Đa thức và nhân tử hóa chương iv vành đa thức

CHUONG IV VANH DA THUG §1 VANH DA THHC MOT AN Vann da then mOt 6n nal niem da flak la khai niam ma ta da lam quen it nhiau a ph6 thong 111 got la da Oak, met tdng c6 (bang a, + aix + + am?" n, la nbitng s6 that va x la met chlt d6 cat a, i = Phe-p Ong va phep nhan da thdc la (a, + a ix + + amx") + (b0 + b ix + a, + b o + + (am + em).?" + +b/) = in +1 + (a, + a ix + + anin )(60 + b iz + + bnx") = + (aobk + I + (Lo b, + (aob i + aib0)x + + akb„)xk + + amen?" d6 ta gift sit n + bnxn + Tn es day thong ta hay dinh nghia da thdc met each tdng quat Jadn va chinh xac hon Gilt ad A la met vanh giao hoan, cd don vi ki hieu la Goi P la tap help the day (a,, , , , ) cac al e A vai moi i E N va bang tat ca trit met s6 hem han Mau vey P la mot be phan via lay thaa de` cam 441.! 97 Tai ngay!!! Ban co the xoa dong chu nay!!! 169900260 (ch I, §1, 15) al dinh nghia phep ceng va phep nhan P nhu sau (1) (an, air -, + (be, (a0 + be, + = an + b e , , ae, )(be, b p , (2) (a0, = (co) e V - c —) vet E akbo = abi, k = 0, 1, 2, i+j=k ek = aobk VI cdc al va bi bang tat ca trit met s6 huu han nen cdc ci ming bang tat ca trit met se huu han, cho nen a1 + bi va (1) va (2) cho to hat phep toim P Ta hay chdng minh P la met vanh giao hoan cc) don vi Trttec Mt hi4n nhien pile)) Ong la giao holm M kat hop Phan tit khong la day (0, 0, , 0, ), Agri tit dal cila day (a0, a,,, ) la day (— a0, — a r , , — an, ) Vey P la met nh6m Ong giao hoan VI A la giao hoan, nen i+j=k i+j=k do pile)) Men la giao hoan Do phdp Man A c6 tinh chat kat hop va phan phtli dal vai phep Ong, nen vol mei m = 0, 1, 2, -ta cd the viat Ea Eb E ahbi ) E (alibi) =j+1=m h +i+j=m ‘1.1-4-i=1 h it/ ) = G ah(bfj ) h+k=m +j=kh+i+j=m = tit dd to al Agri nhan P la kat hop Day (1, 0, , 0, ) 98 la !Than tis don vi cim P Vay P la 1111# vi nhdm nhan giao hoan t Cu6iclngphadtroAcenavig E cti(bi + = i+j=k E alb,/ + E aPj i+j=k i+j=k vol moi k = 0, 1, 2, ., ta suy tit lust phan phdi - P Bay gib ta hay Kat day x = (0, 1, at co then quy tac nhan (2) x2 k3 = = (0, 0, 1, Q (0, 0, 0, 1, • • = (0, 0, , 0, 1, 0, n lh quy valc vi&t x° = (1, 0, Mat khac ta xet anh xa Av P a (a, 0„ , 0, ) Xnh xa hign nhien la mat don eau (vanh) Do tit gib ta clang nhat plign to a E A veil day (a, 0, , 0, ) E P, tit vi vay A la mat vanh coo vanh P Vi mai phan to cim P la mat day (%,ate , ) the bang tat ca trit mat so` hitu hen, cho nen mai phalli to caa P co clang (cto, alp , ac 0, ) , cid a a vOi (a, 0,0 , E A kh6ng nhat thigt khac \riga (long nhat 0, ) va viac dua vao day x cho phop ta vigt 99 (a., %, 0, ) (a., 0, ) + (0, a1, 0, ) + + (0, a., 0, .) = (a., 0, ) + (a i , 0, ) (0, 1, 0, .) + + + (an, 0, ) (0, 0, 1, 0, ) = a + aix + + ax" = = anx° + st lx + + a n x" Nob( to thubng ki hieu cac phan td elm P vier dual clang + a ix + + an? bang f(x), Dinh nghia Vinh P goi la heath da thhc ctia tin x idly he hi A, hay van tat vInh da thac cua an x tren A, va hiOu la A /X/ Cite Orin tit cua vanh dd goi la da thee oda x lay he A Trong mat da thdc an f(x) = ox + a ix + + an? cac i = 