]k/df kf7ok':tsdf lbOPsf] h:tf] Pp6f ju{ sf6g'xf]; / To;nfO{ tn b]vfOP
h:tf] u/L sf6]/ k|To]s 6'qmfsf] k5fl8 vfU;Lsf] 6'qmf 6f“:g'xf]; .
(ii) nfg]n af]8{df ;a} 6'qmfx¿ ldnfP/ 6f“:g'xf]; . ljBfyL{x¿nfO{ k|Zg ub}{ a2 af6 b2 sf]
6'qmfnfO{ x6fO{ lbg'xf]; .
(iii) ca af“sL /x]sf] efu a2 – b2 nfO{ cfotfsf/ agfO{ If]qkmn lgsfNg] af/]df 5nkmn ub{}{
kf7ok':tsdf b]vfOPsf lrq cg';f/sf] 6'qmfx¿ ldnfP/ cfotfsf/ 6'qmfsf] nDafO{ a+b /
rf}8fO{ a-b ePsf] lgisif{df k'¥ofpg'xf]; .
49 Ull0ft – (, lzIfs lgb]{lzsf](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-1-320.jpg)
![(iv) ca cfotsf] If]qkmn A = l × b sf] ;"q cg';f/ cfotfsf/ 6'qmfsf] If]qkmn = (a+b) (a-b)
cfPsf] lgisif{df k'¥ofpg'xf]; .
2. oxL tl/sfn] tn b]vfOP cg';f/sf 6'qmfx¿ Knfg]n af]8df 6f“;]/ 5nkmn ub} ;"qx¿ kQf nufpg
{
nufpg'xf]; .
5fof“ kfl/Psf] 6'qmfx¿ lgsfNbf,
Pp6f 6'qmfsf] If]qkmn = ab
csf]{ 6'qmfnfO[ klg a × b agfpg b2 yk
ug'{k5{ .
af“sL 6'qmfsf] If]qkmn = a2 – ab – ab + b2
To;}n] (a-b)2 = a2 – 2ab + b2 eof] .
3. kf7ok':tsf] k]h 74 sf] pbfx/0f 1, 2, 3, 4 nfO{ ;d"xdf 5nkmn u/fpg] -k|lt a]~rsf] ;d"x agfpg
;lsG5 _
4. kf7ok':tsf] k]h 74 sf] cEof; 8.1.1 sf] g+= 1, 5 / 16 -jf cfjZostf cg';f/ cGo_ sIffsfo{sf]
¿kdf u/fO{ af“sL ;d:ofx¿ u[xsfo{sf] ¿kdf ug{ nufpg'xf]; .
5. a3 + b3 / a3 - b3 :j¿ksf cleJo~hssf] v08Ls/0f
a3 + b3 / a3 - b3 :j¿ksf] v08Ls/0f ug]{ ;DaGwdf kf7ok':tssf] cEof; 8.1.2 5nkmn ug]{ /
pknAw ePdf tn b]vfOP cg';f/sf] df]8n k|of]u u/L ;"q k|dfl0ft ug{ nufpg]÷ u/]/ b]vfpg] .
a3 + b3 sf] v08Ls/0f
oxf“ l;ªuf] 3gsf] nDafO, rf}8fO / prfO ;a} a ;]=dL= 5
cyf{t cfotg a3 5 . dWoefuaf6 b ;]=dL= lrGx nufO lrqdf
b]vfOPcg';f/ sf6bf oxf“ klg cf7 6'qmf aGb5g . o;df 7"nf]
3g a3 df Pp6f ;fgf] 3g b3 yk]/ 6'qmfx¿ ldnfp“b} hf“bf
cGtdf prfO (a+b) ePsf] / cfwf/sf] If]qkmn a2 – ab / b2
ePsf] 3gfsf/ j:t' tof/ x'G5 .
;j{kyd Ps ;fO8sf rf/cf]6f 6'qmfx¿ ;a} x6fpg'xf]; . o;/L x6fp“bf Pp6f r]K6f], b'Ocf]6f nfDrf]
|
/ Pp6f 3gfsf/ 6'qmf lg:sG5 .
