Đa thức và hàm hữu tỷ

Chu . o . ng 2 D - ath´u . cv`ah`amh˜u . uty ’ 2.1 D - ath´u . c . 44 2.1.1 D - ath´u . c trˆen tru . `o . ng sˆo ´ ph´u . c C . 45 2.1.2 D - ath´u . c trˆen tru . `o . ng sˆo ´ thu . . c R . 46 2.2 Phˆan th´u . ch˜u . uty ’ . 55 2.1 D - ath´u . c Dath´u . cmˆo . tbiˆe ´ nv´o . ihˆe . sˆo ´ thuˆo . c tru . `o . ng sˆo ´ P d u . o . . cbiˆe ’ udiˆe ˜ nd o . n tri . du . ´o . ida . ng tˆo ’ ng h˜u . uha . n Q(x)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n (2.1) trong d ´o z l`a biˆe ´ n, a 0 ,a 1 , .,a n l`a c´ac sˆo ´ ;v`amˆo ˜ itˆo ’ ng da . ng (2.1) dˆe ` u l`a d ath´u . c. K´yhiˆe . u: Q(z) ∈P[z]. Nˆe ´ u a 0 ,a 1 , .,a n ∈ C th`ı ngu . `o . i ta n´oi r˘a ` ng Q(z)l`ad ath´u . c trˆen tru . `o . ng sˆo ´ ph´u . c: Q(z) ∈ C[z]. Nˆe ´ u a 0 ,a 1 , .,a n ∈ R th`ı Q(z)l`ada th´u . c trˆen tru . `o . ng sˆo ´ thu . . c: Q(z) ∈ R[z]. 2.1. D - ath´u . c 45 Nˆe ´ u Q(z) =0th`ıbˆa . ccu ’ a n´o (k´y hiˆe . u degQ(z)) l`a sˆo ´ m˜u cao nhˆa ´ t cu ’ amo . i lu˜y th`u . acu ’ a c´ac sˆo ´ ha . ng =0cu ’ ad ath´u . cv`ahˆe . sˆo ´ cu ’ asˆo ´ ha . ng c´o lu˜yth`u . a cao nhˆa ´ td ´o g o . il`ahˆe . sˆo ´ cao nhˆa ´ t. Nˆe ´ u P (z)v`aQ(z) ∈P[z] l`a c˘a . pd ath´u . cv`aQ(z) =0th`ıtˆo ` nta . i c˘a . pd ath´u . c h(z)v`ar(z) ∈P[z] sao cho 1 + P = Qh + r, 2 + ho˘a . c r(z) = 0, ho˘a . c degr<degQ. D - i . nhl´yB´ezout. Phˆa ` ndu . cu ’ aph´ep chia d ath´u . c P (z) cho nhi . th´u . c z − α l`a h˘a ` ng P (α) (r = P (α)). 2.1.1 D - ath´u . c trˆen tru . `o . ng sˆo ´ ph´u . c C Gia ’ su . ’ Q(z) ∈ C[z]. Nˆe ´ u thay z bo . ’ isˆo ´ α th`ı ta thu d u . o . . csˆo ´ ph´u . c Q(α)=a 0 α n + a 1 α n−1 + ···+ a n−1 α + a n . D - i . nh ngh˜ıa 2.1.1. Nˆe ´ u Q(α) = 0 th`ı sˆo ´ z = α d u . o . . cgo . il`anghiˆe . m cu ’ ad ath´u . c Q(z) hay cu ’ aphu . o . ng tr`ınh d a . isˆo ´ Q(z)=0. D - i . nh l´y Descate. D ath´u . c Q(z) chia hˆe ´ t cho nhi . th´u . c z − α khi v`a chı ’ khi α l`a nghiˆe . mcu ’ ad ath´u . c P (z) (t´u . cl`aP (α)=0). D - i . nh ngh˜ıa 2.1.2. Sˆo ´ ph´u . c α l`a nghiˆe . mbˆo . i m cu ’ ad ath´u . c Q(z) nˆe ´ uv`achı ’ nˆe ´ u Q(z) chia hˆe ´ tcho(z − α) m nhu . ng khˆong chia hˆe ´ tcho (z − α) m+1 .Sˆo ´ m du . o . . cgo . il`abˆo . i cu ’ a nghiˆe . m α. Khi m = 1, sˆo ´ α go . i l`a nghiˆe . md o . n cu ’ a Q(z). Trong tiˆe ´ t 2.1.1 ta biˆe ´ tr˘a ` ng tˆa . pho . . psˆo ´ ph´u . c C d u . o . . clˆa . pnˆenb˘a ` ng c´ach gh´ep thˆem v`ao cho tˆa . pho . . psˆo ´ thu . . c R mˆo . t nghiˆe . ma ’ o x = i cu ’ a phu . o . ng tr`ınh x 2 +1=0v`amˆo . t khi d˜a gh´ep i v`ao th`ı mo . iphu . o . ng tr`ınh d ath´u . cd ˆe ` uc´onghiˆe . mph´u . c thu . . csu . . .Dod ´o khˆong cˆa ` n pha ’ i s´ang ta . o thˆem c´ac sˆo ´ m´o . id ˆe ’ gia ’ iphu . o . ng tr`ınh (v`ı thˆe ´ C c`on d u . o . . cgo . i l`a tru . `o . ng d ´ong da . isˆo ´ ). D - i . nh l´y Gauss (d i . nh l´y co . ba ’ ncu ’ ad a . isˆo ´ ). 46 Chu . o . ng 2. D - ath´u . c v`a h`am h˜u . uty ’ Mo . idath´u . cd a . isˆo ´ bˆa . c n (n  1) trˆen tru . `o . ng sˆo ´ ph´u . cd ˆe ` u c´o ´ıt nhˆa ´ tmˆo . t nghiˆe . mph´u . c. T`u . d i . nh l´y Gauss r´ut ra c´ac hˆe . qua ’ sau. 1 + Mo . idath´u . cbˆa . c n (n  1) trˆen tru . `o . ng sˆo ´ ph´u . cd ˆe ` uc´od´ung n nghiˆe . mnˆe ´ umˆo ˜ i nghiˆe . md u . o . . c t´ınh mˆo . tsˆo ´ lˆa ` nb˘a ` ng bˆo . icu ’ an´o,t´u . cl`a Q(x)=a 0 (z − α 1 ) m 1 (z − α 2 ) m 2 ···(z − α k ) m k , (2.2) trong d ´o α i = α j ∀ i = j v`a m 1 + m 2 + ···+ m k = n. D ath´u . c (2.1) v´o . ihˆe . sˆo ´ cao nhˆa ´ t a 0 =1du . o . . cgo . il`ad ath´u . c thu go . n. 2 + Nˆe ´ u z 0 l`a nghiˆe . mbˆo . i m cu ’ adath´u . c Q(z)th`ısˆo ´ ph´u . cliˆen ho . . p v´o . in´o z 0 l`a nghiˆe . mbˆo . i m cu ’ adath´u . c liˆen ho . . p Q(z), trong d´o d a th´u . c Q(z)du . o . . c x´ac d i . nh bo . ’ i Q(z) def = a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n . (2.3) 2.1.2 D - ath´u . c trˆen tru . `o . ng sˆo ´ thu . . c R Gia ’ su . ’ Q(z)=z n + a 1 z n−1 + ···+ a n−1 z + a n (2.4) l`a d ath´u . c quy go . nv´o . ihˆe . sˆo ´ thu . . c a 1 ,a 2 , .,a n . D ath´u . c n`ay c´o t´ınh chˆa ´ td ˘a . cbiˆe . t sau dˆa y . D - i . nh l´y 2.1.1. Nˆe ´ usˆo ´ ph´u . c α l`a nghiˆe . mbˆo . i m cu ’ ad ath´u . c (2.4) v´o . i hˆe . sˆo ´ thu . . c th`ı sˆo ´ ph´u . c liˆen ho . . pv´o . in´o α c˜ung l`a nghiˆe . mbˆo . i m cu ’ a d ath´u . cd ´o. Su . ’ du . ng d i . nh l´y trˆen dˆay ta c´o thˆe ’ t`ım khai triˆe ’ ndath´u . cv´o . ihˆe . sˆo ´ thu . . c Q(z) th`anh t´ıch c´ac th`u . asˆo ´ .Vˆe ` sau ta thu . `o . ng chı ’ x´et d a th´u . cv´o . ihˆe . sˆo ´ thu . . cv´o . ibiˆe ´ nchı ’ nhˆa . n gi´a tri . thu . . cnˆen biˆe ´ nd ´o t a k ´y hiˆe . ul`ax thay cho z. 2.1. D - ath´u . c 47 D - i . nh l´y 2.1.2. Gia ’ su . ’ d ath´u . c Q(x) c´o c´ac nghiˆe . m thu . . c b 1 ,b 2 , .,b m v´o . ibˆo . itu . o . ng ´u . ng β 1 ,β 2 , .,β m v`a c´ac c˘a . p nghiˆe . mph´u . cliˆen ho . . p a 1 v`a a 1 , a 2 v`a a 2 , .,a n v`a a n v´o . ibˆo . itu . o . ng ´u . ng λ 1 ,λ 2 , .,λ n . Khi d´o Q(x)=(x− b 1 ) β 1 (x− b 2 ) β 2 ···(x − b m ) β m (x 2 + p 1 x + q 1 ) λ 1 × × (x 2 + p 2 x + q 2 ) λ 2 ···(x 2 + p n x + q b ) λ n . (2.5) D - i . nh l´y 2.1.3. Nˆe ´ ud ath´u . c Q(x)=x n + a 1 x n−1 + ···+ a n−1 x + a n v´o . ihˆe . sˆo ´ nguyˆen v`a v´o . ihˆe . sˆo ´ cao nhˆa ´ tb˘a ` ng 1 c´o nghiˆe . mh˜u . uty ’ th`ı nghiˆe . md ´o l`a sˆo ´ nguyˆen. D ˆo ´ iv´o . id ath´u . cv´o . ihˆe . sˆo ´ h˜u . uty ’ ta c´o D - i . nh l´y 2.1.4. Nˆe ´ u phˆan sˆo ´ tˆo ´ i gia ’ n  m (, m ∈ Z,m>0) l`a nghiˆe . m h˜u . uty ’ cu ’ a phu . o . ng tr`ınh v´o . ihˆe . sˆo ´ h˜u . uty ’ a 0 x n +a 1 x n−1 +···+a n−1 x+ a n =0th`ı  l`a u . ´o . ccu ’ asˆo ´ ha . ng tu . . do a n v`a m l`a u . ´o . ccu ’ ahˆe . sˆo ´ cao nhˆa ´ t a 0 . C ´ AC V ´ IDU . V´ı d u . 1. Gia ’ su . ’ P (z)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n .Ch´u . ng minh r˘a ` ng: 1 + Nˆe ´ u P (z) ∈ C[z]th`ıP (z)=P (z). 2 + Nˆe ´ u P (z) ∈ R[z]th`ıP (z)=P (z). Gia ’ i. 1 + ´ Ap du . ng c´ac t´ınh chˆa ´ tcu ’ a ph´ep to´an lˆa ´ y liˆen ho . . p ta thu d u . o . . c p(Z)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n = a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n = a 0 (z) n + a 1 (z) n−1 + ···+ a n−1 z + a n = P (z). 48 Chu . o . ng 2. D - ath´u . c v`a h`am h˜u . uty ’ 2 + Gia ’ su . ’ P (z) ∈ R[z]. Khi d ´o P (z)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n = a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n = a 0 (z) n + a 1 (z) n−1 + ···+ a n−1 z + a n = a 0 (z) n + a 1 (z) n−1 + ···+ a n−1 z + a n = P (z). T`u . d ´oc˜ung thu du . o . . c P (z)= P (z)v`ı P(z)=P(z).  V´ı d u . 