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I?ia?aiia ainoaea?noaaiiiai yecaiaia ii iaoaiaoeea

aeey nooaeaioia iaoaiaoe/aneiai oaeoeueoaoa

Iineianeiai ai?iaeneiai iaaeaaiae/aneiai oieaa?neoaoa

Aeaaa?a e oai?ey /enae

1. A?oiiu; i?eia?u e i?inoaeoea naienoaa yeaiaioia a?oiiu.

2. Eieueoea e iiey; i?eia?u e i?inoaeoea naienoaa yeaiaioia.

3. A?eoiaoe/aneea ooieoeee: ((n), ((n), ((n).

4. Aeai?eoi Aaeeeaea e aai i?eiaiaiey.

5. N?aaiaiey e eo naienoaa. Oai?aiu Yeea?a e Oa?ia.

6. Aacen e ?acia?iinoue aaeoi?iiai i?ino?ainoaa.

7. Iniiaiua oai?aiu i nenoaiao eeiaeiuo o?aaiaiee.

8. Ei?ie iiiai/eaia, oai?aia Aaco, noaia Ai?ia?a.

9. ?aceiaeaiea iiiai/eaia iaae iieai a i?iecaaaeaiea

iai?eaiaeeiuo iiiaeeoaeae e aai aaeeinoaaiiinoue.

10. Oai?aia i no?iaiee i?inoiai aeaaa?ae/aneiai ?anoe?aiey.

1. A?oiiu; i?eia?u e i?inoaeoea naienoaa yeaiaioia a?oiiu

10. Ii?aaeaeaiea a?oiiu.

Anthaeo a aeaeueiaeoai caienue (G, () icia/aao, /oi ia iaionoii
iiiaeanoaa G caaeaia iia?aoeey “(”.

Ii?aaeaeaiea. Iiiaeanoai (G, () iacuaaaony a?oiiie, anee auiieiaiu
neaaeothuea oneiaey:

(1) iia?aoeey “(” annioeeaoeaia, o.a. ((x, y, z(G) (x(y)(z = x((y(z);

(2) iiiaeanoai G iaeaaeaao iaeo?aeueiui yeaiaioii ioiineoaeueii
iia?aoeee (:

((e(G)((x(G) x(e = e(x = x;

(3) eaaeaeue yeaiaio iiiaeanoaa G iaeaaeaao neiiao?e/iui yeaiaioii:

((x(G) ((y(G) x(y = y(x = e.

20. I?eia?u a?oii: /eneiaua a?oiiu, a?oiiu neiiao?ee aaiiao?e/aneeo
oeao?, a?oiiu iiaenoaiiaie, iao?e/iua a?oiiu.

I?eia?u a?oii aanueia ?aciiia?aciu. Ia?a/eneei iaeioi?ua ec ieo.

1. *eneiaua a?oiiu (a?oiiu, yeaiaiou eioi?uo yaeythony eiiieaeniuie
/eneaie).

a) Aaeaeeoeaiua a?oiiu oeaeuo /enae Z, ?aoeeiiaeueiuo /enae Q,
aeaenoaeoaeueiuo /enae R, eiiieaeniuo /enae C.

a) Ioeueoeieeeaoeaiua a?oiiu iaioeaauo ?aoeeiiaeueiuo /enae Q*,
iaioeaauo aeaenoaeoaeueiuo /enae R*, iaioeaauo eiiieaeniuo /enae C*,
iieiaeeoaeueiuo ?aoeeiiaeueiuo /enae Q+, iieiaeeoaeueiuo
aeaenoaeoaeueiuo /enae R+.

2. A?oiiu iiaenoaiiaie S(X) e Sn, aeaenoaothueo ia iiiaeanoaa X, a
/anoiinoe, ia iiiaeanoaa {1, 2, . . . , n}.

3. A?oiiu aeaeaeaiee aaiiao?e/aneeo oeao?. Ionoue O – eaeay-ieaoaeue
aaiiao?e/aneay oeao?a ia ieineinoe, O(O) – iiiaeanoai aeaeaeaiee
ieineinoe, ia?aaiaeyueo oeao?o O ia naay. Iiiaeanoai O(O) ioiineoaeueii
iia?aoeee eiiiiceoeee (iineaaeiaaoaeueiiai auiieiaiey) aeaeaeaiee
yaeyaony a?oiiie. Yeaiaiou iiiaeanoaa O(O) /anoi iacuaathony neiiao?eyie
oeao?u O.

?anniio?ei, iai?eia?, a?oiio neiiao?ee i?aaeeueiiai o?aoaieueieea.

; eo oaeiaii iaicia/eoue (, (, (. Aeey iienaiey oiiiaeaiey yeaiaioia
a?oiiu (G, () iiaeii eniieueciaaoue oae iacuaaaioth oaaeeoeo Eyee
(oaaeeoeo oiiiaeaiey a?oiiu).

Aeey a?oiiu neiiao?ee i?aaeeueiiai o?aoaieueieea oaaeeoea Eyee eiaao
aeae:

( ( ( ( ( (

( ( ( (

( ( ( (

( ( ( (

(

(

( ( (

(

( (

(

(

Caiaoei, /oi a?auaiey ia?aiiiaeathony ii i?aaeeo (2 = (, (3 = (.
Aeaeaa, eaaae?ao ethaiai io?aaeaiey ?aaai (.

Eaaei i?iaa?eoue, /oi(( = (, (( = (. E?iia oiai, (( = (.

Inoaeueiua i?iecaaaeaiey a oaaeeoea eaaei ainnoaiiaeoue, eniieuecoy,
iai?eia?, a?oiiiaoth no?oeoo?o iia?aoeee. A /anoiinoe, eiaai:

(( =(((() = (2( = (( = (;

(( =(((() = ((()( = (2 = (.

4. A?oiiu aaiiao?e/aneeo i?aia?aciaaiee. A?oiiu a?auaiee, iiaeiaee,
aiiioaoee n caaeaiiui iauei oeaio?ii, ia?aeeaeueiuo ia?aiinia.

5. Iao?e/iua a?oiiu. Oeaaeai ia aeaa aaaeiaeoea iao?e/iua a?oiiu:

GLn(R) – iieiay eeiaeiay a?oiia (a?oiia ia?aoeiuo iao?eoe),

SLn(R) – niaoeeaeueiay eeiaeiay a?oiia

(a?oiia iao?eoe n aaeeie/iui ii?aaeaeeoaeai),

30. A?eoiaoeea a?oiiu: ia?aoiua yeaiaiou, noaiaie n oeaeui iieacaoaeai.

I?e iienaiee oaaeeoeu Eyee a?oiiu neiiao?ee i?aaeeueiiai o?aoaieueieea
iu eniieueciaaee oae iacuaaaiua a?eoiaoe/aneea naienoaa yeaiaioia
a?oiiu. Ioiaoei aaaeiaeoea ec ieo a neaaeothuae oai?aia.

Oai?aia. Ionoue (G,() – a?oiia. Oiaaea aeey aa yeaiaioia ni?aaaaeeeau
?aaainoaa:

(a) (xy)(zt) = x(y(zt) = ((xy)z)t;

(a) (xy)-1 = y-1x-1;

(a) (xp)q = xpq; xpxq = xp+q aeey ethauo oeaeuo p, q.