Ull0ft – (, lzIfs lgb]{lzsf 50](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-2-320.jpg)
![oL 6'qmfx¿ lgsflnPkl5 af“sL /x]sf] efu
ca lgsflnPsf] Pp6f nfDrf]nfO{ lgsfn]/ af“sL /x]sf] efu dfly g} yKg'xf]; . o;f] ubf{ ;f]sf] prfO
a+b aGg k'U5 .
af“sL /x]sf] Pp6f nfDrf] / Pp6f 3gdf csf]{ 3g b3 yk]/ dflysf]
glhs prfO a+b x'g] u/L 78ofP/ /fVg'xf]; .
ca ;f] 3gfsf/ j:t'sf] prfO a+b / cfwf/sf] If]qkmn a2 – ab +
b2 ePsf]n] cfotg (a+b)( a2 – ab + b2 ) eof] .
a3 - b3 sf] v08Ls/0f
oxf“ l;ªuf] 3gsf] nDafO, rf}8fO / prfO ;a} a ;]=dL= 5 cyf{t cfotg a3 5 . dWoefuaf6 b
;]=dL= lrGx nufO lrqdf b]vfOPcg';f/ sf6bf oxf“ klg cf7 6'qmf aGb5g . o;df 7"nf] 3g a3 df
Pp6f ;fgf] 3g b3 yk]/ 6'qmfx¿ ldnfp“b} hf“bf cGtdf prfO (a-b) ePsf] / cfwf/sf] If]qkmn a2
+ ab / b2 ePsf] 3gfsf/ j:t' tof/ x'G5 .
51 Ull0ft – (, lzIfs lgb]{lzsf](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-3-320.jpg)
![;j{k|yd l;ªuf] 3g a3 af6 ;fgf] 3g b3 x6fpg'xf]; . o;/L x6fp“bf a3 - b3 af“sL /xG5 .
ca af“sL /x]sf] a3 - b3 sf] dfly kl§sf 6'qmfx¿ -b'O{cf]6f nfDrf] / Pp6f r]K6f]_ nfO{ lgsfn]/
;fO8df ldnfP/ /fVg'xf]; . o;f] ubf{ ;f]sf] prfO a-b aGg k'U5 .
ca ;f] 3gfsf/ j:t'sf] prfO a-b / cfwf/sf] If]qkmn a2 + ab + b2 ePsf]n] cfotg
(a-b)( a2 +ab + b2 ) eof] .
6. ax2 + bx + c :j¿ksf cleJo~hssf] v08Ls/0f
s_ ax2 + bx + c :j¿ksf cleJo~hssf] v08Ls/0f ug]{ k|lqmof ;'¿ ug'{eGbf klxn] kb ljR5]bg
lgod (distributive law) sf] af/]df s]xL 5nkmn ug{ pko'Qm xf]nf . h:t} M a(b±c) = ab ± ac .
o; ;DaGwL s]xL wf/0ff al;;s]kl5 To; ljR5]lbt kbx¿af6 cl3Nnf] l:yltdf Nofpg ;femf lng'kg]{
l:yltsf] aofg ug]{, h:t} M ab± ac = a(b ± c)
gf]6 M w]/}h;f] ljBfyL{x¿n] a(b × c) = ab × ac ug]{ x'“bf ;f] ug{ x'G5 jf x'“b}g . 5nkmn ug{
nufpg'xf]; .
dfly ;fdu|Lsf] g++ iii df pNn]v ul/Pcg';f/sf ;fdu|Lx¿ ljt/0f ul/;s]kl5 To;sf] aLhLo
cleJo~hs n]Vg nufpg'xf]; . h;sf] ju{sf] nDafO÷rf}8fO x PsfO cfotsf] nDafO x rf}8fO 1
Ull0ft – (, lzIfs lgb]{lzsf 52](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-4-320.jpg)
![PsfO tyf ;fgf] ju{sf] nDafO÷rf}8fO 1/1 PsfOsf 5g . x2 +4x + 3 . tL cf7 6'qmf sfuhx¿af6
Pp6f k"0f{ cfot agfpg nufpg'xf]; .-sfuhsf 6'qmfx¿ rnfP/_ . sIffdf k|ltof]lutf ug{
nufpg'xf]; ls s;n] ;a}eGbf klxn] ;f] cfotsf] nDafO / rf}8fO eGg ;S5 . (x+3)(x+1) h'g
To; cleJo~hssf] u'0fgv08x¿ x'g . lrq o; k|sf/ x'g]5 .