2. Ch´u . ng minh r˘a ` ng nˆe ´ u a l`a nghiˆe . mbˆo . i m cu ’ ad ath´u . c P (z)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n ,a 0 =0 th`ı sˆo ´ ph´u . c liˆen ho . . p a l`a nghiˆe . mbˆo . i m cu ’ adath´u . c P (z)=a 0 z n + a 1 z n−1 + ···+ a n−1 z + a n (go . il`adath´u . c liˆen ho . . pph´u . cv´o . id ath´u . c P (z)). Gia ’ i. T`u . v´ıdu . 1 ta c´o P (z)=P (z). (2.6) V`ı a l`a nghiˆe . mbˆo . i m cu ’ a P (z)nˆen P (z)=(z − a) m Q(z),Q(a) = 0 (2.7) trong d ´o Q(z)l`adath´u . cbˆa . c n − m.T`u . (2.6) v`a (2.7) suy ra P (z)=P (z)=(z − a) m Q(z)=(z − a) m Q(z). (2.8) Ta c`on cˆa ` nch´u . ng minh r˘a ` ng Q(a) = 0. Thˆa . tvˆa . y, nˆe ´ u Q(a)=0th`ı b˘a ` ng c´ach lˆa ´ y liˆen ho . . pph´u . cmˆo . tlˆa ` nn˜u . a ta c´o Q(a)=Q(a)=0 ⇒ Q(a)=0. D iˆe ` u n`ay vˆo l´y. B˘a ` ng c´ach d˘a . t t = z,t`u . (2.8) thu d u . o . . c P (t)=(t − a) m Q(t), Q(a) =0. 2.1. D - ath´u . c 49 D˘a ’ ng th´u . c n`ay ch´u . ng to ’ r˘a ` ng t = a l`a nghiˆe . mbˆo . i m cu ’ adath´u . c P (t).  V´ı d u . 3. Ch´u . ng minh r˘a ` ng nˆe ´ u a l`a nghiˆe . mbˆo . i m cu ’ ad ath´u . cv´o . i hˆe . sˆo ´ thu . . c P (z)=a 0 z n + a 1 z n−1 + ···+ a n (a 0 = 0) th`ı sˆo ´ ph´u . c liˆen ho . . p a c˜ung l`a nghiˆe . mbˆo . i m cu ’ ach´ınh dath´u . cd ´o. Gia ’ i. T`u . v´ıdu . 1, 2 + ta c´o P (z)=P (z) (2.9) v`a do a l`a nghiˆe . mbˆo . i m cu ’ a n´o nˆen P (z)=(z − a) m Q(z) (2.10) trong d ´o Q(z)l`adath´u . cbˆa . c n − m v`a Q(a) =0. Ta cˆa ` nch´u . ng minh r˘a ` ng P (z)=(z − a) m Q(z),Q(a) =0. (2.11) Thˆa . tvˆa . yt`u . (2.9) v`a (2.10) ta c´o P (z)= (z − a) m Q(z)=(z − a) m · Q(z) =  (z − a)  m Q(z)=(z − a) m Q(z) v`ı theo (2.9) Q( z)=Q(z) ⇒ Q(z)=Q(z). Ta c`on cˆa ` nch´u . ng minh Q( a) = 0. Thˆa . tvˆa . yv`ı Q(a) =0nˆen Q(a) =0v`adod´o Q(a) =0v`ıdˆo ´ iv´o . id ath´u . cv´o . ihˆe . sˆo ´ thu . . cth`ı Q(t)=Q(t).  V´ı du . 4. Gia ’ iphu . o . ng tr`ınh z 3 − 4z 2 +4z − 3=0. Gia ’ i. T`u . d i . nh l´y 4 suy r˘a ` ng c´ac nghiˆe . m nguyˆen cu ’ aphu . o . ng tr`ınh v´o . ihˆe . sˆo ´ nguyˆen d ˆe ` ul`au . ´o . ccu ’ asˆo ´ ha . ng tu . . do a = −3. Sˆo ´ ha . ng tu . . do 50 Chu . o . ng 2. D - ath´u . c v`a h`am h˜u . uty ’ a = −3 c´o c´ac u . ´o . cl`a±1, ±3. B˘a ` ng c´ach kiˆe ’ m tra ta thu d u . o . . c z 0 =3 l`a nghiˆe . m nguyˆen. T`u . d ´o z 3 − 4z 2 +4z − 3=(z − 3)(z 2 − z +1) =(z − 3)(z − 1 2 + i √ 3 2  z − 1 2 − i √ 3 2  hay l`a phu . o . ng tr`ınh d ˜a cho c´o ba nghiˆe . ml`a z 0 =3,z 1 = 1 2 − i √ 3 2 ; z 2 = 1 2 + i √ 3 2 ·  V´ı d u . 