Aeieacaoaeuenoai. I?iaa?ei oieueei ioieo (a). Eiaai:

(xy)(y-1x-1) = x(yy-1)x-1 = x(1)x-1 = 1,

(y-1x-1)(xy) = y-1(x-1x)y = y-1(1)y = 1;

ioeoaea e iieo/aai o?aaoaiia ooaa?aeaeaiea. (

40. ?aoaiea a a?oiiao eeiaeiuo o?aaiaiee. A ea/anoaa i?eiaiaiey
i?inoaeoeo naienoa i?eaaaeai neaaeothuee i?inoie ?acoeueoao.

Oai?aia. A i?iecaieueiie ioeueoeieeeaoeaiie a?oiia G iaeiicia/ii
?ac?aoeii eaaeaeia ec o?aaiaiee:

ax = b, ya = b, aaea a, b – oeene?iaaiiua yeaiaiou a?oiiu.

Aeieacaoaeuenoai. Aeiionoei, /oi yeaiaio g oaeiaeaoai?yao ?aaainoao ag =
b. Oiaaea oiiiaeay iaa /anoe ?aaainoaa neaaa ia yeaiaio ia?aoiue e g,
iieo/ei

a-1(ag) = a-1b, ioeoaea iaoiaeei g = a-1b. Eaaei i?iaa?eoue, /oi yeaiaio
a-1b yaeyaony ?aoaieai o?aaiaiey ax = b, o.a. ni?aaaaeeeai ?aaainoai
a(a-1b) = b.

Aiaeiae/ii aeieacuaaaony ?ac?aoeiinoue aoi?iai o?aaiaiey. (

I?eia?u. 1. ?aoeoue o?aaiaiea (12)x = (13) a a?oiia iiaenoaiiaie S3.

Eiaai: x = (12)(13) = (123).

2. ?aoeoue o?aaiaiea (x = ( a a?oiia neiiao?ee i?aaeeueiiai
o?aoaieueieea.

Eiaai: x = ( -1( = (, iineieueeo (( yaeyaony io?aaeaieai e

C((() = (C()( = B( = C.

a a?oiia GL2(R).

Eiaai:

.

2. Eieueoea e iiey; i?eia?u e i?inoaeoea naienoaa yeaiaioia

10. Ii?aaeaeaiea eieueoea e iiey.

Ii?aaeaeaiea. Iaionoia iiiaeanoai A, ia eioi?ii caaeaiu iia?aoeee
neiaeaiey e oiiiaeaiey, iacuaaaony eieueoeii, anee auiieiaiu neaaeothuea
aeaa oneiaey:

a) (A, +) – aaaeaaa a?oiia;

a) oiiiaeaiea aeeno?eaooeaii ioiineoaeueii neiaeaiey, o.a. aeey ethauo
yeaiaioia x, y, z ec A auiieiaiu ?aaainoaa: (x + y)z = xz + yz; x(y + z)
= xy + xz.

Ii?aaeaeaiea. Eieueoei iacuaaaony eiiiooaoeaiui, anee iia?aoeey
oiiiaeaiey a iai eiiiooaoeaia; eieueoei iacuaaaony annioeeaoeaiui, anee
iia?aoeey oiiiaeaiey a iai annioeeaoeaia. Eieueoei iacuaaaony eieueoeii
n aaeeieoeae, anee iii iaeaaeaao iaeo?aeueiui yeaiaioii ioiineoaeueii
oiiiaeaiey.

Ii?aaeaeaiea. Ionoue A – annioeeaoeaiia eieueoei n aaeeieoeae 1. Yeaiaio
a(A iacuaaaony ia?aoeiui, anee nouanoaoao yeaiaio b(A oaeie, /oi ab = ba
= 1.

Eaaei i?iaa?eoue, /oi yeaiaio b, i eioi?ii eaeao ?a/ue iaoiaeeony
iaeiicia/ii, iiyoiio ii iaicia/aaony a-1 e iacuaaaony yeaiaioii ia?aoiui
e a.

Aaaeiaeoei oeiii eieaoe yaeythony iiey.

Ii?aaeaeaiea. Annioeeaoeaii-eiiiooaoeaiia eieueoei n aaeeieoeae
iacuaaaony iieai, anee a iai anyeee iaioeaaie yeaiaio ia?aoei.

20. I?eia?u eieaoe: /eneiaua eieueoea, eieueoea iiiai/eaiia, eieueoea
iineaaeiaaoaeueiinoae e ooieoeee, eieueoea iao?eoe, eieueoea au/aoia.

Anee a?oiiu iiyaeythony, i?aaeaea anaai, eae a?oiiu ia?aoeiuo
ioia?aaeaiee, oi aicieeiiaaiea iiiyoey eieueoea naycaii n eco/aieai
aaaeiaeoeo /eneiauo nenoai e iiiai/eaiia.

1. *eneiaua eieueoea (eieueoea, yeaiaiou eioi?uo yaeythony eiiieaeniuie
/eneaie):

a) (eeanne/aneea /eneiaua eieueoea) eieueoei oeaeuo /enae Z, eieueoei
?aoeeiiaeueiuo /enae Q, eieueoei aeaenoaeoaeueiuo /enae R, eieueoei
eiiieaeniuo /enae C.

a) eieueoei Z[i] oeaeuo aaonniauo /enae aeaea a + bi, aaea a, b – oeaeua
/enea;

n oeaeuie a, b.

2. Eieueoea iiiai/eaiia R[x], Q[x], Z[x], C[x] io iaeiie ia?aiaiiie x n
aeaenoaeoaeueiuie, ?aoeeiiaeueiuie, oeaeuie e eiiieaeniuie
eiyooeoeeaioaie.

3. Eieueoea iineaaeiaaoaeueiinoae e ooieoeee. N?aaee yoeo eieaoe
auaeaeei iniai:

a) eieueoei iineaaeiaaoaeueiinoae aeaenoaeoaeueiuo /enae n iau/iuie
iia?aoeeyie neiaeaiey e oiiiaeaiey iineaaeiaaoaeueiinoae;

a) eieueoei ia?aie/aiiuo iineaaeiaaoaeueiinoae aeaenoaeoaeueiuo /enae;

a) eieueoei ooiaeaiaioaeueiuo iineaaeiaaoaeueiinoae;

a) eieueoei iai?a?uaiuo aeaenoaeoaeueii-cia/iuo ooieoeee ia io?acea [0 ,
1].

4. Eieueoea iao?eoe. N?aaee ?aciiia?aciuo iao?e/iuo eieaoe auaeaeei
neaaeothuea:

a) iieiia iao?e/iia eieueoei Mn(A) iaae eieueoeii A eee eieueoei
eaaae?aoiuo iao?eoe ii?yaeea n n yeaiaioaie ec eieueoea A, a ea/anoaa
eieueoea eiyooeoeeaioia A iiaeii ?anniao?eaaoue, a /anoiinoe, ethaia
/eneiaia eieueoei;

a) eieueoei Dn(A) aeeaaiiaeueiuo iao?eoe, o.a. iao?eoe, o eioi?uo aia
aeaaiie aeeaaiiaee iaoiaeyony oieueei ioeaaua yeaiaiou;

a) eieueoei TNn(A) ieeueo?aoaieueiuo iao?eoe, o.a. o?aoaieueiuo iao?eoe
n ioeyie ia aeaaiie aeeaaiiaee.

Eieueoea Mn e TNn yaeythony iaeiiiooaoeaiuie, a eieueoea TNn iao
aaeeieoeu.