gf]6M ;Dej ePdf To:t} u/L x2 + 2x – 3 sf] u'0fgv08x¿ kQf nufpg x e'hf ePsf juf{sf/
sfuh ljt/0f u/L ;f] ju{df x × 1 PsfOsf cfotx¿ b'Ocf]6f hf]8]/ To; cfotaf6 1 × 1
{
PsfOsf tLgcf]6f ju{x¿ 36fpg nufpg] .
ca To; cfs[lt -5fof gkfl/Psf]_ af6 Pp6f k"0f{ cfot agfpg s] ug'{knf{ < 5nkmn ug{
nufpg'xf]; .
o;sf] nDafO (x + 3) / rf}8fO (x – 1) x'G5 .
v_ x2 – 3x +2 :j¿ksf] v08Ls/0f ug{ x2 sf ju{x¿ ljt/0f ul/;s]kl5 nDafO × PsfO / rf}8fO
1 PsfOsf 3 cfotx¿ To; ju{af6 sf6g nufpg'xf]; / 1 × 1 PsfOsf ju{x¿ 2 cf]6f To;
af“sL cfotdf hf]8g nufpg'xf]; . h;sf] :j¿k o;k|sf/ x'g]5 M
o;nfO{ s;/L k"0f{ cfotsf] ¿kdf agfpg] xf], 5nkmn ug'{xf]; . lrqdf em}“ o;nfO{ nDafOlt/af6 /
PsfO rf}8fO x'g] u/L sf6L cfot agfp“bf of] :j¿ksf] x'g]5 .
o;sf] nDafO (x-3+2 = x – 1) / rf}8fO (x-2) x'G5 .
u_ o;/L k|of]ufTds tl/sfaf6 wf/0ff alg;s]kl5 v08Ls/0f ug]{ k|lqmofsf] nflu kf7ok':tssf] 8.1.3
df lbOPsf] tl/sf tyf pbfx/0fx¿ 5nkmn ug{ nufpg] . cEof; 8.1.3 sf] g+= 1,2,3,4 / 5
sIffsfo{ lbg] / g+= 9,11,15 / 23 ;d"xsfo{ lbg'xf]; . af“sL u[xsfo{ lbg] .
53 Ull0ft – (, lzIfs lgb]{lzsf](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-5-320.jpg)
![7. a4 + a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f
-s_ a4 + a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f ug{ cl3Nnf] sIffdf a2 + ab + b2 :j¿ksf
cleJo~hssf] k"0f{ ju{ agfpg ckgfOPsf] k|lqmofaf/] 5nkmn ug]{ .
ca a4 + a2b2 + b4 df klxnf] / clGtd b'O{ kbx¿ ju{ kb ePsfn] k"0f{ ju{ agfpg ;lsg] s'/f
af]w u/fpg] . lbPsf] cleJo~hs = (a2)2 + a2b2 +(b2)2 ln“bf o; cleJo~hsdf (a+b)2 sf] ;"q
k|of]u x'gsf] nflu lgDgfg';f/ kb yk 36 u/L ldnfpg nufpg] / v08Ls/0fsf nflu cfjZos
r/0fx¿ n]Vg ;xof]u ug]{ .
(a2)2 + 2(a2)(b2) +(b2)2 - a2b2 -oxf“ cleJo~hsnfO{ b'O{ ju{kbsf] cGt/sf] ¿kdf
nfg vf]lhPsf] 5 ._
= (a2 + b2)2 - (ab)2 [(a2 + b2)2 = (a2)2 + 2a2b2 + (b2)2 ]
= [(a2 + b2) + ab] [(a2 + b2) - ab] -a2 - b2 sf] ;"qsf] k|of]u_
= (a2 + ab + b2) (a2 - ab + b2) (a sf] 3ftfªssf] 36bf] qmddf /fvL ldnfp“bf_
∴ a4 + a2b2 + b4 = (a2 + ab + b2) (a2 - ab + b2)
b|i6Jo M a4 - a2b2 + b4 cleJo~hsnfO{ k"0f{ ju{ agfpg ;lsP tfklg o;sf] :j¿k
(a2 + b2)2 - (ab)2 x'G5 h;nfO{ v08Ls/0f ug{ g;lsg] x'“bf a4 - a2b2 + b4 :j¿ksf
cleJo~hssf] v08Ls/0f ug{ ldNb}g egL a'emfpg] .