0, 1, n gal la cac he hi caa da thdc Cac as goi la cac hang hi mist da thitc, dim biOt ace = o goi la hang hi hi Bac ctla m01 da thac Xet mat day (an, a , an, ) thuoc vanh P Vi cac ar bang tat ca trb mot s6 hitu hart nen nau thi bao gib cUng cd mat chi s6 n cho a.x ye a, = I > n Theo nhu tren, to vita (an, ., an, 0, ) = + aix + an? Dinh nghin Elbe caa da thud khac f(x) = aox° + +an _ + vbi a * 0, n > 0, la n He to a goi is Ile to cao nlilit cila fix) 100 NMI vhy ta chi dinh nghia bac coa mat da flit khac D61 vii da thdc ta bao no khOng c6 bhc Dinh li GU/ sit f(x) ua g(x) Ia hai da Mac kheic (i) Neu bac f(x) khac bac g(x), thi ta co f(x) + g(x) m ua bac (f(x) + g(x)) = max (bac f(x), bac g(x)) Neu be f(x) = bac g(x), on neu them nha f(x) + g(x) # 0, thi ta co bac (f(x) +g(x)) s max (bat f(x), t4c g(x)) (ii) Neu f(x) g(x) x 0, thi ta co bac (f(x) g(x)) Lc bac f(x) + bac g(x) Viac chung minh khOng c6 gi kh6 khan, xin nhubng cho ban doe Dinh li Neu A la mot mien nguyen f(x) tia g(x) la hai da Mac kluic cart vanh A[x], thi f(x) g(x) # Mr bac (f(x) g(x)) = bac f(x) + bac g(x) Chang mink Gia sd f(x), g(x) E A[x] la hai da thdc khac f(x) = ac + + amain (am # 0) g(x) = bo + + (bn # 0) Theo quy tac nhan da thdc ta f(x)g(x)= aobo + +(a.bk + +akb o)xk + + amboxn+m am va b Winn 0, nen amb # (A khOng Tide cda kliOng), d6 f(x) g(x) # va bac (f(x) g(x)) = m + n = bac f(x) + bac g(x)1 HO qua Neu A la mien nguyen, thi A [x] cling In mien nguyen Phep chia vii du Trong muc to da thdy ngu A la mat mien nguyen thi A[x] clang la mat mien nguyen Ta td dat eau hei : ngu A la mat tutting thi A [x] co phai la mat tuning khOng ? Cau hOi duqc 101 tit lbi hie khac, A[x] khong phai la met truing vi da three_ x chang han khong co nghich &to Thy fly truing hop A[x] la met mien nguyen dew Met, no la met vanh oclit (eh V, §2) nghla la met vanh di( cc( phep chic voi du Dinh 11 Gid sit A la mot trilling, f(x) va g(x) x la hai da three ezia vanh A[x] ; thd thz bao gib wing co hai da three day nhat q(x) va r(x) thitec A silo cho f(x) = g(x) q(x) + r(x), noi bdc r(x) < bdc g(x) nen r(x) # O Cluing minh TrUtIc hdt ta hay chling minh tinh nha set fix) = g(x) q'(x) + bac r'(x) < be g(x) neu r'(x) # Ta suy = g(x) (q(x) - ex)) + r(x) - r'(x) IsIdu r(x) = r'(x), ta co g(x) (q(x) - q'(x)) = 0, vi g(x) # va A[x] la met mien nguyen, nen , suy m q(x) - q'(x) = We la q(x) = q'(x) Gia sii r(x) # r'(x), vay bac (r(x) - r'(x)) = her (g(x)(q(x) - qrx)) = bac g(x) + bac (q(x) - &fix)) (dinh If 2) Mat khac theo gia thiet va dinh II bac (r(x) - r'(x)) S max (bac r(x), bac r'(x)) < bac g(x) bac g(x) + bac (q(x) - qlx)), &du mau thuan voi clang thile tren Chu y : nau met hai da thdc r(x) va r'(x) bang thi ta khong thd nit deli bac nhung then khong anh hureng Uri viec chdng minh, , vi 10c bac (r(x) - r'(x)) bang W.c r(x) nen r'(x) = va bang bac r'(x) ndu r(x) = Con sit ton tai cua q(x) va r(x) thi suy tii thuat Wan dud! day Tim q(x) va r(x) goi la three hien phep chia f(x) cho g(x) Da Utak q(x) goi la thuong, da thttc r(x) la du cua f(x) cho g(x) Viec tim thuong va du la