30. I?eia?u iieae.

] .

, aaea f(x), g(x) – iiiai/eaiu n aeaenoaeoaeueiuie eiyooeoeeaioaie,
i?e/ai iiiai/eai g(x) iaioeaaie. Iia?aoeee neiaeaiey e oiiiaeaiey
ae?iaae iau/iua.

3. Iiea au/aoia Zp ii i?inoiio iiaeoeth p. Iai?eia?, aeey p=7
ooaa?aeaeaiea iieo/aaony ec neaaeothueo ?aaainoa a eieueoea Z7: 2(4 =
3(5 = 6(6 = 1.

40. A?eoiaoeea eieaoe e iieae. Aaaeiaeoea a?eoiaoe/aneea naienoaa
yeaiaioia eieaoe e iieae i?eaaaeaiu a oai?aiao.

Oai?aia. Aeey ethauo yeaiaioia eieueoea ni?aaaaeeeau ?aaainoaa:

(a) 0(x = x(0 = 0;

(a) i?aaeei ciaeia: x(- y) = (-x)y = -(xy);

(a) (aeeno?eaooeaiinoue oiiiaeaiey ioiineoaeueii ?aciinoe)

(x – y)z = xz – yz, x(y – z) = xy – xz;

aaea ?aciinoue ii?aaeaeyaony iau/iui ia?acii x – y := x + (- y).

Aeieacaoaeuenoai. (a) Eiaai: 0(x = (0 + 0)(x = 0(x +0(x, ioeoaea 0(x =
0. Aiaeiae/ii i?iaa?yaony e aoi?ia ?aaainoai x(0 = 0.

(a) Eiaai: 0 = x(0 = x((y + (-y)) = x(y +x((-y), ioeoaea x((-y) =
-(x(y).

(a) Eiaai: (x – y)z =(x + (- y))z = x(z + (-y)(z = x(z – y(z. (

:= a(b-1, anee a, b – yeaiaiou iiey, i?e/ai b ( 0.

Oai?aia. A iiea ni?aaaaeeeau iau/iua i?aaeea ?aaiou n ae?iayie:

;

;

;

, anee ab ( 0;

a /anoiinoe, ni?aaaaeeeai ecaanoiia i?aaeei aeaeaiey ae?iaae.

.

.

Aiaeiae/ii i?iaa?ythony e aeaa inoaaoeony ioieoa. (

3. A?eoiaoe/aneea ooieoeee: ((n), ((n), ((n).

10. Iieiay ioeueoeieeeaoeaiinoue.

Ii?aaeaeaiea. *eneiaie (a?eoiaoe/aneie) ooieoeeae iacuaaaony ooieoeey,
ii?aaeaeaiiay ia iiiaeanoaa Z+ oeaeuo iieiaeeoaeueiuo /enae e
i?eieiathuay eiiieaeniua cia/aiey.

*eneiaay ooieoeey ( iacuaaaony aiieia ioeueoeieeeaoeaiie, anee auiieiaiu
oneiaey:

(1) ((x) ((x)(0,

(2) aeey ethauo acaeiii i?inouo /enae x e y

((xy)= ((x) ((y).

Caiaoei, /oi iaiin?aaenoaaiii ec ii?aaeaeaiey auoaeaao ?aaainoai

((1)=1.

A naiii aeaea, ((1)(0, oae eae eia/a aeaiiay ooieoeey ( auea au ioeaaie;
((1)= ((1(1)= ((1) ((1), neaaeiaaoaeueii, ((1)=1.

Eaaei i?iaa?eoue, /oi eaaeaeay ec neaaeothueo ooieoeee

((x)=1, ((x)= x, ((x)= x-1,

aiieia ioeueoeieeeaoeaia.

Neaaeothuay oai?aia iicaieyao nouanoaaiii ?anoe?eoue caian aiieia
ioeueoeieeeaoeaiuo ooieoeee.

Oai?aia. I?iecaaaeaiea aiieia ioeueoeieeeaoeaiuo ooieoeee yaeyaony
aiieia ioeueoeieeeaoeaiie ooieoeeae.

Aeieacaoaeuenoai. Ionoue /enea x e y acaeiii i?inou, a ooieoeee f e g
aiieia ioeueoeieeeaoeaiu. Oiaaea, iaicia/ea /a?ac h i?iecaaaeaiea
ooieoeee f e g, eiaai:

h(xy)=f(xy)g(xy)=f(x)f(y)g(x)g(y)=[f(x)g(x)][f(y)g(y)]=

=h(x)h(y).

Neaaenoaea. Aeey ethaiai oeaeiai k ooieoeey ((x)= xk aiieia
ioeueoeieeeaoeaia.

20. Noiia cia/aiee ooieoeee ii anai aeaeeoaeyi a?aoiaioa.

Aaaaeai a ?anniio?aiea, ia?yaeo n ooieoeeae ((x), ooieoeeth

,

?aaioth noiia anao cia/aiee ooieoeee ((d) i?e oneiaee, /oi ia?aiaiiay d
i?iaaaaao ana aeaeeoaee /enea x.

, oi

.

oaeaea aiieia ioeueoeieeeaoeaia.

Aeieacaoaeuenoai. ?anniio?ei i?iecaaaeaiea noii, iaoiaeyuaany a i?aaie
/anoe o?aaoaiiai ?aaainoaa:

=

.

Inoaeinue caiaoeoue, /oi aeey eaaeaeiai iaai?a ((1, (2,…, (k ) oeaeuo
iaio?eoeaoaeueiuo /enae (i, ia i?aainoiaeyueo ai, a noiia

, iieo/aai

.

Naienoai iieiie ioeueoeieeeaoeaiinoe ?anniao?eaaaiie ooieoeee
iaiaaeeaiii auoaeaao ec oiai, /oi acaeiii i?inoua /enea niaea?aeao
?acee/iua i?inoua niiiiaeeoaee. (

30. *enei aeaeeoaeae ((x) e noiia aeaeeoaeae ((x).

?anniio?ei neaaeothuea aiieia ioeueoeieeeaoeaiua ooieoeee:

, aaea ((x)=1, – /enei aeaeeoaeae /enea x,

, aaea ((x) = x, – noiia aeaeeoaeae /enea x.

Oai?aia. Ni?aaaaeeeau oiaeaeanoaa:

)=(a1 + 1)( a2 + 1)…( ak + 1),

.

Aeieacaoaeuenoai. a) Ec ii?aaeaeaiey ooieoeee ((x) iaiaaeeaiii neaaeoao
oeacaiiia oiaeaeanoai, iineieueeo a neeo iniiaiiai oiaeaeanoaa eaaei
iiaen/eoaoue /enei neaaaaiuo, eaaeaeia ec eioi?uo ?aaii 1, a eaaeaeie ec
neiaie niioaaonoaothuaai i?iecaaaeaiey.

a) Yoi oiaeaeanoai iieo/aaony ec iniiaiiai oiaeaeanoaa e oi?ioeu noiiu
/eaiia aaiiao?e/aneie i?ia?annee:

.(

40. Ooieoeey Yeea?a. Iaeiie ec aaaeiaeoeo /eneiauo ooieoeee yaeyaony
neaaeothuay ooieoeey, aia?aua aaaaeaiiay a ?anniio?aiea Yeea?ii.

Ii?aaeaeaiea. *a?ac ((x) iaicia/aaony eiee/anoai /enae ?yaea

1, 2, …, x, (*)

acaeiii i?inouo n /eneii x.