cEof; 8.1.4 sf s]xL ;d:ofx¿sf] xn
2. p4 + 4
= (p2)2 + (2)2 -b'j} kbx¿ k"0f{ju{ 5g _
= (p2)2 + 2.p2. 2 + (2)2 - 2.p2. 2 (2p2, 2 yKg] / 36fpg]_
= (p2 + 2)2 – 4p2
= (p2 + 2 + 2p)( p2 + 2 - 2p)
= (p2 + 2p + 2)( p2 - 2p + 2)
Ull0ft – (, lzIfs lgb]{lzsf 54](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-6-320.jpg)
![13. x4 + 23x2 + 256
= (x2)2 + 23x2 + (16)2
= (x2)2 + 2.(x)2.16 + (16)2 – 9x2
= (x2 + 16)2 – (3x)2
= [(x2 + 16) + 3x] [(x2 + 16) - 3x]
= (x2 + 16 + 3x) (x2 + 16 - 3x) -b'O{ ju{kbsf] cGt/df n]lvof]_
= (x2 + 3x + 16) (x2 - 3x + 16)
18. x2 – 10x + 24 + 6y - 9y2 -oxf“ klg cleJo~hsnfO{ b'O{ ju{kbsf cGt/df JoQm
ug'{kb{5_
= x2 – 2x.5 + 25 -1 + 6y - 9y2
= (x – 5)2 – (9y2 - 6y + 1)
= (x – 5)2 – (9y – 1)2 -b'O{ ju{kbsf] cGt/df n]lvof]_
= [(x – 5) + (3y – 1)] [(x – 5) - (3y – 1)]
= (x – 5 + 3y – 1) (x – 5 - 3y + 1)
= (x + 3y – 6) (x - 3y – 4)
12 2
= 2. .
ca ljBfyL{nfO{ ug{ nufpg] .
13. 1
55 Ull0ft – (, lzIfs lgb]{lzsf](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-7-320.jpg)
![= 7 1
= 2 .1 1 2 7
= 1 9
= 1 -b'O{ ju{kbsf] cGt/df n]lvof]_
= 1 1
= 1 1
b|i6Jo M cleJo~hsdf ;dfg¿ksf] ljleGg 3ftfªs ePsf] kb ePsf] cj:yfdf To:tf] ¿knfO{ gof“
gfd /fv]/ v08Ls/0f ubf{ ;lhnf] x'G5 .
h:t} M oxf“ dfgf}“ .
lbOPsf] cleJo~hs = x4 - 7x2 + 1
= (x2)2 + 2.(x)2.1 +1 -2x2 - 7x2
= (x2 + 1)2 – 9x2
= (x2 + 1)2 – (3x)2
= (x2 + 1 + 3x) (x2 + 1 - 3x)
= (x2 + 3x + 1) (x2 - 3x + 1)
k'gM dfg kmsf{p“bf,
7 3 3
1 1 1
17. a8 – b8
= (a4)2 - (b4)2
Ull0ft – (, lzIfs lgb]{lzsf 56](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-8-320.jpg)
![= (a4 + b4)( a4 - b4)
= (a4 + b4)[(a2)2 - (b2)2]
ca ;"q k|of]u ug{ nufO{ ljBfyL{x¿nfO{ ;dfwfg ug{ nufpg] .
%= d"Nofªsg
lqmofsnfksf] cfwf/df d"Nofªsg ug]{ .
