2
2.1.
nPE06PA308AHVIs:l
Bb IP A>KEHVIVI
CbopMyllbl coKpa~eHHoro YMHo>KeHIIls:t
Ll)ur mo6hrx a,
IQHe paBeHCTBa:
b H c H u,eJioro rroJIO:>KHTeJibHoro n
BepHhi cJie.n:yro-
(a±b) 2 = a 2 ±2ab+b2 ;
(a±b) 3 = a 3 ±3a2b+3ab 2 ±b 3 ;
a
2
-b 2 = (a-b)(a+b);
a 3 ±b3 = (a±b)(a 2 +ab+b 2);
(a+b+ci = a 2 +b 2 +c 2 +2ab+2bc+2ac;
( a+b
rz =an +nan-1b + n(;.; 1) an-2b2 + ... +bn.
2.2 . .Qe11eH111e
MHOrOYJleHOB
MllOWtUZello.M CTerreHH n Ha3hmaror Bhrpa:>KeHHe BHI'l:a:
Pn( X) =anX n +an-1X n-1 + ... +a1x+a0 .
L(eJieHHe MHOfO'-IJieHOB BhiiiOJIH5IIOT aHaJIOfH'-IHO .n:eJieHHIO U,eJihiX
qHceJI: .n:eJIHT crapumH: qJieH MHoroqJiena-.n:eJIHMoro na crapumH: qJieH
MHOroqJieHa-.n:eJIHTeJI5I, 3aTeM qaCTHOe YMHO:>KaiOT Ha MHOfOqJieH-I'l:emiTeJib H rroJiyqennoe rrpmi3Be.n:enHe BhiqHraror H3 MHoroqJiena-.n:eJIHMoro. MnoroqJien-rrepBhiH ocTaToK anaJiorHqHhiM o6pa30M .n:eJIHT
na MnoroqJien-.n:emneJih. L(eJiemre rrpo.n:oJI:>KaiOT .n:o rex rrop noKa ne
rroJiyqar ocraroK 0 HJIH crerrenh MnoroqJiena-ocraTKa ne 6y.n:er Menhrne crerremi MnoroqJiena-.n:emneJIH.
HanpU.Mep,
4
3
2
_x -3x +5x +21 x 2 +x-2
x 4 +x3 -2x 2
x 2 -4x+11
3
2
-4x + 7x +2
-4x 3 - 4x 2 + 8x
11x 2 -8x+2
11x 2 +11x-22
-19x+24
21
Pe3yJioTaT !J:CJICHHR 3al1HChiBaiOT cJie~J:yiOrn;HM o6pa3oM:
x 4 -3x 3 +5x 2 +2 = (x 2 +x-2)( x -4x+11 )+( -19x+24).
2
Teope.Ma Ee3y. OCTaTOK rOT !J:CJICHYIR MnoroqJiena Pn(x) na IJ:BY'I(x-c) paBeH 3Ha'ICHHIO MHOrO'lJiena npH x = c, TO eCTb r = Pn(c).
C.neiJcm6ue 1. )lJIR !J:C.JIHMOCTH Mnoro'IJieHa Pn(x) ria !J:BY -IJICH
(x- C) HC06XO!J:HMO H ,ll;OCTaTO'IHO, 'IT06b! 'IHCJIO C 6b!JIO KOpHCM MHOrO'IJICHa Pn(x).
C.neiJcm6ue 2. EcJIH c 1 , c2 , ... ,en nee KopHH MHoro'IJiena
Pn( X) =anX n +an-1X n-1 +... +a1x+ao,
TO Pn(x) =an (x-c 1 )(x -c2 ) ... (x-cn).
TeopeMa o ~e.nbtx uopHRX. BcRKHH ~eJihiH Kopeno MHOroLIJiena c
JICH
1
~CJib!MH K03cpqrHI.J,HCHTaMH 5IBJI5ICTC5I IJ:CJIHTCJICM CB060lJ:HOrO '!JICHa.