tic Mac ndu b4c f(x) < bac g(x) Ta 102 chi can dat q(x) = 0, r(x) = f(x) Trong twang hap trai lai ta dung nhan xet eau day : Neu ta biet mot da attic h(x) cho fi(x) = f(x) - g(x) h(x) cc) bac tittle stir be hen bac cua f(x) thi bai Man tra don gian tam : tim thadng va du cha fi (x) cho g(x) That vay, nee fi (x) = g(x) qt (x) + ri(x), ta guy f(x) = g(x) (h(x) + q i(x)) + r i (x) tit q(x) = h(x) + q i(x), r(x) = r i (x) Trong tittle tihn, vat f(x) = + + + ao g(x)=bne +bn _ixn + + be , br, m va n m to nhan xet am thi da thtic fi (x) = f(x) - g(x) h(x) rang, My h(x) = ce bac tittle sit be him bac cfm f(x), hoac fi (x) bang Trong tniang hop f (x) = 0, dv r(x) = va thitang q (x) = h(x) Ngu fi (x) x ta tip tut vai fi (x), ta dude f2(x) Day da tilde fi(x), f2(x) ce bac giam dam Khi ta di den mot da tittle ce bac flute sit be ban bac mitt g(x) thi da thIc d6 chinh la du r(x.) Ngu mot da thtic cfm day bang thi dv r(x) = De nhin thay 1.6 hon ta hay vast the bade ma ta da thuc hien d6 duqc day fi (x), fi (x) = f(x) - g(x) h(x) f2 (x) = ft(x) - g(x) h i (x) fk(x) = fk_ (x) - g(x) hk_ k (x) vai fk(x) = hoac bac fk(X) < be g(x) Ding va yea ve cite clang thtic de 1ai, ta duqc 103 f(x) = g(x)(h(x) + h i (x) + + hk _ 1(x)) + fk(x), tit q(x) = h(x) + h i(x) + + hk _ (x), r(x) = fk(x) n Vi du Trong time flan cla time hien chic fix) cho g(x), no.tbi to sap did nhu sau da lap day fi(x), f2(x) 1) A la tntbng 56 him tl —x3 — 7x2 + 2x — —x3 + x2 — — x — 2x2 + 2r — 1 x +4 —8x2 + — x — — 8x2 +8r —4 11 x Tv d6 —x3 — 7x2 + 2x —4 = (-2x2 + 2x — 1) x + ) — x 2) A la trubng cac s6 nguyan mod 11 —1x3 — 7x2 + ix — - Tx3 + 1x2 - 5x - 8X2 - 8X2 aX gX - - aX2 aX - 6x + g g Vay —1 — 7x2 2x — = (-2x + 2x — 1)(6x + 4) 104 • Tit (hob nghia (ch III, §1, 2) va djnh 11 ta co the khAc qua H qua f(x) chia hit cho g(x) va chi *hi du phep chin f(x) cho g(x) bang NghiOm cga mOt da thtic Dinh nghia GM si c la mOt phan tit tag g eim vanh A, /la/ = an 4- apc + + an i la mOt da thlic thy 3, cim vanh Aix] ; phiin tit f(c) = an + a i c + + an e e A dttqc bang each thay x bai c goi la gin tri caa f(x) tai c Ngu f(c) = thi c goi la nghigm caa f(x) Tim nghiem ciut f(x) A goi la gidi phuong trinh dui s6 b4c tt ant + + an L (an * A Dinh H Gid sit A la mdt &yang, c E A, f(x) E Aix] The can pile)) chia f(x) cho x - c In f(c) Ching mink Ngu ta chia f(x) cho x - c, du hoac bang hoac la mat da tilde bac vi bac (x - c) bang Slay du la mat phan tii r E A Ta cd f(x) = (x - c) q(x) + r Thay x bang c, to dupe f(c) = q(c) + r, vfly r = f(c) n HO quA c la nghiqm caa f(x) vet chi f(x) chin hit cho x - c Thqc Man phep chia f(x) = anx" + cho + + an c, ta dude cac he td cua da tilde thing q(x) = b oxn i + b izji + + n_ 105 cho belt cac cOng flute bo = ao bi = + cbi , -1 va du r = an + _ Vi r = f(c), ta say mat pluming phdp (phuong plulp Hoocne) dg tinh f(c) bang so dc" eau day : a a, chb a n -1 rani phan to taa clang thd nhi doge bang each cOng vao phan tit taring ang mla ding thil nit& tich cita c voi phan to dilng trot& clang tht? nhi Dinh nghia Gia sit A la mat tniang, c E A, fix) E