Ni?aaaaeeeaa neaaeothuay oai?aia, eioi?oth i?eaaaeai aac
aeieacaoaeuenoaa.

, oi

.

Neaaenoaea. Ooieoeey Yeea?a aiieia ioeueoeieeeaoeaia e

.

.

,

. (

4. Aeai?eoi Aaeeeaea e aai i?eiaiaiey

10. Aeai?eoi Aaeeeaea. Iaeaieueoee iauee aeaeeoaeue /enae a, b iiaeii
iaeoe n iiiiuueth aeai?eoia Aaeeeaea, eioi?ue ninoieo a neaaeothuai.

Ionoue b>0. ?acaeaeei a ia b, oiaaea ii oai?aia i aeaeaiee n inoaoeii:

a = bq1 + r1.

Anee r1 = 0, oi IIAe(a, b) = b.

Anee r1 ( 0, oi ?acaeaeei b n inoaoeii ia r1:

b = r1q2 + r2.

Anee r2 = 0, oi i?ioeann aeaeaiey caeii/ei, a anee r2 ( 0, oi ?acaeaeei
r1 n inoaoeii ia r2 :

r1 = r2q3 + r3.

I?iaeieaeay aeaeaa oaeei aea ia?acii, iu caeii/ei i?ioeann aeaeaiey eae
oieueei iieo/eony inoaoie ?aaiue 0.

Caiaoei, /oi oaeie inoaoie iaycaoaeueii iieo/eony. A naiii aeaea,
inoaoie anaaaea iaiueoa aeaeeoaey, iiyoiio b > r1 > r2 > r3 > . . . e
/enei iieo/aaiuo inoaoeia ia i?aainoiaeeo b.

Eoae, a ?acoeueoaoa oeacaiiiai aeai?eoia iieo/ei, /oi:

a = bq1 + r1 ,

b = r1 q2 + r2 ,

r1 = r2 q3 + r3 , (1)

. . . . . . . . . . . . .

rn-2 = rn-1 qn-1 + rn ,

rn-1 = rn qn .

Oiaaea ia iniiaaiee naienoa 20 e 10 :

IIAe(a, b) = IIAe(b, r1) = IIAe(r1, r2) = . . . = IIAe(rn-1, rn) = rn.

Neaaeiaaoaeueii, iaeaieueoee iauee aeaeeoaeue /enae a e b niaiaaeaao n
iineaaeiei iaioeaaui inoaoeii rn a aeai?eoia Aaeeeaea aeey /enae a e
b.

I?eia?. Iaeoe IIAe(160, 72).

I?eiaiei e aeaiiui /eneai aeai?eoi Aaeeeaea:

160 = 72(2 + 16, 72 = 16(4 + 8, 16 = 8(2. (2)

Neaaeiaaoaeueii, IIAe(160, 72) = 8.

20. Oai?aia (i eeiaeiii i?aaenoaaeaiee IIAe). Anee d – iaeaieueoee
iauee aeaeeoaeue /enae a e b, oi nouanoaotho oaeea oeaeua /enea x e y,
/oi auiieiyaony ?aaainoai: d = xa + yb.

( Aeiionoei, /oi /enea a e b naycaiu neaaeothueie niioiioaieyie:

a = bq1 + r1 ,

b = r1 q2 + r2 ,

r1 = r2 q3 + r3 ,

. . . . . . . . . . . . .

rn-2 = rn-1 qn-1 + rn .

Aeieaaeai, /oi eaaeaeia ec /enae rk eeiaeii au?aaeaaony /a?ac a e b n
oeaeuie eiyooeoeeaioaie. Aeey r1 ooaa?aeaeaiea o?eaeaeueii: r1 = a – bq1
. N/eoay, /oi eaaeaeia ec /enae r1 , r2 , . . . , rn-1 yaeyaony
oeaei/eneaiiie eeiaeiie eiiaeiaoeeae /enae a e b (rk = (k a + (k b),
eiaai

rn = (n-2 a + (n-2 b – ((n-1 a + (n-1 b) qn-1 = ((n-2 – (n-1) a + ((n-2
– (n-1 qn-1)b. (

I?eia?. Iaeoe eeiaeiia i?aaenoaaeaiea IIAe(160, 72).

?aoaiea. Ec aoi?iai ?aaainoaa nenoaiu (2) neaaeoao, /oi 8 = 72 – 16(4, a
ec ia?aiai ?aaainoaa iieo/ei, /oi 16 = 160 – 72(2. Ec aeaoo iieo/aiiuo
?aaainoa iaoiaeei: 8 = 72 – 16 ( 4 = 72 – (160 – 72 ( 2) ( 4 = (-4) (
160 + 9 ( 72.

Oaeei ia?acii, eneiiia i?aaenoaaeaiea IIAe eiaao aeae:

8 = (-4) ( 160 + 9 ( 72.

. Aeey ?aceiaeaiey /enea ( a iai?a?uaioth oeaiioth ae?iaue iiaeii
ainiieueciaaoueny aeai?eoiii Aaeeeaea:

Iai?a?uaiua ae?iae iiaeii eniieueciaaoue aeey ?aoaiey ?acee/iuo
oai?aoeei-/eneiauo caaea/.

1. Eeiaeiia i?aaenoaaeaiea iaeaieueoaai iauaai aeaeeoaey

I?eia? 1. Iaeoe eeiaeiia i?aaenoaaeaiea iaeaieueoaai iauaai aeaeeoaey
/enae (59, 163).

:

= [2; 1, 3, 4, 1, 2].

Ceaaeiaaoaeueii, iiaeii oaia?ue caiieieoue oaaeeoeo:

qs 2 1 3 4 1 2

Ps 1 2 3 11 47 58 163

Qs 0 1 1 4 17 21 59

(s +1 -1 +1 -1 +1 -1

Ionthaea iieo/aai 59 ( 58 – 163 ( 21 = -1 eee 59 ( (-58) + 163 ( 21 = 1.

2. ?aoaiea eeiaeiuo aeeioaioiauo o?aaiaiee

Eae i?aeoe/anee iaoiaeeoue eaeia-ieaoaeue ?aoaiea eeiaeiiai
iaii?aaeaeaiiiai o?aaiaiey

ax + by = c i?e (a, b)=1, c=1 ?

, ioeoaea aQn – bPn = (-1)n .

I?eia?. ?aoeoue aeeioaioiai o?aaiaiea 163x + 59y = 1.

?aoaiea. Iu i?iaa?eee ?aiueoa, /oi 163 ( 21 + 59 ( (-58) = 1,
neaaeiaaoaeueii, iauaa ?aoaiea eiaao aeae:

6. Aacen e ?acia?iinoue aaeoi?iiai i?ino?ainoaa

= (1e1 + . . . + (nen, aaea (i – /enea, ei – aaeoi?u ec i?ino?ainoaa V,
iacuaaaony eeiaeiie eiiaeiaoeeae aaeoi?ia ei; /enea (i iacuaathony
eiyooeoeeaioaie eeiaeiie eiiaeiaoeee.

Ii?aaeaeaiea. Eeiaeiie iaiei/eie nenoaiu aaeoi?ia E = (e1, . . . , en)
iacuaaaony iiiaeanoai anaaiciiaeiuo eeiaeiuo eiiaeiaoeee aaeoi?ia
aeaiiie nenoaiu; iaicia/aiea L(E). Oaeei ia?acii,

.