C.neiJcmBue H3 TeopeMbi
o ueJihiX KopnRx. I1p11 OTbiCKanHH QeJrhrx
KOpHCH MIIOrO'IJICHa C l(CJibiMH K03cpcpH~HCHTaMH lJ:OCTaTO'IHO pacCMOTpCTb IJ:CJIHTCJIH CB060lJ:HOrO l.JJICHa.
2.3.
CTeneHIII 111 ap111CpMeTIII'-feCKIIIe KOpHIII
IlpOH3BCIJ:CHHC
n
COMHQ)KHTCJICH, Ka)J(ll,hiH H3 KOTOpbiX paBeH
cmeneHb10 ttuc;za a.
ApurjJ.MemUtteCKUM KOpHe.M n-HOII
a,
na3biBaiOT n-noi1
l.JHCJia
CTCIICHH H3 HCOTpHI(aTCJibHOrO
a Ha3biBaiOT TaKOC HCOTpHQaTCJibHOC 'IHCJIO b, n-aR CTCllCHb KO-
TOporo paBna a:
r;/ci = b
<:::>
bn =a .
CsoiicTBa cTerreHei1 H apncpMeTnqecKHX KOpHeii:
(2.1)
(2.2)
(2.3)
(2.4)
22
(2.9)
1
1
a-; = r;/ci; a2 = .Ja
(2.10)
(2.11)
(2.12)
(an)m
(ab)n
=anm;
=anbn;
(ar
b = an
bn;
a
-n
1
=-·
an '
(2.5)
(2.6)
1
~=~=(1Y;
(2.13)
2~=1al;
(2.14)
2n+<j a2n+ 1
(2.7)
(2.15)
1
(2.8)
!fjiFa = nn.;J(i = a mn .
PaseHCTBa 2.2-2.9 sepHbi t~:JIR JII06brx
b.
=a ;
(2.16)
n II m II rroJio:>KIITeJibHhiX a II
n II m II JII06hrx
PaseHCTBa 2.1II 2.10-2.16 sepnhi t~:JIR HaTypaJibHbiX
HeOTpiiU:aTeJibHbiX a II
b.
BHCCCHHC MHO:mHTCJICH IIOP: 3HaK KOpH.SI H BbiHCCCHHC MHO)KHTCJICH H3-IIO,l.l; 3HaKa KOpH.SI
B
CJiy<Iae He'IeTHOH CTerreHH KOpH51 BC51KHH MHO:>KIITeJih MO:>KHO
BHCCTI1 IIOp: 3HaK :'JTOrO KOpH51, a eCJIII CTCIICHb MHO:>KHTeJI51 Bbiiiie CTeIIeHH KOpH51, TO II BbiHeCTH H3-IIOP: 3HaKa ::nora KOpH5I.
B CJiy<Iae 'IeTHOH CTerreHII KOpH51
Heo6xop:HMO IIOMHIITb, 'ITO:
1) IIpH 8fteCeHUU .MHOJICUmeJlJl noa 3HaJC KeaapamHOlO 'KOpHJl (B o6IIJ:eM cJiy<Iae rro;a: 3HaK KOpHR <IeTno:H cTerrenii) neo6xop:HMO yqHThiBaTh 3HaK ::noro MHO:>KHTeJI51:
aJb =
{.J:Jb, ecmr a > 0,
-.J:Jb, ecmr a < 0;
2) IIpH U38JletteflUU KeaapanntOlO KOpflJl (KOpH51 qeTHOi1: CTeiieHH)
II3 rrpoH3Bep:eHH51 neo6xop:IIMO y<IIITbnaTh, 'ITO KopeHb orrpep:eJien II
B
CJiy~rae, eCJIH o6a MHO:>KHTeJI51 llOJIO:>KHTeJibHbl, H B CJiy<Iae, eCJIII o6a
MHO:>KHTeJIR OTPHU:aTeJibHhi:
Jab=
{J;F-a·Jb;...J-E,
ecrrrr a> 0, H b > 0,
ecrrrr a< 0, II b < 0.
TeeT .Qns:. npoeepK111 TeopeTW·IeCKII'IX
3H8HII1111
YcmaHoeume coomeemcmeue (1-5):
1.