Aix] va m la mat ea to nhien 1, c IA nghiem bei cap m nau va chi nal' f(x) chia hat cho (x - cfn va f(x) khOng chia hgt cho (x Trong trUang hop m = noted ta can goi c IA nghiem don, In = thi c la nghiem Jeep Ngtted ta coi mat da thttc ea mat nghiem bed cap m nhu mat da thole c6 m nghigm trimg vai Pit W dai sd ve phiin td• sieu vi5t WA sit A la mat truong cim mat truitmg K, c E K va f(x) = ao + aix + + ane la mat da tittle dm vanh AM Lac do, vi a o, ., an E A nen ao, a1, , an E K, de ta co thg coi f(x) la mat da thim lay he tit K va f(c) la mat phan to thuOc K Ngu f(c) = thi c goi la nghiem cka met da this° lay he to A Dinh nghia Gial sit A IA mat truting caa mat truting K Mat phan to c E K goi la dai so tren A nau c la nghiem dm mat da Glue khan lay lag to A ; c goi la sieu Met tren A trttang hop trai 106 fly u va u i la hai nghiem pinning trinh bac hai 2 — Y1 — = ° vdi he s6 thoc, dd it, va u l la phtic lien hop Ta suy ra, (8), ode nghiem y , y2 , y3 M that Prong true:mg hop my vi A * nen D * 0, do 5/1 , y2 , y3 I& ba nghiem phan Met Cac s6 Yt y2 , y3 la thoc, nhung muon tinh cluing theo cling ante Cac-da-n6 thl lai phai lay can bac cna nhung so phtic Ngubi ta da thong minh dude rang, trtiOng hap A < 0, khang thd bidu thi the nghiem cua phuong trinh (3) bang Mc can thUe vai luring that duoi can Phuong trinh bGc b6n Phep giai mot phuong trinh bac ban x4 + ax3 + bx2 + ex +d = vOi he s6 pink toy se dtta ve phdp giai mat phuong trinh bac ba goi la phuong trinh giai bac ba 1a flan hanh nhu sau : , Chuydn ba hang tit cudi sang vg phai rdi ceng a2 x2 vao ca hai vd, ta duoc a ry ( — x2 — cx — d (x2 + t Sau dd ta Ong vao hai vd cua phuong trinh tong ax ( x2 ± +4 dd y la met an mth, ta dttoc (11) 166 (x2 + (X ax + Y = ( a — + y x2 + ay y2 + - x += - d 'Pa hay Itta chgn an phu y cho vg phai lh mat chinh phuong Mien thg thi chi vide lam trial tieu bide sd dm tam thitc bac hai din voi x vg phhi ( a-g- — a2 (4 Y b Y ) ( -4- - =° hay Y3 - by2 + (ac - 4d)y - [d(a2 - 4b) + c21 = DO la mat phuang trinh bac ba, gal la phuang trinh gidi cita phuang trinh da cho Gia sit yo la mat nghiem cua phuong trinh de Dien yo van phuong trinh (10), ta dugc vg phai elm phuong trinh la mat chinh phuang ax ( x n +2 = (ax 0) ) de x2 + ax -2- -2- - Yo p,x2 + aX Hai phuong trinh bac hai d6 se cho tat ca bon nghidm elm phuong trinh bac ban Vay phep giai mat phuong trinh bac bOn da dugc dua ye phep giai mat phuong trinh bac ba vit hai phuong trinh bac hai Ta say tit rang phuong trinh bac bon gihi dvtgc bang can thdc Vi dg Dial phudng trinh bac bon x _ 3x3 ax2 _ +2 Chuyan ba hang tit cute sang vg phai _ 3x3 _3x2 + 3x _ Sau de cang vito hai vg 9x2 , ta dude 3x = —3x2 + 3x - (x — (x 167 + Y7 Cuai cang Ong vito hai ve t ng - + )') = (12) (x2 = (y - -4) +3x ( + =42 -2 Ta flu eau y lam that flan biet s6 9(1- 4(y y2 ( - 2) = 0, hay sau khai trien Y2 - 3) + 3' - = 3)(Y2 ÷ 1) = O Chon y = 3, phuang trinh (121 tra 3x ( X2 3x Tir d6 T 12 T 3x - 3x hay 3x • 2' 3x x2 - 3x + = , x2 + = - Hai pinning trinh chip ta ben nghiem 1, 2, i, -1 Su thuc &tieing hop cu the