Caiaoei, /oi eeiaeiay iaiei/ea nenoaiu aaeoi?ia yaeyaony eeiaeiui
iiaei?ino?ainoaii.

Aiai?yo, /oi aaeoi? ( eeiaeii au?aaeaaony /a?ac nenoaio E, anee ( (
L(E).

Ioiaoei i?inoaeoea naienoaa eeiaeiuo iaiei/ae:

(a) Anee W – iiaei?ino?ainoai a V, E ( W, oi L(E) ( W;

(a) Eeiaeiay iaiei/ea L(E) niaiaaeaao n ia?ana/aieai anao eeiaeiuo
iiaei?ino?ainoa, niaea?aeaueo nenoaio E;

(a) L(E ( G) = L(E) + L(G), aaea noiia iiaei?ino?ainoa U e W
ii?aaeaeyaony ?aaainoaii U + W := { u + w( u ( U, w ( W }.

20. Eeiaeii iacaaeneiua nenoaiu.

Eeiaeiay eiiaeiaoeey aaeoi?ia iacuaaaony o?eaeaeueiie, anee ana aa
eiyooeoeeaiou ?aaiu 0. Cia/aiea o?eaeaeueiie eeiaeiie eiiaeiaoeee ?aaii
0.

Ii?aaeaeaiea. Nenoaia aaeoi?ia iacuaaaony eeiaeii iacaaeneiie, anee
anyeay aa iao?eaeaeueiay eeiaeiay eiiaeiaoeey ioee/ia io ioey.

Caiaoei, /oi aeey aeieacaoaeuenoaa eeiaeiie iacaaeneiinoe nenoaiu
aeinoaoi/ii i?e?aaiyoue e ioeth i?iecaieueioth aa eeiaeioth eiiaeiaoeeth
e auaanoe ec yoiai ?aaainoai ioeth anao aa eiyooeoeeaioia.

E?iia oiai, nenoaia aaeoi?ia yaeyaony eeiaeii caaeneiie, anee iaeioi?ay
aa iao?eaeaeueiay eeiaeiay eiiaeiaoeey ?aaia 0.

Iai iio?aaothony a aeaeueiaeoai neaaeothuea aeaa eaiiu, eioi?ua iu
i?eaaaeai aac aeieacaoaeuenoaa.

Eaiia 1. Anee nenoaia E eeiaeii iacaaeneia, a nenoaia E(s (iieo/aiiay
i?eniaaeeiaieai aaeoi?a s e nenoaia E) eeiaeii caaeneia, oi s eeiaeii
au?aaeaaony /a?ac E.

Eaiia 2 (iniiaiay eaiia i eeiaeiie caaeneiinoe).

“Aieueoay“ nenoaia eeiaeii caaeneia, anee iia eeiaeii au?aaeaaony /a?ac
“iaeaiueeoth“.

30. Aacen eeiaeiiai i?ino?ainoaa.

Ii?aaeaeaiea 1. Nenoaia E iacuaaaony aacenii eeiaeiiai i?ino?ainoaa V
(iaicia/aiea B(V)), anee auiieiaiu oneiaey:

(a) E eeiaeii iacaaeneia;

(a) V = L(E), o.a. anyeee aaeoi? i?ino?ainoaa V eeiaeii au?aaeaaony
/a?ac E.

Ia?yaeo n aeaiiui ii?aaeaeaieai iiaeii i?eaanoe e ae?oaea yeaeaaeaioiua
ii?aaeaeaiey.

Ii?aaeaeaiea 2. Iaeneiaeueiay eeiaeii iacaaeneiay nenoaia E iacuaaaony
aacenii eeiaeiiai i?ino?ainoaa V.

Ii?aaeaeaiea 3. Nenoaia E iacuaaaony aacenii eeiaeiiai i?ino?ainoaa V,
anee anyeee aaeoi? i?ino?ainoaa V iaeiicia/ii caienuaaaony a aeaea
eeiaeiie eiiaeiaoeee aaeoi?ia nenoaiu E.

Caiaoei, /oi oeacaiiua ii?aaeaeaiey ?aaiineeueiu.

40. ?acia?iinoue eeiaeiiai i?ino?ainoaa.

Ii?aaeaeaiea. Eeiaeiia i?ino?ainoai iacuaaaony eiia/iiia?iui, anee iii
iaeaaeaao eiia/iui aacenii.

Ii?aaeaeaiea. *enei yeaiaioia a eaeii-ieaoaeue aacena eeiaeiiai
i?ino?ainoaa V iacuaaaony aai ?acia?iinoueth; iaicia/aiea dimV. Ioeaaia
i?ino?ainoai eiaao ii ii?aaeaeaieth ionoie aacen e ioeaaoth
?acia?iinoue.

Ioiaoei i?aaeaea anaai oai?aio i ei??aeoiinoe ii?aaeaeaiey ?acia?iinoe.

Oai?aia. Anyeea aeaa aacena iaeiiai eiia/iiia?iiai i?ino?ainoaa
niaea?aeao iaeeiaeiaia /enei aaeoi?ia.

Aeieacaoaeuenoai. Ionoue E e G – aeaa aacena i?ino?ainoaa V. Yoe nenoaiu
aaeoi?ia eeiaeii yeaeaaeaioiu, o.a. iie eeiaeii au?aaeathony ae?oa /a?ac
ae?oaa. Anee au iaeia nenoaia auea “aieueoie”, a ae?oaay “iaeaiueeie”,
oi “aieueoay” nenoaia ieacaeanue au eeiaeii caaeneiie a neeo iniiaiie
eaiiu i eeiaeiie caaeneiinoe, cia/eo, iaa iie niaea?aeao iaeeiaeiaia
/enei aaeoi?ia. (

Neaaenoaea.

(a) ?acia?iinoue eeiaeiie iaiei/ee L(E) ?aaia ?aiao nenoaiu E (?aia
nenoaiu – iaeneiaeueiia /enei aa eeiaeii iacaaeneiuo aaeoi?ia): dim L(E)
= r(E).

(a) Anyeay nenoaia aaeoi?ia n-ia?iiai eeiaeiiai i?ino?ainoaa,
niaea?aeauay aieaa n yeaiaioia eeiaeii caaeneia.

50. I?eia?u.

1. Eii?aeeiaoiia i?ino?ainoai kn eiaao noaiaea?oiue aacen ec aaeeie/iuo
aaeoi?ia ei := (0, . . . , 0, 1, 0, . . . , 0) ( aaeeieoea iaoiaeeony ia
ianoa n iiia?ii i), neaaeiaaoaeueii, dim kn = n. Iiaeii aeieacaoue, /oi
nenoaia ec n aaeoi?ia-no?ie ia?acoao aacen i?ino?ainoaa kn (
ii?aaeaeeoaeue yoie nenoaiu ioee/ai io ioey.

2. Aacen i?ino?ainoaa ?aoaiee iaeii?iaeiie nenoaiu eeiaeiuo o?aaiaiee –
yoi ooiaeaiaioaeueiay nenoaia ?aoaiee.

eiaao noaiaea?oiue aacen ec iao?e/iuo aaeeieoe Eij (aaeeieoea
iaoiaeeony ia ianoa n iiia?ii (i, j), neaaeiaaoaeueii,

= nm.