<I>opMyJihi coKpamennoro yMno:>KeHIIR:
a) n 3 +m 3 +3nm(n+m);
2) (m-n)(m+n);
6) m2 +n 2 ;
23
3) (m-n)(m 2 +mn+n 2 );
2
4) (m+n)( m -mn+n
5) (m+n-p)
6) (m+n)
3
7)(n-m)
3
2
2
;
);
B) m 2 -n2
;
2
r) (n+m) ;
11:) (n-m)
3
3
2
;
;
e) n +m
;
:/K) n 3 -m 3 +3mn(n-m);
8) m2 -2mn+n 2 .
;
3) m3 -n 3 ;
n) m2 +n 2 + p 2 +2(mn-mp-np);
K)
m 2 +n 2 + p 2 +2mn+2mp+2np.
2. CBOMCTBa CTerreHei1:
1) xn+m;
a) (xyt;
n
2) £.
y n'
3) Xn·yn;
4) x-n;
1
1) b~;
6) b;
Wn;
B) bJb;
2
4)~;
24
1 .
bmn'
.!!:.
2) bm;
3)
a)
r)~·m;
1
n) lbl;
5) b2;
6)
rfiJb;
IK) .Jb ;
3) '(jj.
2n+-<j b2n+ 1 .
e)
4. BbmeceHIIe MHOJKIITeJieH: II3-non 3HaKa KopHJI:
BbiPA)KEHI1E
PE3YJibTAT YI1POIUEHI15I
1)~;
a) -abc 2 ~;
2)
.J-a3b4c6 ;
3)~;
4) ~-a 3 b4 c 6
6) -ab 2 lcl
3
Ja;
B) -ab2lc13 J=a ;
.
r) ab21cl3 ,fa ;
n) abc 2 ~;
e) -abc 2 ~.
5. BHeceHwe MHOJKIITeJieH: non 3HaK KopH51:
BbiPA)KEHI1E
PE3YJibTAT YI1POIUEHI15I
1) ab 2 c2~, c<O;
a)~;
2
2) ab c2~, c > 0 ;
6) V32a 6b10 c6
3) -2ab 2c'$1C;
B)-~;
4) -2ab 2c!ifaC.
r) ~-32a 6 b 10 c 6
n)
-~16a 5 b 8 c 5
;
;
;
e) ~-16a 5 b 8 c 5 .
HoMep 3a,l(aHHJI
HoMep
rrpaBHJ!bHOfO
OTBeTa
2
3
1
1- r, 2- B, 1- B, 2-6, 1- e, 2-3,
3-3,4- e, 3- a, 4- ,n:, 3- ,n:, 4- r,
5- H, 6- a, 5- )!(, 6- r. 5- )!(, 6- 6.
7- )!(, 8- )'(.
4
1- r,
2- B,
3-.n:,
4-e
5
1- B,
2 -a,
3- r,
4-.n:
25
nplr1Mepb1
IlpHMep 1.
Yrrpocnne Bbrpa:iKeHMe
4
. 1
1
a+-1- · a+b- 1 4- 1b(abc+a+c) ·
b+c- 1
Pewemte. Br,moiiHMM rrocJie,n:oBaTeJibHO CJie,n:yrom:Me ,n:eikTBM5I:
t)
4
= 4 = 4(bc+1)
_ 4(bc+1) .
a+
c
a+bcc+1 (bc+i)(a+-c-)- abc+a+c'
b+~)
bc+1
c(
2) _1_=
a+i
b
=-b_.
b(a+i) ab+1'
2
3
4
4(bc+1) . _b_= 4(ab c+bc+ab+1).
b(abc+a+c) '
) abc+a+c · ab+ 1
4( ab 2c +bc+ab+ 1)-4
b(abc+a+c)
)
4b(abc +c+a)
= b(abc+a+c)
4
·
0mBem:4.
IIpnMep 2.
(
YrrpocTMTe Bbipa:>r<eHHe
1
+
2x
+
1
x 2 +3x+2 x 2 +4x+3 x 2 +5x+6 )
Pewe11ue.
2
. (3-x) +12x.