ta nhinthay nghiem cart phuang trinh bac ben da cho vi tang s6 cac he s6 bang Do &I ta co' the vier M lam n6i bat nghiem x4 - 3x3 + 3x2 - 3x + = (x - 1)(x3 - 2x2 + x - 2) Vay van de bay gib la giai.phuang trinh x3 - 2x2 + x - =0 ma ta cei the yin x (x - 2) + x -2 = (x - 2)(x + 1) =0 Ta 168 d6,7 ta suy cac nghiem cfut phuang trinh Nguiri to chdng minh rang khong the giai bang can tilde cac pinning trinh tdng quest bac Ion hon bon (Aben, Galoa) Hon the nda, Galoa da tim &toe ties chufin dd Nat mot phuong trinh da cho gial dugc bang can tilde hay khOng BAI TAP Trong vanh Crxl (C la tniang s6 Mlle) hay plan tich cac da thdc : x2 + x° — — x2 — 4i + 3, x7 — — i5 met tich nhfing da thdc Mt kha quy Bidu din hinh hoc cat nghiftm cues da thdc f(x) = xP — , g(x) = (x — — b (a x 0) Tit dd suy rang f(x) va g(x) cd khong qua hai nghiOrn chung Tim cac nghi6m plate cua da thim f(x) = (1 — x2)3 + 8r3 Phan tich da thdc f(x) deb nhttng da thdc Mt UM quy vei M s6 thy° Trong vanh C[x] chting minh rang da thdc f(x) chia hat cho da thdc g(x) va chi moi nghiOm eta g(x) dau la nghiOm cna f(x) va mcd nghiOm bOi cap k eta g(x) cling lit nghiam boi cap Ion hon k eta f(x) Trong vanh Q[x] (Q la trubing s6 hitu ti), chtIng- minh rang da tilde f(x) = x31` + x3/+I X312+2 ehia hot cho da thdc g(x) = x2 + x + vdi k, 1, n la nhting s6 tu nhian thy y Trong vanh Q[x], chiing minh rang da thdc f(x) = x3 - - 3n2 x + n3 , vdi n la mot ad to nhien khac 0, la mot da thdc bat MA guy: 169 GM sit a = + i a) Bigu din a dual clang Wang giac Tim madun va ae-gu-men dm a" W a' b) Vigt an va a-fi dual clang a + bi c) Bigu din hinh hoe can gia tri a" va vol n d) Chting mirth rang MO so phtle z yd@u c6 thg bigu din (Wan mot each nhat dyed clang z = x + ya vat x, y la nhang so thgc e) Chung minh rang anh xa f : C M z=x+y •al—o[ x Y —2y x + 2y W vanh s6 phtie clan vanh the ma trap than vuOng cap la mot Bang eau Tit suy rang tap hap the ma trail, thge,vuOng cap clang [ -2y x + 2y] la mat tniong f) TInh [ —2 21 Giai the phuang trinh bac ba sau day : a) 4y3 — 36y2 + 84y — 20 = b) x3 — x — = c) x3 + las + 15 = d)x3 + 3x2 — 6x + = Ch'ing mink (bang da thdc xang) : (xt _ x2)2 (x2 _ x3) (xi _ x3)2 = _4p vai xt , x2 , x3 la the nghiam caa phgung trinh X3 170 px +q = 27q2 10 Giai the phuong trinh a) x4 - 3x3 + x2 + 4x -6 = b) x4 - 4x3 + 3x2 + 2x - = c) x4 + 2x3 + 8x2 + 2x + = 4) x + 6x3 6x2 8=0 §2 DA THOC VI% HE SO HOU Ti NghiOm hUu ti ctia mot da thtic ved h0 s6 hem ti Trade St ta nhan xet rang Su f(x) = an xn + + an (a n m 0) la met da than vol he s6 hdu ti thi f(x) cd the vial (lured clang + cto) = g(x) f(x) = tri (an ? + dd b la man s6 chung cua the pilau s6 va the la nhung s6 nguyen Vi f(x) va g(x) chi khan met nhan to bac nen cac nghiem Ma f(x) la Mc nghiem Mut g(x) Vay viec tim nghiem eila met da tilde vdi he s6 him ti dude cliia ve viec tim nghlOm mla met da thdc vdi he s6 nguyen Mat khac ta tong nhan xet rang Su a la nghiem cim da thdc g(x) a di + a ce1-1 + + a a + a = n ta cd sau nhan hai ve yea a: (an at' + _ I (an - I + + a: -2 (an a) + ao 7b cd # = an a la nghiem