4. I?ino?ainoaa iiiai/eaiia Qn[x] n ?aoeeiiaeueiuie eiyooeoeeaioaie
noaiaie ia i?aainoiaeyuae n eiaao neaaeothuea aacenu:

a) noaiaea?oiue aacen aeaea 1, x, x2, . . . , xn;

a) aacen Oaeei?a “a oi/ea c”:

1, (x – c), (x – c)2, . . . , (x – c)n , aaea c – iaeioi?ia /enei;

a) [aacen Eaa?aiaea “a oi/ea (c1, . . . , cn+1)”:

gi(x) = {(x – c1) . . . (x – ci)^ . . . (x – cn+1)}/ {(ci – c1) . . .
(ci – ci)^ . . . (ci – cn+1)},

aaea c1, . . . , cn+1 – iiia?ii ?acee/iua neaey?u, a ciae ^ icia/aao
ionoonoaea oeacaiiiai iiiaeeoaey.]

Eii?aeeiaou iiiai/eaia f(x)

ioiineoaeueii noaiaea?oiiai aacena – yoi aai eiyooeoeeaiou;

;

[ioiineoaeueii aacena Eaa?aiaea – yoi no?iea (f(c1), . . . , f(cn+1)).]

5. Aauanoaaiiia eeiaeiia i?ino?ainoai C eiaao noaiaea?oiue aacen (1, i).

7. Iniiaiua oai?aiu i nenoaiao eeiaeiuo o?aaiaiee

10. Enneaaeiaaiea nenoaiu eeiaeiuo o?aaiaiee.

Ionoue caaeaia nenoaia eeiaeiuo o?aaiaiee: Ax = b, aaea A- iniiaiay
iao?eoea, x- noieaaoe ia?aiaiiuo, b – noieaaoe naiaiaeiuo /eaiia. N
iiiiuueth yeaiaioa?iuo i?aia?aciaaiee no?ie a iniiaiie iao?eoea iiaeii
iino?ieoue iaeneiaeueioth nenoaio aaeeie/iuo noieaoeia. E?iia oiai,
oaeaeei ec ?anoe?aiiie iao?eoeu ioeaaua no?iee. Oiaaea iiaeii n/eoaoue,
/oi ?anoe?aiiay iao?eoea nenoaiu o?aaiaiee eiaao aeae:

,

aaea a iineaaeiae no?iea aaaeouee yeaiaio iaicia/ai /a?ac (.

Aeey iaioeaaiai /enea ( aiciiaeiu aeaa neo/ay:

(a) ( iaoiaeeony aei /a?ou, o.a. eaaeeo a iniiaiie iao?eoea.
Neaaeiaaoaeueii, a yoii neo/aa iu iiaeai iaienaoue iauaa ?aoaiea
niaianoiie nenoaiu. Caiaoei, /oi ana ia?aiaiiua aoaeoo naycaiu ( ?aia
iniiaiie iao?eoeu ?aaai /eneo ia?aiaiiuo nenoaiu.

(a) ( iaoiaeeony iinea /a?ou; oiaaea nenoaia ianiaianoia e ?aia
iniiaiie iao?eoeu iaiueoa ?aiaa ?anoe?aiiie iao?eoeu ia aaeeieoeo.

Oai naiui, iu aeieacaee oai?aio.

Oai?aia. Ionoue ( – aaaeouee yeaiaio iineaaeiae no?iee i?eaaaeaiiie
nooiai/aoie iao?eoeu. Oiaaea

(a) nenoaia niaianoia ( ( iaoiaeeony aei /a?ou;

(a) nenoaia ianiaianoia ( ( iaoiaeeony iinea /a?ou;

(a) nenoaia yaeyaony ii?aaeaeaiiie ( ( iaoiaeeony aei /a?ou e ana
ia?aiaiiua naycaiiua;

(a) nenoaia yaeyaony iaii?aaeaeaiiie ( ( iaoiaeeony aei /a?ou e eiaaony
oioy au iaeia naiaiaeiay ia?aiaiiay.

20. E?eoa?ee niaianoiinoe e ii?aaeaeaiiinoe.

Ec i?eaaaeaiiie oai?aiu iaiaaeeaiii auoaeatho neaaeothuea aeaa e?eoa?ey.

E?eoa?ee niaianoiinoe (oai?aia E?iiaeea?a-Eaiaeee). Nenoaia Ax = b
eeiaeiuo o?aaiaiee yaeyaony niaianoiie ( ?aia iniiaiie iao?eoeu ?aaai
?aiao ?anoe?aiiie iao?eoeu, o.a. r(A) = r(A(b).

E?eoa?ee ii?aaeaeaiiinoe. Nenoaia Ax = b eeiaeiuo o?aaiaiee io n
ia?aiaiiuo yaeyaony ii?aaeaeaiiie ( ?aia iniiaiie iao?eoeu ?aaai ?aiao
?anoe?aiiie iao?eoeu e ?aaai /eneo ia?aiaiiuo a nenoaia, o.a. r(A) =
r(A(b) = n.

30. Naycue iaaeaeo ?aoaieyie niaianoiie iaiaeii?iaeiie e naycaiiie n iae
iaeii?iaeiie nenoaiaie eeiaeiuo o?aaiaiee.

Aeiionoei, /oi aeaia niaianoiay nenoaia eeiaeiuo o?aaiaiee:

Ax = b. (1)

Ionoue (0, (1, (2 – /anoiua ?aoaiey nenoaiu (1), ( – aa iauaa ?aoaiea.
Oiaaea ni?aaaaeeeau ?aaainoaa A(1t = b, A(2t = b. Au/eoay ii/eaiii ec
ia?aiai aoi?ia, ia iniiaaiee ecaanoiuo naienoa, iieo/aai: 0 = A(1t –
A(2t = A((1t – (2t) = A((1 – (2)t, o.a. ?aciinoue iaaeaeo aeaoiy
/anoiuie ?aoaiey nenoaiu (1) yaeyaony ?aoaieai naycaiiie n iae
iaeii?iaeiie nenoaiu

Ax = 0. (2)

Anee oaia?ue ( – iauaa ?aoaiea nenoaiu (2), oi eiaai A( t = 0,
neaaeiaaoaeueii,

b = b + 0 = A(0t + A( t = A((0t +( t) = A((0 +( )t,

o.a. noiia /anoiiai ?aoaiey nenoaiu (1) e iauaai ?aoaiey nenoaiu (2)
yaeyaony ?aoaieai nenoaiu (1).

Oaeei ia?acii, ni?aaaaeeeaa

Oai?aia. Iauaa ?aoaiea niaianoiie iaiaeii?iaeiie nenoaiu (1) yaeyaony
noiiie /anoiiai ?aoaiey nenoaiu (1) e iauaai ?aoaiey nenoaiu (2).

Iineieueeo iauaa ?aoaiea iaeii?iaeiie nenoaiu iiaeao auoue caienaii a
aeaea eeiaeiie eiiaeiaoeee ON?, oi iieo/aai, /oi iauaa ?aoaiea nenoaiu
(1) iiaeii caienaoue a neaaeothuae ia?aiao?e/aneie oi?ia:

( = (0 + (1(1 + (2(2 + . . . + (m(m,

aaea (0 – eaeia-ieaoaeue /anoiia ?aoaiea nenoaiu (1); (1, (2, . . . ,
(m – ON? nenoaiu (2),

(1, (2, . . . , (m – aeaenoaeoaeueiua ia?aiao?u; m = n – r(A).

