VI xI x, nen xI ab = x Theo b) xIa hoac xI b Ned
c) Tim tat ca cac to dang cau cia X lit do say ra rang tap hop cac to dang c&u cua X Itt mat nhdm xyclic dui v6i phep
CHLIONG VI
DA THUG TREN TRUONG SO
§1. DA THUG VoI He 436 THUG VA PITOC
1. Tnstmg so phtic
Cho mot da thdc v6i hg s6 th0c thi chtia chAc da than do - co nghigm trong trubng ad th0c, cu th6 da thdc x 2 + 1 kh6ng co nghigm trong trubng s6 thorn Dual day ta se thgy moi da tilde bac n vol hg s6 phac co dung n nghigm phdc. D4 chting minh ra hay dua vao cat 66 de sau day :
Be de 1. Moi da thdc yea 114 sd thgc co bac le co it nhdt
mot nghient Sec. Chong mirth. Gia ad
f(x) = a/ + an _r 1 + + an an m 0, n 14.
Qua giao trinh giai doh ta bigt rang vol nhting gia tri dining
va am mitt x kha 16n ye gia tri tuygt d6i, ham s6 f(x) co can dgu trai nhau. Vgy rb nhang gia tri thge So, x, a va b chAng han, sao cho
f(a) < 0, f(b) > 0.
Mat khan ham so f(x) la lien tuc, vi v&y co. mgt gift tri c cua x/nam gicia a va b, sao cho f(c) = 0. n
Ta chug rang phep cluing minh cfia b6 de 1 khong dai sd, no da 14 nhang kat qua dm giai tfch. Them naa ye ngen ngil ta da goi anh xa
R -• R c c
la ham s6 f(x).
B6 de 2. Mpi da thetc bec hai ax 2 + bx + c, ooi he s6 phdc, bao gia cling co hat nghiem phetc.
Cluing mink. Gni co l va co 2 la hai can bac hai cue. b 2 - 4cic, -b + 1 -la + co 2
ta cd hai nghigm cda da th/c la 2a va . n
2a
B6 de 3. Mot da the& b4e ldn han 0 Si he a trate co it nhdt met nghiem 'plate. •
Cluing mink. Mof sd to nhien it > 0 dolt cd the via duel dang it = in la met se to nhien va n' la met s6 tp nhien le. Ta hay cluing minh bang quy nap theo m. V6i m = 0 khgng dinh la dung theo be de 1. Ta gia sit khgng dish IA dung cho m - 1 Ira chang minh nd dung cho m. Murin vgy, ta hay xet met da th/c f(x) cd bac n = (m > 0) veil he s6 thoc. Coi f(x) nhu met da thdc cita vanh v6i C la trobng se
ph/c, ta hay xet met trurbng m6 Ong E cua C sao cho f(x) cd
dung it nghiem a l , a 2 ,..., trong E (ch V, §2, 2, ha qua cim (huh If 3). Gia sit c la met se time tuy j, at
(1) pi; = + e(ai cei), i s j.
Ta cd Cn2 phan tit thanh lap boi cac t6 hop chap 2 dm n phan tit a t , , an . Xet da th/c
(2) g(x) = fit) - fits). • = + a txt 1 + int
Vi bac cast g(x) bang se Mc nhan ti x - nen ta cd
n(n - I) - 1) _ 1
/ = Cn - 2 - 2 - 2n1
vei q = n'(en' - 1) ; vi re' va. en' - 1 la nhung se Id nen q
Ta hay cluing mink the he tit a l , ...al tha g(x) deu la that
ca. Trude het the he W dd la ciic da thuc el& sung co ban cunt
the Ai Vey theo (1) chung la nhting da thtic eua a i , ..., a n vtli he s6 Wye, vi c la met s6 thee. Han naa cluing con la nhting da thitc d61 icing cart a 1 , ..., a. That thy nett ta thay a i bang ai a i bang a i thi pl. van gift nguyen con the P hi va Ai; thi
tin° del cho nhau (h i va j), dieu nay chat than khang
lam thay del the he to at, ..., a 1 . Do dci theo (ch V, §2, 2, chid
3I) CAC he tit do la nhung se thne. Vey g(x) la met da thdc veii he s6 thac ed ben 2v` -1q, vol q la 14. Theo gia thiet quy nap
g(x) ed it nhat met nghiem phdc dang (1), We la cti It nhat
met cap chi se (i, j) sao cho phan to pii = cticej + c(a i + ai )
thuen truting se phde C.
Neu to gan cho c met gia tri them kluic thi ta se dude met da thde g(x) khan loaf (2), thata nhan met nghiem phdc dang (1), nhung n6i chung voi met cap chi se (i, j) khac. Tur
n6u ta gan cho clan luot Cn + 1 gia tri phan Met thi nhat dinh phai chicle hai nghiem pink dang (1) vOl ding met cap chi s6
j) vi ta chi th td hop chap 2 tide n phan tit. Vay the nao
thug co hai se tilde c i va c 2 khac nhau sao cho the phan to
a. a + c
I (a. + a.)
an) + c (a. + a.)
ling
2
ved cling met cap chi s6 i, j deu la pink. Dat
a = anal + c i (ai + b = a.a. + c,(a. + a) ). Ta cd a, b E C. Ta suy ra a - b a. + a j c — C2 a - b aIa) tie I — C2 157
Vey a + a. era i dell la Ta suy re ai ai a nghiem cim da thdc hec hair
x2 — (ai adx +
voi he s6 phlic. Theo 14 de 2, a i va ai la Onto. Nhu fly to da cluing minh de thdc f(x) khong nhung S met, ma cd hei nghiem phtic.•
Dinh li 1. MQi da thdc bee tun hen 0 ()di he 86 phitc co it nit& met nghiem phdc.
Cluing mink. Dia sit f(x) la mot da thdc bez n > 0
veil he sti phdc. Dar
Fx7 = + +
vai cat ai la cat lien hop rim Mc a. i = 0, , n. X6t da thdc :
g(x) = f(x)f(x). Ta cd g(x) = bo + b ix + . + b 2nx2" voi bk = E k 0,1, 2n i+j=k vi Slc = I EPj = bk bv=k
nen the he ad bk la thuc. Theo b6 de 3, g(x) co it nhgt met nghiem phfic z = s + it
g(z) = f(z) ,(17 = 0.
Do d6 hoar f(z) = 0 hoer irjz = 0. NOu F27 = 0 = "do + z + = o -
thi ao + a tz + + anzn = no + + + anr = 0
We la fq) 0
114 quit 1. Cdc da thuc bdt khd quy cat' yank Chi), C la twang ad phdc, l¢ cac da thzic bac nhdt.
Cluing mink. Theo (Ch V §2, 2), cac da thee bac nhgt la Mt kha quy. Dia sit f(x) la met da th/c cfia C[i] co bac lon han