thew da thdc h(x) + an _ i x' + + ann x + ao 171 veil he stS nguyen va he s6 cao nhat bang Do dd mudn S the nghiem etag(x) ta chi vies tim eac nghiem eta h(x) Ta dal van de a day lit tim cac nghiam hau ti etaeac da thac f(x) cri clang f(x) = + an _ i xn I + + x + ao vdi the nguyen Gilt sit a la mat nghiem hitu ti eta f(x), the+hi theo ("nth II eta (Ch V, §1, 2), a phki a nguyen Matt !chive, to + an _ian + + aka + an = ta co thg vi a(an1 + an _ian-2 + + = dd al a Rini vey the nghiem nguyen aka f(x), neu cd, phai la nhang It& eta ao Cho nen muen tim the nghiem nguyen etc fix) ta xet the tide eta s6 hang ao, va sau dd, thit xem the tuft de S phai la nghiem eta fix) hay khang De hart the se lan th8 nguea ta chia the then xet sau day us Gia sit a la mat nghiem nguyen aim f(x) The thi f(x) Chia het cho x - a f(x) = (x - a)q(x) va theo ad de Rode ne (Ch IV, §1, 4), q(x) la mat da thde vai he se nguyen Do q(1) va q(-1) la nhang s6 nguyen vb /(1) - (1) a = q(1)' + a Cr( 1) ngu a klthe va -1 Vi thy trade het ta tinh f(1) va f(-1) ctg xem va -1 ed phai lit nghiem eta f(x), sau ta xet the vac a ±-1 Sa a cho f( 1) - a v- + a la nhang so nguyen de tha xem chUng ed phai la nghiem aka f(x), va do se Ian thit eta to bat di mil Chung 172 Vi du Tim nghiem hitu ti Mut da thile f(x) = x5 - 8x4 + 20x3 - 20x + 19x - 12 Trude het ta ed tong cac he se eiut f(x) bang nen la nghiem cda f(x) Bang so de Hodc ne (Ch IV, §1, 4), ta tinh eat he s6 cua da thtie thuong g(x) phep chia f(x) cho x-1 -8 20 -20 19 -12 -7 13 -7 12 dd eac nghiem lai caa f(x) la cac nghiem cua g(x) g(x) = x4 - 7x3 + 13x2 - 7x + 12 'Pa nhan xet rang g(c) > v6i m9i c c 0, nen g(x) khOng cd nghiem am Vi vay ta chi xet cat u6c thing dm s6 hang tti : 1, 2, 3, 4, 6, 12 Ta cd g(1) = 12, g(-1) = 40 VI eat s6 40 40 40 3' 7' 13 khOng phai la nguyen, nen 2, 6, 12 khOng Olaf la nghiem cua g(x) Cite s6 ye deulam cho g(1) va g(-1) -a 1+ a nguyen, nen clung ed the la nghiem cult g(x) Mu6n their xem clung cd phai la nghiem cent g(x), ta chi viec tie') tau vao bang tie)) nhu sau : -8 20 -20 19 -12 1 -7 13 -7 12 -4 -4 173 Vey ta co f(x) = (x - /)(x - 3)(x - 4)(x2 + 1) Cae nghiem nguyen cua f(x) la 1, 3, Da thee bat arid quy ads vanh Q[x] D6i vdi trifling s6 thvc It va tntbng s6 phdc C, van de xet xern met da tilde da cho cfia vanh FtkJ hay C[x] co Mt kith quy hay khang rat don glint (01, 1) ; nhvng vanh Q[x] Q la trttling s6 hdu ti thi van d4 phdc tap him nhieu D6i vdi cac da thdc bee hai va ba cua 411[4, viec xet item S Mt kith quy hay kitting &roc dtta ve viec dm nghiem hUu ti gda cac da thdc (eh V, §1, bill tap 2) : CAC da thdc bec hai va bee ba cim Q[x] la bat kha quy ya chi ehting kh6ng S nghiem hUu ti D6i veri cac da tilde Mc Ian hon ba thi van dg phdc tap Min nhigu Chang him da t hdc x4 + 2x2 + = (x2 + 1)2 rilang)(hocemitunw,hgocmetWC thvc sit x2 + 1, vey khang phai la bat kha quy Trong mac ta da nhen set ring mci da thde fix) vdi he s6 hdu ti deu S the vigt dudi clang f(x) = 6-1 g(x) dci 1Ft mot s6 nguyOn Mute 0, g(x) la met da thile veil he se nguyen Trong vanh Q[x], f(x) va