57 Ull0ft – (, lzIfs lgb]{lzsf](https://image.slidesharecdn.com/unit8edited-130320002155-phpapp01/85/Unit-8-edited-9-320.jpg)
Unit 8 edited
- 1. 8 aLhul)ftLo cleJo~hs (Algebraic Expression) kf7306L M 6 != kl/ro aLhul0ftLo cleJo~hsx¿nfO{ b'O{ jf ;f] eGbf a9L cleJo~hs jf kbx¿sf] u'0fgsf] ¿kdf JoQm ug]{ k|lqmofnfO{ v08Ls/0f elgG5 . o; PsfOdf ljleGg :j¿ksf aLhul0ftLo cleJo~hsx¿sf] v08Ls/0f ug]{ ljlwx¿ l;sfpg] k|of; ul/Psf] 5 . @= p2]Zo of] PsfOsf] cGTodf ljBfyL{x¿ lgDglnlvt s'/fx¿df ;Ifd x'g]5g M 1. a2 – b2 :j¿ksf aLhLo cleJo~hsx¿sf] v08Ls/0f ug{ . 2. a3 + b3 tyf a3 - b3 :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ . 3. ax2 + bx + c :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ . 4. a4 + a2b2 + b4 :j¿ksf cleJo~hsx¿sf] v08Ls/0f ug{ . #= z}lIfs ;fdu|L sf8{af]8{ k]k/, s}“rL, HofldtL afs;, a3 + b3 / a3 - b3 sf] df]8n, nfg]n af]8{, vfU;L, wfuf], ud, cleJo~hs n]lvPsf cfotfsf/ sf8{x¿ x2 + 5x + 6 x2 - 5x + 6 x2 + x - 6 x2 - x - 6 2x2 - 5x + 6 2x2 - 7x + 6 $= lzIf0fl;sfO lqmofsnfk 1. a2 – b2 :j¿ksf] cleJo~hssf] v08Ls/0f lgDgfg';f/ l;sfpg'xf]; M (i) df]6f] sf8{af]8{ k]k/df kf7ok':tsdf lbOPsf] h:tf] Pp6f ju{ sf6g'xf]; / To;nfO{ tn b]vfOP h:tf] u/L sf6]/ k|To]s 6'qmfsf] k5fl8 vfU;Lsf] 6'qmf 6f“:g'xf]; . (ii) nfg]n af]8{df ;a} 6'qmfx¿ ldnfP/ 6f“:g'xf]; . ljBfyL{x¿nfO{ k|Zg ub}{ a2 af6 b2 sf] 6'qmfnfO{ x6fO{ lbg'xf]; . (iii) ca af“sL /x]sf] efu a2 – b2 nfO{ cfotfsf/ agfO{ If]qkmn lgsfNg] af/]df 5nkmn ub{}{ kf7ok':tsdf b]vfOPsf lrq cg';f/sf] 6'qmfx¿ ldnfP/ cfotfsf/ 6'qmfsf] nDafO{ a+b / rf}8fO{ a-b ePsf] lgisif{df k'¥ofpg'xf]; . 49 Ull0ft – (, lzIfs lgb]{lzsf
- 2. (iv) ca cfotsf] If]qkmn A = l × b sf] ;"q cg';f/ cfotfsf/ 6'qmfsf] If]qkmn = (a+b) (a-b) cfPsf] lgisif{df k'¥ofpg'xf]; . 2. oxL tl/sfn] tn b]vfOP cg';f/sf 6'qmfx¿ Knfg]n af]8df 6f“;]/ 5nkmn ub} ;"qx¿ kQf nufpg { nufpg'xf]; . 5fof“ kfl/Psf] 6'qmfx¿ lgsfNbf, Pp6f 6'qmfsf] If]qkmn = ab csf]{ 6'qmfnfO[ klg a × b agfpg b2 yk ug'{k5{ . af“sL 6'qmfsf] If]qkmn = a2 – ab – ab + b2 To;}n] (a-b)2 = a2 – 2ab + b2 eof] . 