5. Biˆe ’ udiˆe ˜ nd ath´u . c P 6 (z)=z 6 − 3z 4 +4z 2 − 12 du . ´o . ida . ng: 1 + t´ıch c´ac th`u . asˆo ´ tuyˆe ´ n t´ınh; 2 + t´ıch c´ac th`u . asˆo ´ tuyˆe ´ n t´ınh v´o . i tam th´u . cbˆa . c hai v´o . ihˆe . sˆo ´ thu . . c. Gia ’ i. Tat`ımmo . i nghiˆe . mcu ’ ad ath´u . c P (z). V`ı z 6 − 3z 4 +4z 2 − 12 = (z 2 − 3)(z 4 +4) nˆen r˜o r`ang l`a z 1 = − √ 3,z 2 = √ 3,z 3 =1+i, z 4 =1− i, z 5 = −1+i, z 6 = −1 − i. T`u . d ´o 1 + P 6 (z)=(z− √ 3)(z + √ 3)(z−1−i)(z−1+i)(z +1−i)(z +1+i) 2 + B˘a ` ng c´ach nhˆan c´ac c˘a . p nhi . th´u . c tuyˆe ´ n t´ınh tu . o . ng ´u . ng v´o . i c´ac nghiˆe . mph´u . c liˆen ho . . pv´o . i nhau ta thu d u . o . . c P 6 (z)=(z − √ 3)(z + √ 3)(z 2 − 2z + 2)(z 2 +2z +2).  V´ı d u . 6. T`ım d ath´u . chˆe . sˆo ´ thu . . cc´olu˜yth`u . a thˆa ´ p nhˆa ´ t sao cho c´ac sˆo ´ z 1 =3,z 2 =2− i l`a nghiˆe . mcu ’ a n´o. 2.1. D - ath´u . c 51 Gia ’ i. V`ıdath´u . cchı ’ c´o hˆe . sˆo ´ thu . . cnˆen c´ac nghiˆe . mph´u . c xuˆa ´ thiˆe . n t`u . ng c˘a . p liˆen ho . . pph´u . c, ngh˜ıa l`a nˆe ´ u z 2 =2− i l`a nghiˆe . mcu ’ an´oth`ı z 2 =2+i c˜ung l`a nghiˆe . mcu ’ a n´o. Do d´o P (z)=(z − 3)(z − 2+i)(z − 2 − i)=z 3 − 7z 2 +17z − 15.  V´ı du . 7. Phˆan t´ıch d ath´u . c (x +1) n − (x − 1) n th`anh c´ac th`u . asˆo ´ tuyˆe ´ n t´ınh. Gia ’ i. Ta c´o P (x)=(x +1) n − (x − 1) n =[x n + nx n−1 + .]− [x n − nx n−1 + .]=2nx n−1 + . Nhu . vˆa . y P (x)l`ad ath´u . cbˆa . c n − 1v´o . ihˆe . sˆo ´ cao nhˆa ´ tb˘a ` ng 2n.D ˆo ´ i v´o . id ath´u . c n`ay ta d ˜abiˆe ´ t(§1) nghiˆe . mcu ’ a n´o: x k = icotg kπ n ,k=1, 2, .,n− 1. Do d ´o (x +1) n − (x − 1) n =2n  x − icotg π n  x − icotg 2π n  ···  x − icotg (n − 1)π n  . Khi phˆan t´ıch d ath´u . c trˆen tru . `o . ng P th`anh th`u . asˆo ´ ta thu . `o . ng g˘a . pnh˜u . ng d ath´u . c khˆong thˆe ’ phˆan t´ıch th`anh t´ıch hai d ath´u . c c´o bˆa . c thˆa ´ pho . ntrˆenc`ung tru . `o . ng P d ´o. Nh˜u . ng d ath´u . cn`ayd u . o . . cgo . il`ad a th´u . cbˆa ´ t kha ’ quy. Ch˘a ’ ng ha . n: d ath´u . c x 2 − 2l`akha ’ quy trˆen tru . `o . ng sˆo ´ thu . . cv`ı: x 2 − 2=(x − √ 2)(x + √ 2) 52 Chu . o . ng 2. D - ath´u . c v`a h`am h˜u . uty ’ nhu . ng bˆa ´ t kha ’ quy trˆen tru . `o . ng sˆo ´ h˜u . uty ’ . Thˆa . tvˆa . y, nˆe ´ u x 2 − 2=(ax + b)(cx + d); a, b, c, d ∈ Q th`ı b˘a ` ng c´ach d ˘a . t x = − b a ta c´o b 2 a 2 − 2=0⇒ √ 2=± b a v`a √ 2 l`a sˆo ´ h˜u . uty ’ . Vˆo l´y. V´ı d u . 8. Phˆan t´ıch d ath´u . c x n − 1 th`anh t´ıch c´ac dath´u . cbˆa ´ t kha ’ quy trˆen R. Gia ’ i. D ˆa ` u tiˆen ta khai triˆe ’ ndath´u . cd ˜a cho th`anh t´ıch c´ac th`u . a sˆo ´ tuyˆe ´ n t´ınh x n − 1=(x − ε 0 )(x − ε 1 )···(x − ε n−1 ), ε k = cos 2kπ n + i sin 2kπ n ,k= 0,n− 1 v`a t´ach ra c´ac nhi . th´u . c thu . . c. Ta c´o ε k ∈ R nˆe ´ u sin 2kπ n =0⇒ 2k . . . n, 0  k<n− 1. T`u . d ´o 1 + Nˆe ´ u n l`a sˆo ´ le ’ th`ı diˆe ` ud´o(2k . . . n)chı ’ xˆa ’ y ra khi k =0(v`ık<n) v`a khi d ´o ε 0 =1. 2 + Nˆe ´ u n l`a sˆo ´ ch˘a ˜ n(n =2m) th`ı nghiˆe . m ε k chı ’ thu . . c khi k =0 v`a k = m.Dod ´o ε 0 =1,ε m = −1. Dˆo ´ iv´o . i c´ac gi´a tri . k c`on la . i ε k khˆong l`a sˆo ´ thu . . c. D ˆo ´ iv´o . i c´ac gi´a tri . k n`ay ta c´o sin 2(n − k)π n = sin  2π − 2kπ n  = − sin 2kπ n v`a do d ´o ε n−k = ε k ⇒ ε 1 = ε n−1 , ε 2 = ε n−2 , . 2.1. D - ath´u . c 53 M˘a . t kh´ac (x − ε k )(x − ε k )=x 2 − (ε k + ε k )x + ε k ε k = x 2 − x · 2 cos 2kπ n +1. Do d ´o x n − 1=                (x − 1) n−1 2  k=1  x 2 − x · 2 cos 2kπ n +1  nˆe ´ u n l`a sˆo ´ le ’ , (x − 1)(x +1) n−2 2  k=1  x 2 − x · 2 cos 2kπ n +1  nˆe ´ u n l`a sˆo ´ ch˘a ˜ n.  B ` AI T ˆ A . P 1. Ch´u . ng minh r˘a ` ng sˆo ´ z 0 =1+i l`a nghiˆe . mcu ’ adath´u . c P 4 (z)=3z 4 − 5z 3 +3z 2 +4z − 2. T`ım c´ac nghiˆe . m c`on la . i. (D S. z 1 =1− i, z 2 = −1+ √ 13 6 , z 3 = −1 − √ 13 6 ) 2. Ch´u . ng minh r˘a ` ng sˆo ´ z 0 = i l`a nghiˆe . mcu ’ adath´u . c P 4 (z)=z 4 + z 3 +2z 2 + z +1. T`ım c´ac nghiˆe . m c`on la . i. (D S. z 1 = −i, z 2 = −1+ √ 3i 2 , z 3 = −1 − i √ 3 2 ) 3. X´ac d i . nh bˆo . icu ’ a nghiˆe . m z 0 =1cu ’ adath´u . c P 4 (z)=z 4 − 5z 3 +9z 2 − 7z +2. (DS. 3) 4. X´ac d i . nh bˆo . icu ’ a nghiˆe . m z 0 =2cu ’ adath´u . c P 5 (z)=z 5 − 5z 4 +7z 3 − 2z 2 +4z − 8. (DS. 3)