g(x) la lift Mt vity f(x) la Mt ad quy va chi g(x) lit bat kha quy Do dri tieu chum Aidenstaind ma ta dim m chafe day dg xet met da thdc cua Q[x] co Mt kha quy hay•khang la tieu chum cho cac da th1e veil he se nguyen De chuan bi cho viec chting minh tIeu chuifn ay, trout Mt ta gidi thieu khai niem da Dixie nguyen ban va ehtIng minh hai b6 de Dinh nghla Gia sit f(x) la mot da auk vol he sec nguyen, f(x) gal IA nguyen thin ngu Se he se oda f(x) khdng co vdc Chung nail khdc ngoai ±1 Cho mot da t h dc vdi he s6 Nguyen f(x) C z[x], Id !lieu bang a vac chung Idn nhat Sit cdc he s6 cda f(x), ta cel 174 f(x) = ac(x) vdi ((x) e Z[xl va the he se cda r(x) kh6ng ed Lida Chung nit° khac ngoal ±1, the la r(x) nguygn ban Ndu f(x) E Q[x] thi to ed thg vigt f(x) dttdi dang f(x) = b COO fa(x) nguygn ban va a, b E Z nguyen t6 ding Vi du f(x) = 6x + 2x2 = 176 (20x3 15 )(60x + 24x - 75) - -2- = 8x2 28) ' Cac se 20, 8, -25 Itheng S doe chung nao kink ngthi va -1 Be de Tick cua hai da thdc nguyen ban la mat da thdc nguyen bdn Cluing mink Gitt sit f(x) = a + a ix + + amx m g(x) = b + b ix + + b rix" la hai da axle nguygn ban 1k chi can e.hting rainh rang cho met se nguyen td p thy p khang chia het the he s6 cita da thtic tich f(x)g(x) 136 rang p khong chia Mt the he se eita f(x) va g(x) Gia sit p chia Mt cte, , as_ i , be , bs_ i va p Elblag ehia Mt as va bs 11k xet h@ s6 c s+e cua da thile tich f(x)g(x) + ) p chia crvs = ( + ar_ ib.E1 ) + a rb s + (arkibri Mt the tdng cad dgu ngoac, nhvng Wing chia Mt fiat arbs vi p la nguyen to Do chi p khOng ehia Mt c m • ,, 116 de Neu f(x) la mdt da tilde yea he se nguyen co bac Ian hon utt f(x) khong bdt khd quy Q[xl, thi f(x) phdn tich duos thiznh mat tick nhung da Hale bac kluic yeti he ad nguyen Cluing mink Gih sit fix) khong bat kita quy Q[xl, thg thl f(x) ed thg vigt 175 f(x) w(x)y(x) voi p(x) va 7;tax) la nhang talc Mac sa coa f(x) tro ng Q[x] Theo nha tren da nhan xet, to cd thd viet w(x)= b g(x), y(x) = h(x) dd g(x), h(x) lit nhang da thac nguyen ban va a, b, c, d la nhang s6 nguyen Do dci f(x) = g(x)h(x) P _ — = — va p, q nguyen to ding Ta H hiau cache se q bc1 da thac tich g(x) h(x) bang the thi theo bd de 1, g(x) h(x) caa la nguyen ban, the nen cite khOng cd utdc chung nito khan ngoai Pei ±1 Mat kluic vi f(x) E Z[x] nen the s6 — phal IA nguyen, do q chia het vi q nguyen vdi p Ta suy m q = ±1, Mc la f(x) = -±p g(x) h(x) Vi p(x) va ydx) la nhang udc there set ona f(x) nen g(z) va h(x) la !Mang da tilde bac khan dm Z[x] n Tien chutin Aidenstaina GM sd f(x) = ao + a ix + + aorn (n > 1) Id mOt da th,Zc obi M s6 nguyen, oh girl sit c6 mOt s6 nguyen t6 p cho p khang chic) hit he s6 cao nit& ate , nhung p chin hat Mc he ad IM p khOng chia s6 hang ht a o Tha thi da th,2c f(x) l¢ bat kid quy Q[4 Cluing mink Gib s8 f(x) cd nhang ado thy° sa Q[x] Theo bd de 2, f(x) od the vier f(x) = g(x) h(x), dd 176 g(x) = bo b ix + + b E Z, h(x) = co + c rx + + csx' c, E Z, < s c n r< a0 = b c a t = b l co + b oc I ta cd a k = b kC o b k-I C I + " + an = b r e Theo gilt thigt p chia Mt a = bee ; vay vi p lit