3. kf7ok':tsf] k]h 74 sf] pbfx/0f 1, 2, 3, 4 nfO{ ;d"xdf 5nkmn u/fpg] -k|lt a]~rsf] ;d"x agfpg ;lsG5 _ 4. kf7ok':tsf] k]h 74 sf] cEof; 8.1.1 sf] g+= 1, 5 / 16 -jf cfjZostf cg';f/ cGo_ sIffsfo{sf] ¿kdf u/fO{ af“sL ;d:ofx¿ u[xsfo{sf] ¿kdf ug{ nufpg'xf]; . 5. a3 + b3 / a3 - b3 :j¿ksf cleJo~hssf] v08Ls/0f a3 + b3 / a3 - b3 :j¿ksf] v08Ls/0f ug]{ ;DaGwdf kf7ok':tssf] cEof; 8.1.2 5nkmn ug]{ / pknAw ePdf tn b]vfOP cg';f/sf] df]8n k|of]u u/L ;"q k|dfl0ft ug{ nufpg]÷ u/]/ b]vfpg] . a3 + b3 sf] v08Ls/0f oxf“ l;ªuf] 3gsf] nDafO, rf}8fO / prfO ;a} a ;]=dL= 5 cyf{t cfotg a3 5 . dWoefuaf6 b ;]=dL= lrGx nufO lrqdf b]vfOPcg';f/ sf6bf oxf“ klg cf7 6'qmf aGb5g . o;df 7"nf] 3g a3 df Pp6f ;fgf] 3g b3 yk]/ 6'qmfx¿ ldnfp“b} hf“bf cGtdf prfO (a+b) ePsf] / cfwf/sf] If]qkmn a2 – ab / b2 ePsf] 3gfsf/ j:t' tof/ x'G5 . ;j{kyd Ps ;fO8sf rf/cf]6f 6'qmfx¿ ;a} x6fpg'xf]; . o;/L x6fp“bf Pp6f r]K6f], b'Ocf]6f nfDrf] | / Pp6f 3gfsf/ 6'qmf lg:sG5 . Ull0ft – (, lzIfs lgb]{lzsf 50
- 3. oL 6'qmfx¿ lgsflnPkl5 af“sL /x]sf] efu ca lgsflnPsf] Pp6f nfDrf]nfO{ lgsfn]/ af“sL /x]sf] efu dfly g} yKg'xf]; . o;f] ubf{ ;f]sf] prfO a+b aGg k'U5 . af“sL /x]sf] Pp6f nfDrf] / Pp6f 3gdf csf]{ 3g b3 yk]/ dflysf] glhs prfO a+b x'g] u/L 78ofP/ /fVg'xf]; . ca ;f] 3gfsf/ j:t'sf] prfO a+b / cfwf/sf] If]qkmn a2 – ab + b2 ePsf]n] cfotg (a+b)( a2 – ab + b2 ) eof] . a3 - b3 sf] v08Ls/0f oxf“ l;ªuf] 3gsf] nDafO, rf}8fO / prfO ;a} a ;]=dL= 5 cyf{t cfotg a3 5 . dWoefuaf6 b ;]=dL= lrGx nufO lrqdf b]vfOPcg';f/ sf6bf oxf“ klg cf7 6'qmf aGb5g . o;df 7"nf] 3g a3 df Pp6f ;fgf] 3g b3 yk]/ 6'qmfx¿ ldnfp“b} hf“bf cGtdf prfO (a-b) ePsf] / cfwf/sf] If]qkmn a2 + ab / b2 ePsf] 3gfsf/ j:t' tof/ x'G5 . 51 Ull0ft – (, lzIfs lgb]{lzsf
- 4. ;j{k|yd l;ªuf] 3g a3 af6 ;fgf] 3g b3 x6fpg'xf]; . o;/L x6fp“bf a3 - b3 af“sL /xG5 . ca af“sL /x]sf] a3 - b3 sf] dfly kl§sf 6'qmfx¿ -b'O{cf]6f nfDrf] / Pp6f r]K6f]_ nfO{ lgsfn]/ ;fO8df ldnfP/ /fVg'xf]; . o;f] ubf{ ;f]sf] prfO a-b aGg k'U5 . ca ;f] 3gfsf/ j:t'sf] prfO a-b / cfwf/sf] If]qkmn a2 + ab + b2 ePsf]n] cfotg (a-b)( a2 +ab + b2 ) eof] . 