8. Ei?ie iiiai/eaia; noaia Ai?ia?a; oai?aia Aaco

10. Ei?ie iiiai/eaia.

Ii?aaeaeaiea. *enei c iacuaaaony ei?iai iiiai/eaia f, anee f(c)=0.

Ae?oaeie neiaaie, /enei c yaeyaony ei?iai iiiai/eaia f, anee

a0cn + a1cn-1 + … + an – 1c + an = 0.

Yoi ?aaainoai icia/aao, /oi /enei c yaeyaony ei?iai o?aaiaiey

a0 xn + a1xn-1 + … + an – 1 x + an = 0,

i?e iiaenoaiiaea aianoi x /enea c iieo/aaony aa?iia ?aaainoai. Iiyoiio
ei?aiue iiiai/eaia f e ei?aiue niioaaonoaothuaai o?aaiaiey f(x) = 0 –
yoi iaeii e oi aea.

Noaia Ai?ia?a iicaieyao i?iaa?youe, yaeyaony ee aeaiiia /enei c ei?iai
aeaiiiai iiiai/eaia eee iao: n aa iiiiuueth iu eae ?ac e au/eneyai
cia/aiea f(c).

Anee o?aaoaony i?iaa?eoue ianeieueei cia/aiee c, oi aeey yeiiiiee
aueeaaeie no?iyo ia o?e ioaeaeueiua noaiu, a iaeio – iauaaeeiaiioth.
Iai?eia?, aeey iiiai/eaia

f = 3×5 – 5×4 – 7×2 + 12

e /enae c = 1,-1,2 ninoaaeyaony oaaeeoea

3 -5 0 -7 0 12

1 3 -2 -2 -9 -9 3

-1 3 -8 8 -15 15 -3

2 3 1 2 -3 -6 0

Eiia/ii, i?e caiieiaiee o?aoueae e /aoaa?oie no?iee oaaeeoeu ?aaioaao”
oieueei ia?aay no?iea – no?iea eiyooeoeeaioia iiiai/eaia f.

Iu aeaeei, a /anoiinoe, /oi ec o?ao ?anniio?aiiuo /enae oieueei c = 2
yaeyaony ei?iai aeaiiiai iiiai/eaia.

20. Oai?aia Aaco.

Oai?aia Aaco. Ionoue f – iiiai/eai, c – iaeioi?ia /enei.

1. f aeaeeony ia aeao/eai x – c oiaaea e oieueei oiaaea, eiaaea /enei c
yaeyaony aai ei?iai.

2. Inoaoie io aeaeaiey f ia x – c ?aaai f(c).

Aeieacaoaeuenoai. Nia/aea iu aeieaaeai aoi?ia ooaa?aeaeaiea. Aeey yoiai
?acaeaeei f c inoaoeii ia x – c:

f = (x – c)q + r;

ii ii?aaeaeaieth inoaoea, iiiai/eai r eeai ?aaai 0, eeai eiaao
noaiaiue, iaiueooth noaiaie x – c, o.a. iaiueooth 1.

Ii noaiaiue iiiai/eaia iaiueoa 1 oieueei a neo/aa, eiaaea iia ?aaia 0,
e iiyoiio a iaieo neo/ayo r ia naiii aeaea yaeyaony /eneii – ioeai eee
ioee/iui io ioey.

Iiaenoaaea oaia?ue a ?aaainoai f = (x – c)q + r cia/aiea x = c, iu
iieo/ei

f(n) = (n – c)q(n) + r = 0,

oae /oi aeaenoaeoaeueii r = f(c), e ia?aia ooaa?aeaeaiea aeieacaii.

Oaia?ue ia?aia ooaa?aeaeaiea ii/oe i/aaeaeii. A naiii aeaea,
ooaa?aeaeaiea “f aeaeeony ia x – c” icia/aao, /oi inoaoie io aeaeaiey
?aaai 0. Ii inoaoie, ii aeieacaiiiio, ?aaai f(c), oae /oi “f aeaeeony
ia x – c” icia/aao oi aea naiia, /oi e f(c) = 0. (

Oai?aia Aaco aeaao aiciiaeiinoue, iaeaey iaeei ei?aiue iiiai/eaia,
eneaoue aeaeaa ei?ie iiiai/eaia, noaiaiue eioi?iai ia 1 iaiueoa:
anee f(c) = 0, oi f = (x – c)q, e inoaaony ?aoeoue o?aaiaiea q(x) = 0.
Eiiaaea yoei i?eaiii – ii iacuaaaony iiieaeaieai noaiaie – iiaeii
iaeoe ana ei?ie iiiai/eaia. A /anoiinoe, iiaeia?aa iaeei ei?aiue
eoae/aneiai o?aaiaiey, iiaeii aai iieiinoueth ?aoeoue – iinea
iiieaeaiey noaiaie aeinoaoi/ii ?aoeoue iieo/aiiia eaaae?aoiia o?aaiaiea.

?aoei a ea/anoaa i?eia?a o?aaiaiea

x4 – x3 – 6×2 – x + 3 = 0.

Oeaeua ei?ie iiiai/eaia f = x4 – x3 – 6×2 – x + 3 aeieaeiu auoue
aeaeeoaeyie naiaiaeiiai /eaia, oae /oi yoi iiaoo auoue oieueei /enea
(1 e (3. I?e yoii 1 ia yaeyaony ei?iai iiiai/eaia f, iineieueeo
noiia aai eiyooeoeeaioia, i/aaeaeii, ia ?aaia 0.

I?e x = -1: eiaai noaio

1 -1 -6 -1 3

-1 1 -2 -4 3 0

Iu aeaeei, /oi -1 – ei?aiue f , e a /anoiii iieo/aaony iiiai/eai

g = x3 – 2×2 – 4x +3.

Cia/aiea x = 1 aoi?ie ?ac i?iaa?youe iaca/ai: anee au /enei 1 auei
ei?iai g, oi iii auei au e ei?iai f, /oi iaaa?ii. A -1 i?iaa?eoue
iaycaoaeueii – ie/oi ia iaoaao ae auoue oaeaea e ei?iai /anoiiai g:

1 -2 -4 3

-1 1 -3 -1 4

Neaaeiaaoaeueii, g(-1) ( 0.

Ninoaaei noaio Ai?ia?a aeey x = 3:

)/2.

)/2.

30. Neaaenoaey ec oai?aiu Aaco. Oai?aia Aaco iicaieyao /anoe/ii
ioaaoeoue e ia aaaeiue oai?aoe/aneee aii?in – Neieueei ei?iae iiaeao
eiaoue iiiai/eai?

Oai?aia. Iiiai/eai noaiaie n eiaao a ethaii iiea ia aieaa n ei?iae.

Aeieacaoaeuenoai. Ionoue iiiai/eai f noaiaie n eiaao k ei?iae, e c
-iaeei ec aai ei?iae. I?aaeiieiaeei i?ioeaiia – ionoue k>n.

Ii oai?aia Aaco, f = (x – c)g, e /anoiia g eiaao noaiaiue n – 1. Anyeee
ei?aiue f, ioee/iue io c, yaeyaony iaeiia?aiaiii e ei?iai g: anee f(a)
= 0, oi (a – c)g(a) = 0, ioeoaea g(a) = 0, oae eae a( c. Ae?oaeie
neiaaie, iiiai/eai g eiaao, ii iaiueoae ia?a k – 1>n – 1 ei?aiue, o.a.
/enei aai ei?iae oaeaea aieueoa aai noaiaie.