nguyen t6, nen Mac p chia Mt b o hoac p chia hgt c_ Gia sit p chia b 0, the' thi p khOng chia c o, vi neu th0g thi p se chia Mt a = b oco, trai vii gia thigt p khOng thg chia Mt myi he s6 coa g(x), vi ngu thg thi p se chia Mt a = bre , trai via gilt thigt Vay giA sii b k la he s6 dau tign cum- g(x) :hong chia hgt cho p Ta hay xet + bock, ak = bkco +bk_IC I a k, bk_ , , 60deu chia het cho p Vay bkco phai chia cho p Vi p lit nguyen t6, ta suy hoac k chia cho p, hoac c dila Mt cho p, may than Atli gia thigt ye b k va co • Vi du 1) Da thdc x4 + 6x3 — 18x2 + 42x + 12 la bat kha quy Q[x] That fly ta cd the ap dung tigu than Aidenstaind vdi p = 2) Da thdc xn + + + +p yea p la met s6 nguygn to' thy yr, lit bat kha quy Q(x] BAI TAP Tim nghigm hitu ti cart cac da thdc a) x3 — 6x2 + 15x — 14 b) 2x3 + 3x2 + 6x — c) x6 —6x5 +11x4 —x3 —18x3 + 20x —8 d) x5 + 2x4 + 6x3 + 3x2 — 42x — 48 177' Gi/t sii -q , vdi p, q E Z nguyen t6 citng nhau, cua da thdc an.xn + nghiam + + ao vol s6 nguyen Chung minh rang a) p I aq Ira q I a n b) p - mq la Mc cim f(m) voi m nguyen ; dac Met p - q lit Mc cua f(1), p + q la Mc ctia f(- 1) dung bai tap 2) de tinh nghiem hvu ti cda da thdc 10x5 - 81x4 + 90x3 - 10212 + 80x - 21 Chung minh rang da tittle f(x) vet he sg'nguyen kheng cu nghiem nguyen ngu f(0) va f(1) la nhilng s6 le Gia sit p(x) la mat da tilde vol he s6 nguyen va p(x) bat kha quy Zhcl Chung minh rang, Z[x], Mu p(x)If(x)g(x), thl huge p(x)I f(x) hoc p(x)I g(46 Trong vanh Z[x] clidng minh ring mai da thitc, Mule va khan ±1, dgu cd the vigt dud) dung Mich acing da thdc Mt quy Dung Heti chugn Aidenstaina dg cluing minh rang cac da thdc sau day la Mt kha quy QM a) x4 - 8x3 + 12+2 -fix +3 b) x4 -x3 + 2r + c) + xP -2 + + x + vati p nguyen tg FluOng don : dgit x = y + Tim digu kian can va du da da tilde x4 + px2 + q la Mt kha quy (Hal Gia su fix) = (x - cti)(x - a2) (x - an ) - 1, vat cac a1 laMitngs6uyepblo.Chngmitafx)lbkh quy QDcl 178, MUC LUG Lai n6i Trang Chiterng I - TAP 1-10P VA QUAN Ht §I- T4p htyp va Soh o 1Chai them tap hop BO phan caa met tap hop lieu cua hai tap hop T0p hop rang Tap hop met, hai phin to Tap hop cac N) phan ctla mot tap hop a cac caa hai tap hOp Hop va giao caa hai Op hop Tich 7 8 9 Anh xo 10 Anh va too anh 13 11 Don anh - Toan anh - Song anh 12 Tich anh o 14 13.Thu het:, va and Ong anh 14 TOp hop chi so 15 flOp, giao, Lich de cac caa met ho tap hOp Bai tap 15 17 18 19 20 §2 Quan hf I Quan he hai ngdi 23 Quan he lilting &king 24 Quan he Oil III MI tap 25 Scr cac 28 nen de au If thuyit tap hqp 179 Chuang - NUA NHOM VA NHOM §1 Nfra /them ?hop Loan hai ng61 37 NOM rthem 39 Bai tGp 42 §2 alham Nhom 43 Nh6m 47 NhOm chuOn lac va Dion, thIldng 52 cdu DEng 58 D6i 'ding NM 64 BM tap 68 Chuang Ip - VANTI VA TRUONC §1 Vanh va mien nguyen Vanh 78 Doc cua khang Mien nguyen Vanh 80 SI Mean va vanh thtiong D6ng 82 85 BM tap 87 §2 Tnr&ug I Traang 91 ' TnIdng 91 Trttong c&c thUong 92 Bai tap 94 Chuang IV - VANH DA Tale §1 Vanh da that mat ho I.' SIM da Mac met An 97 Bac caa mOt da thilc 100 Phep chid vii cht 101 Nghiem Na mOt da Mac 105