6. ax2 + bx + c :j¿ksf cleJo~hssf] v08Ls/0f s_ ax2 + bx + c :j¿ksf cleJo~hssf] v08Ls/0f ug]{ k|lqmof ;'¿ ug'{eGbf klxn] kb ljR5]bg lgod (distributive law) sf] af/]df s]xL 5nkmn ug{ pko'Qm xf]nf . h:t} M a(b±c) = ab ± ac . o; ;DaGwL s]xL wf/0ff al;;s]kl5 To; ljR5]lbt kbx¿af6 cl3Nnf] l:yltdf Nofpg ;femf lng'kg]{ l:yltsf] aofg ug]{, h:t} M ab± ac = a(b ± c) gf]6 M w]/}h;f] ljBfyL{x¿n] a(b × c) = ab × ac ug]{ x'“bf ;f] ug{ x'G5 jf x'“b}g . 5nkmn ug{ nufpg'xf]; . dfly ;fdu|Lsf] g++ iii df pNn]v ul/Pcg';f/sf ;fdu|Lx¿ ljt/0f ul/;s]kl5 To;sf] aLhLo cleJo~hs n]Vg nufpg'xf]; . h;sf] ju{sf] nDafO÷rf}8fO x PsfO cfotsf] nDafO x rf}8fO 1 Ull0ft – (, lzIfs lgb]{lzsf 52
- 5. PsfO tyf ;fgf] ju{sf] nDafO÷rf}8fO 1/1 PsfOsf 5g . x2 +4x + 3 . tL cf7 6'qmf sfuhx¿af6 Pp6f k"0f{ cfot agfpg nufpg'xf]; .-sfuhsf 6'qmfx¿ rnfP/_ . sIffdf k|ltof]lutf ug{ nufpg'xf]; ls s;n] ;a}eGbf klxn] ;f] cfotsf] nDafO / rf}8fO eGg ;S5 . (x+3)(x+1) h'g To; cleJo~hssf] u'0fgv08x¿ x'g . lrq o; k|sf/ x'g]5 . gf]6M ;Dej ePdf To:t} u/L x2 + 2x – 3 sf] u'0fgv08x¿ kQf nufpg x e'hf ePsf juf{sf/ sfuh ljt/0f u/L ;f] ju{df x × 1 PsfOsf cfotx¿ b'Ocf]6f hf]8]/ To; cfotaf6 1 × 1 { PsfOsf tLgcf]6f ju{x¿ 36fpg nufpg] . ca To; cfs[lt -5fof gkfl/Psf]_ af6 Pp6f k"0f{ cfot agfpg s] ug'{knf{ < 5nkmn ug{ nufpg'xf]; . o;sf] nDafO (x + 3) / rf}8fO (x – 1) x'G5 . v_ x2 – 3x +2 :j¿ksf] v08Ls/0f ug{ x2 sf ju{x¿ ljt/0f ul/;s]kl5 nDafO × PsfO / rf}8fO 1 PsfOsf 3 cfotx¿ To; ju{af6 sf6g nufpg'xf]; / 1 × 1 PsfOsf ju{x¿ 2 cf]6f To; af“sL cfotdf hf]8g nufpg'xf]; . h;sf] :j¿k o;k|sf/ x'g]5 M o;nfO{ s;/L k"0f{ cfotsf] ¿kdf agfpg] xf], 5nkmn ug'{xf]; . lrqdf em}“ o;nfO{ nDafOlt/af6 / PsfO rf}8fO x'g] u/L sf6L cfot agfp“bf of] :j¿ksf] x'g]5 . o;sf] nDafO (x-3+2 = x – 1) / rf}8fO (x-2) x'G5 . u_ o;/L k|of]ufTds tl/sfaf6 wf/0ff alg;s]kl5 v08Ls/0f ug]{ k|lqmofsf] nflu kf7ok':tssf] 8.1.3 df lbOPsf] tl/sf tyf pbfx/0fx¿ 5nkmn ug{ nufpg] . cEof; 8.1.3 sf] g+= 1,2,3,4 / 5 sIffsfo{ lbg] / g+= 9,11,15 / 23 ;d"xsfo{ lbg'xf]; . af“sL u[xsfo{ lbg] . 53 Ull0ft – (, lzIfs lgb]{lzsf