Ii n iiiai/eaiii g iiaeii i?iaanoe oa aea ?annoaeaeaiey, e ia aoi?ii
oaao iieo/eoue iiaue iiiai/eai h, /enei ei?iae eioi?iai oaeaea aieueoa
aai noaiaie. I?iaeieaeay oaeei aea ia?acii, iu i?eaeai e iiiai/eaio
noaiaie 2, eiathuaio aieueoa 2 ei?iae, /aai ia iiaeao auoue.

Iieo/aiiia i?ioeai?a/ea iieacuaaao, /oi i?aaeiieiaeaiea k>n iaaa?ii, e
neaaeiaaoaeueii, k ia aieueoa n, /oi e o?aaiaaeinue aeieacaoue.

Ec oai?aiu i /enea ei?iae auoaeatho aeaa eneeth/eoaeueii aaaeiuo e
aeey oai?ee, e aeey i?aeoeee ooaa?aeaeaiey.

Neaaenoaea 1. Aeaa iiiai/eaia noaiaie, ia aieueoae n, i?eieiatho
iaeeiaeiaua cia/aiey i?e n + 1 cia/aiee x oiaaea e oieueei oiaaea,
eiaaea i?e eaaeaeie noaiaie x iie eiatho iaeeiaeiaua eiyooeoeeaiou.

Neaaenoaea 2. Aeaa iiiai/eaia i?eieiatho iaeeiaeiaua cia/aiey i?e anao
cia/aieyo x oiaaea e oieueei oiaaea, eiaaea i?e eaaeaeie noaiaie x iie
eiatho iaeeiaeiaua eiyooeoeeaiou.

9. ?aceiaeaiea iiiai/eaia a i?iecaaaeaiea iai?eaiaeeiuo

iiiaeeoaeae e aai aaeeinoaaiiinoue

10. Iniiaiay oai?aia a?eoiaoeee eieueoea k[x]. Ethaie iiiai/eai
iieiaeeoaeueiie noaiaie iiaeii ?aceiaeeoue a i?iecaaaeaiea iai?eaiaeeiuo
niiiiaeeoaeae, e oaeia i?aaenoaaeaiea aaeeinoaaiii n oi/iinoueth aei
annioeee?iaaiiinoe e ii?yaeea niiiiaeeoaeae.

Aeieacaoaeuenoai. 1. Nouanoaiaaiea. Eiaeoeoeeae ii n aeieaaeai, /oi
eaaeaeue iiiai/eai f noaiaie n ( 1 iiaeii ?aceiaeeoue a i?iecaaaeaiea
iai?eaiaeeiuo niiiiaeeoaeae. Iniiaaieai eiaeoeoeee i?e n = 1 neoaeeo
o?eaeaeueiia ?aceiaeaiea f = f. Naeaeaa eiaeoeoeaiia i?aaeiieiaeaiea,
?anniio?ei iiiai/eai f noaiaie n. Anee f – iai?eaiaeei, oi ?aceiaeaiea
eiaao aeae: f = f; anee aea f – i?eaiaeei, oi aai iiaeii caienaoue a
aeaea f = gh, aaea noaiaie g, h iaiueoa noaiaie f. Ii i?aaeiieiaeaieth
eiaeoeoeee iiiai/eaiu g e h iiaeii ?aceiaeeoue ia iai?eaiaeeiua
niiiiaeeoaee:

g = p1p2 . . . ps, h = q1q2 . . . qt,

iiyoiio

f = p1p2 . . . psq1q2 . . . qt.

2. Aaeeinoaaiiinoue. I?aaeiieiaeei, /oi iaeioi?ue iiiai/eai f eiaao aeaa
?aceiaeaiey ia iai?eaiaeeiua niiiiaeeoaee:

f = p1p2 . . . ps , f = q1q2 . . . qt,

oiaaea

p1p2 . . . ps = q1q2 . . . qt.

Eaaay /anoue iineaaeiaai ?aaainoaa aeaeeony ia p1, cia/eo, i?aaay
/anoue oaeaea aeaeeony ia p1. Ii iniiaiiio naienoao iai?eaiaeeiiai
iiiai/eaia ia p1 aeaeeony eeai q1, eeai q2, . . . , eeai qt. Eciaiyy,
anee iaaei ioia?aoeeth niiiiaeeoaeae, iiaeii n/eoaoue, /oi p1 aeaeeo q1,
e iineieueeo q1 iai?eaiaeei, oi iie annioeee?iaaiu, o.a. aeey iaeioi?iai
/enea c aa?ii p1 = cq1. Cia/eo, nie?auay ia p1 iaa /anoe ?aaainoaa

p1p2 . . . ps = p1q2 . . . qt,

iieo/aai:

p2 . . . ps = (cq2 ) . . . qt.

Iaicia/ei aeaiiia i?iecaaaeaiea /a?ac m, e caiaoei, /oi deg m b1, /oi iaaiciiaeii. Ia?aoiia ooaa?aeaeaiea
i/aaeaeii. (

20. Ionoue eiathony eaiiie/aneea ?aceiaeaiey iiiai/eaiia f e g:

.

Oiaaea

,

aaea ci = min (ai, bi), di = max (ai, bi).

, aaea ci = min (ai, bi). Oiaaea ii naienoao 10 iiiai/eai ( yaeyaony
aeaeeoaeai iiiai/eaiia f e g e anyeee iauee aeaeeoaeue f e g aeaeeo
iiiai/eai (. Neaaeiaaoaeueii, ( = IIAe(f, g).

Aiaeiae/ii aeieacuaaaony e aoi?ia ooaa?aeaeaiea. (

Ec naienoaa 20 iaiaaeeaiii auoaeaao naienoai

30. (Naycue iaaeaeo IIAe e IIE).

IIAe(f, g) ( IIE(f, g) = f ( g.

10. Oai?aia i no?iaiee i?inoiai aeaaa?ae/aneiai ?anoe?aiey

10. Iiiyoea ieieiaeueiiai iiiai/eaia.

Ionoue ( – aeaaa?ae/aneia /enei iaae iieai k, o.a. ei?aiue iaioeaaiai
iiiai/eaia n eiyooeoeeaioaie ec iiey k.

Ii?aaeaeaiea. Ii?ie?iaaiiue iiiai/eai (((, k, x) iaae iieai k iacuaaaony
ieieiaeueiui iiiai/eaiii /enea (, anee auiieiaiu oneiaey:

a) ((x) – iai?eaiaeei iaae iieai k, o.a. ia ?aceaaaaony a i?iecaaaeaiea
iiiai/eaiia iieiaeeoaeueiie noaiaie n eiyooeoeeaioaie ec k;

a) ((() = 0, o.a. ( – ei?aiue iiiai/eaia ((x).

I?eia?u.

(((, Q, x) x2 + 1 x2 – 5 x2 + 2x – 1 x4 – 4×2 + 16

20. Iniiaiua naienoaa ieieiaeueiuo iiiai/eaiia.

1. Anee f(x) ( k[x] e f(() = 0, oi f(x) aeaeeony ia ieieiaeueiue
iiiai/eai ((o) /enea (.

Aeieacaoaeuenoai. A naiii aeaea, i?aaeiieiaeea, /oi f ia aeaeeony ia (,
caieoai

f = (g + r, deg r

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