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Unit 8 edited Unit 8 edited Document Transcript

  •   8  aLhul)ftLo cleJo~hs (Algebraic Expression) kf7306L M 6!= kl/ro aLhul0ftLo cleJo~hsx¿nfO{ bO{ jf ;f] eGbf a9L cleJo~hs jf kbx¿sf] u0fgsf] ¿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 xg]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 lqmofsnfk1. a2 – b2 :j¿ksf] cleJo~hssf] v08Ls/0f lgDgfg;f/ l;sfpgxf]; M (i) df]6f] sf8{af]8{ k]k/df kf7ok:tsdf lbOPsf] h:tf] Pp6f ju{ sf6gxf]; / To;nfO{ tn b]vfOP h:tf] u/L sf6]/ k|To]s 6qmfsf] k5fl8 vfU;Lsf] 6qmf 6f“:gxf]; . (ii) nfg]n af]8{df ;a} 6qmfx¿ ldnfP/ 6f“:gxf]; . ljBfyL{x¿nfO{ k|Zg ub}{ a2 af6 b2 sf] 6qmfnfO{ x6fO{ lbgxf]; . (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] 6qmfx¿ ldnfP/ cfotfsf/ 6qmfsf] nDafO{ a+b / rf}8fO{ a-b ePsf] lgisif{df k¥ofpgxf]; . 49 Ull0ft – (, lzIfs lgb]{lzsf 
  •   (iv) ca cfotsf] If]qkmn A = l × b sf] ;"q cg;f/ cfotfsf/ 6qmfsf] If]qkmn = (a+b) (a-b) cfPsf] lgisif{df k¥ofpgxf]; .2. oxL tl/sfn] tn b]vfOP cg;f/sf 6qmfx¿ Knfg]n af]8df 6f“;]/ 5nkmn ub} ;"qx¿ kQf nufpg { nufpgxf]; . 5fof“ kfl/Psf] 6qmfx¿ lgsfNbf, Pp6f 6qmfsf] If]qkmn = ab csf]{ 6qmfnfO[ klg a × b agfpg b2 yk ug{k5{ . af“sL 6qmfsf] 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{ nufpgxf]; .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 6qmf aGb5g . o;df 7"nf] 3g a3 df Pp6f ;fgf] 3g b3 yk]/ 6qmfx¿ ldnfp“b} hf“bf cGtdf prfO (a+b) ePsf] / cfwf/sf] If]qkmn a2 – ab / b2 ePsf] 3gfsf/ j:t tof/ xG5 . ;j{kyd Ps ;fO8sf rf/cf]6f 6qmfx¿ ;a} x6fpgxf]; . o;/L x6fp“bf Pp6f r]K6f], bOcf]6f nfDrf] | / Pp6f 3gfsf/ 6qmf lg:sG5 .Ull0ft – (, lzIfs lgb]{lzsf 50 
  •   oL 6qmfx¿ lgsflnPkl5 af“sL /x]sf] efu ca lgsflnPsf] Pp6f nfDrf]nfO{ lgsfn]/ af“sL /x]sf] efu dfly g} yKgxf]; . o;f] ubf{ ;f]sf] prfO a+b aGg kU5 . af“sL /x]sf] Pp6f nfDrf] / Pp6f 3gdf csf]{ 3g b3 yk]/ dflysf] glhs prfO a+b xg] u/L 78ofP/ /fVgxf]; . ca ;f] 3gfsf/ j:tsf] 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 6qmf aGb5g . o;df 7"nf] 3g a3 df Pp6f ;fgf] 3g b3 yk]/ 6qmfx¿ ldnfp“b} hf“bf cGtdf prfO (a-b) ePsf] / cfwf/sf] If]qkmn a2 + ab / b2 ePsf] 3gfsf/ j:t tof/ xG5 . 51 Ull0ft – (, lzIfs lgb]{lzsf 
  •   ;j{k|yd l;ªuf] 3g a3 af6 ;fgf] 3g b3 x6fpgxf]; . o;/L x6fp“bf a3 - b3 af“sL /xG5 . ca af“sL /x]sf] a3 - b3 sf] dfly kl§sf 6qmfx¿ -bO{cf]6f nfDrf] / Pp6f r]K6f]_ nfO{ lgsfn]/ ;fO8df ldnfP/ /fVgxf]; . o;f] ubf{ ;f]sf] prfO a-b aGg kU5 . ca ;f] 3gfsf/ j:tsf] 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/0fs_ 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{ pkoQm 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 lngkg]{ 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{ xG5 jf x“b}g . 5nkmn ug{ nufpgxf]; . 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 nufpgxf]; . h;sf] ju{sf] nDafO÷rf}8fO x PsfO cfotsf] nDafO x rf}8fO 1Ull0ft – (, lzIfs lgb]{lzsf 52 
  •   PsfO tyf ;fgf] ju{sf] nDafO÷rf}8fO 1/1 PsfOsf 5g . x2 +4x + 3 . tL cf7 6qmf sfuhx¿af6 Pp6f k"0f{ cfot agfpg nufpgxf]; .-sfuhsf 6qmfx¿ rnfP/_ . sIffdf k|ltof]lutf ug{ nufpgxf]; ls s;n] ;a}eGbf klxn] ;f] cfotsf] nDafO / rf}8fO eGg ;S5 . (x+3)(x+1) hg To; cleJo~hssf] u0fgv08x¿ xg . lrq o; k|sf/ xg]5 . gf]6M ;Dej ePdf To:t} u/L x2 + 2x – 3 sf] u0fgv08x¿ kQf nufpg x ehf ePsf juf{sf/ sfuh ljt/0f u/L ;f] ju{df x × 1 PsfOsf cfotx¿ bOcf]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{ nufpgxf]; . o;sf] nDafO (x + 3) / rf}8fO (x – 1) xG5 .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 nufpgxf]; / 1 × 1 PsfOsf ju{x¿ 2 cf]6f To; af“sL cfotdf hf]8g nufpgxf]; . h;sf] :j¿k o;k|sf/ xg]5 M o;nfO{ s;/L k"0f{ cfotsf] ¿kdf agfpg] xf], 5nkmn ug{xf]; . lrqdf em}“ o;nfO{ nDafOlt/af6 / PsfO rf}8fO xg] u/L sf6L cfot agfp“bf of] :j¿ksf] xg]5 . o;sf] nDafO (x-3+2 = x – 1) / rf}8fO (x-2) xG5 .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{ lbgxf]; . af“sL u[xsfo{ lbg] . 53 Ull0ft – (, lzIfs lgb]{lzsf 
  •  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 bO{ 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 xgsf] 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{ bO{ 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 xG5 h;nfO{ v08Ls/0f ug{ g;lsg] x“bf a4 - a2b2 + b4 :j¿ksf cleJo~hssf] v08Ls/0f ug{ ldNb}g egL aemfpg] .cEof; 8.1.4 sf s]xL ;d:ofx¿sf] xn2. p4 + 4 = (p2)2 + (2)2 -bj} 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 
  •  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) -bO{ ju{kbsf] cGt/df n]lvof]_ = (x2 + 3x + 16) (x2 - 3x + 16)18. x2 – 10x + 24 + 6y - 9y2 -oxf“ klg cleJo~hsnfO{ bO{ 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 -bO{ 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 
  •   = 7 1 = 2 .1 1 2 7 = 1 9 = 1 -bO{ 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] xG5 . 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) kgM dfg kmsf{p“bf, 7 3 3 1 1 117. a8 – b8 = (a4)2 - (b4)2Ull0ft – (, lzIfs lgb]{lzsf 56 
  •   = (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