- 6. 7. a4 + a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f -s_ a4 + a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f ug{ cl3Nnf] sIffdf a2 + ab + b2 :j¿ksf cleJo~hssf] k"0f{ ju{ agfpg ckgfOPsf] k|lqmofaf/] 5nkmn ug]{ . ca a4 + a2b2 + b4 df klxnf] / clGtd b'O{ kbx¿ ju{ kb ePsfn] k"0f{ ju{ agfpg ;lsg] s'/f af]w u/fpg] . lbPsf] cleJo~hs = (a2)2 + a2b2 +(b2)2 ln“bf o; cleJo~hsdf (a+b)2 sf] ;"q k|of]u x'gsf] nflu lgDgfg';f/ kb yk 36 u/L ldnfpg nufpg] / v08Ls/0fsf nflu cfjZos r/0fx¿ n]Vg ;xof]u ug]{ . (a2)2 + 2(a2)(b2) +(b2)2 - a2b2 -oxf“ cleJo~hsnfO{ b'O{ ju{kbsf] cGt/sf] ¿kdf nfg vf]lhPsf] 5 ._ = (a2 + b2)2 - (ab)2 [(a2 + b2)2 = (a2)2 + 2a2b2 + (b2)2 ] = [(a2 + b2) + ab] [(a2 + b2) - ab] -a2 - b2 sf] ;"qsf] k|of]u_ = (a2 + ab + b2) (a2 - ab + b2) (a sf] 3ftfªssf] 36bf] qmddf /fvL ldnfp“bf_ ∴ a4 + a2b2 + b4 = (a2 + ab + b2) (a2 - ab + b2) b|i6Jo M a4 - a2b2 + b4 cleJo~hsnfO{ k"0f{ ju{ agfpg ;lsP tfklg o;sf] :j¿k (a2 + b2)2 - (ab)2 x'G5 h;nfO{ v08Ls/0f ug{ g;lsg] x'“bf a4 - a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f ug{ ldNb}g egL a'emfpg] . cEof; 8.1.4 sf s]xL ;d:ofx¿sf] xn 2. p4 + 4 = (p2)2 + (2)2 -b'j} kbx¿ k"0f{ju{ 5g _ = (p2)2 + 2.p2. 2 + (2)2 - 2.p2. 2 (2p2, 2 yKg] / 36fpg]_ = (p2 + 2)2 – 4p2 = (p2 + 2 + 2p)( p2 + 2 - 2p) = (p2 + 2p + 2)( p2 - 2p + 2) Ull0ft – (, lzIfs lgb]{lzsf 54
- 7. 13. x4 + 23x2 + 256 = (x2)2 + 23x2 + (16)2 = (x2)2 + 2.(x)2.16 + (16)2 – 9x2 = (x2 + 16)2 – (3x)2 = [(x2 + 16) + 3x] [(x2 + 16) - 3x] = (x2 + 16 + 3x) (x2 + 16 - 3x) -b'O{ ju{kbsf] cGt/df n]lvof]_ = (x2 + 3x + 16) (x2 - 3x + 16) 18. x2 – 10x + 24 + 6y - 9y2 -oxf“ klg cleJo~hsnfO{ b'O{ ju{kbsf cGt/df JoQm ug'{kb{5_ = x2 – 2x.5 + 25 -1 + 6y - 9y2 = (x – 5)2 – (9y2 - 6y + 1) = (x – 5)2 – (9y – 1)2 -b'O{ ju{kbsf] cGt/df n]lvof]_ = [(x – 5) + (3y – 1)] [(x – 5) - (3y – 1)] = (x – 5 + 3y – 1) (x – 5 - 3y + 1) = (x + 3y – 6) (x - 3y – 4) 12 2 = 2. . ca ljBfyL{nfO{ ug{ nufpg] . 13. 1 55 Ull0ft – (, lzIfs lgb]{lzsf
- 8. = 7 1 = 2 .1 1 2 7 = 1 9 = 1 -b'O{ ju{kbsf] cGt/df n]lvof]_ = 1 1 = 1 1 b|i6Jo M cleJo~hsdf ;dfg¿ksf] ljleGg 3ftfªs ePsf] kb ePsf] cj:yfdf To:tf] ¿knfO{ gof“ gfd /fv]/ v08Ls/0f ubf{ ;lhnf] x'G5 . h:t} M oxf“ dfgf}“ . lbOPsf] cleJo~hs = x4 - 7x2 + 1 = (x2)2 + 2.(x)2.1 +1 -2x2 - 7x2 = (x2 + 1)2 – 9x2 = (x2 + 1)2 – (3x)2 = (x2 + 1 + 3x) (x2 + 1 - 3x) = (x2 + 3x + 1) (x2 - 3x + 1) k'gM dfg kmsf{p“bf, 7 3 3 1 1 1 17. a8 – b8 = (a4)2 - (b4)2 Ull0ft – (, lzIfs lgb]{lzsf 56