2
·
CorJiacHo rrpaBHJIY pasJio:>r<eHHJ! KBa,n:paTHoro Tpex'IJie-
Ha Ha IIHHeHHbie MHO:iKHTeJIH
x2 -
-2
ax 2 +bx+c =a(x-x1 )(x-x2 ), r.n:e x 1 H
KOpHM KBa,n:paTHoro TpexqJieHa
f(x)
=ax 2 +bx+c, sarrnrneM:
x 2 +3x+2 =(x+1)(x+2); x 2 +4x+3 =(x+1)(x+3);
x 2 +5x+6 =(x+2)(x+3).
Tor.n.a Hcxo,n:Hoe Bbipa)l(eHHe rrpHMeT BH,n::
2
1
)- (x-3/ +12x
1
. 2x
( (x+1)(x+2) + (x+ 1)(x+3) + (x+2)(x+3)
:
2
·
IlpHBO.ll:J! ,n:po6u K o6meMy sHaMeHaTeJIIo H rrpuMeH5!5I ¢opMyiihi
coKpameHHoro yMHO)l(eHH5I, rroJiyqMM:
26
2
2
x+3+2x +4x+x+1)2
=
( ( x + 1) ( x + 2) ( x + 3)
x 2 - 6x -1-9-1- 12x
2
2
2
(x+1) (x+2) (x+3r2 _2(x+1) (x+2) 2 (x+3) 2 _1.
2
2
- (2x 2 +6.x+4/(x 2 +6x+9)- 4(x+1) (x+2) (x+3)
2
-2
Omeem: 0,5.
llpHMep 3.
x 2 -11xy +30y 2
· , ·
x 2 -9xy+20y 2
CoKpanne ,ll,po6h
.
PaccMoTpHM onrocrne.llbiiO x KBa~(paTHhiM TpexqJieH
-11xy+30y 2 . 3na5I, qTo x 1 +x2 =11y H x 1x 2 =30y 2
Peweuue.
f(x)=x
2
(eM. TeopeMy BneTa rr.
5.2),
x 1 =5y, x 2 =6y. AnaJiorHq2
2
TpexqJiena f ( x) = x - 9xy -1- 20y :
rroJiyqHM
no nail,ZJ,eM Kopmr KBa,ZJ,panroro
x 1 =4y , x 2 =5y . Pa3JIO:>KHM TpexqJieHhi
BhiiiOJIHHM coKpamenne ,ZJ,po6n:
na IInnei1Hhie MHO:>KHTeJIH H
(x-Sy)(x- 6y)- x- 6Y
(x-4y)(x-5y)- x-4y ·
. x-6y
Omeem. - --.
x- 4y
llpHMep 4.
Peweuue.
Hai:f,ZJ,nTe KOpHHl\UIOro'-urena
6x 3 -11x 2 +6x-1.
Haii,ZJ,eM u,eJibie KOpHH MnoroqJiena. CorJiacno TeopeMe o
U,eJihiX KOpH5IX MHOfO'-IJieHa, HMII MOfYT 6biTh TOJibKO ,ZJ,eJIHTeJIII CB060,ZJ,HOf0 qJiena, TO ecTb '-IHCJia -1 11 1. Haii,ZJ,eM: ·
'
3
2
P3 (-1)=6·(-1) -11·(-1) +6·(-1)-1=-6-11-6-1=-24
u
p3 (1) =6·1-11·1+6·1-1 =0.
P.1 ( -1) i= 0 , TO qncJio -1 ne 5IBII5IeTC5I r<opneM MnoroqJiena, a TaK KaK P3 ( 1) =0 , TO "lHCJIO 1 - U,eJihiH KOpeHb MHOfO'-IJieHa.
TaK KaK
Tor,ZJ,a corJiacno ciie,ZJ,CTBHIO 113 TeopeMhi
3
6x -11x 2 + 6x -1 ,ZJ,eJIHTC5I na ,ZJ,BY'-IJieH ( x -1) .
Ee3y
MHOfO"IJieH
BhlrrOJIHHM ,ZJ,eJiemre MHoroqJiei-IoB:
3
2
6x -11x + 6.:r -1
6x 3 - 6x 2
-5x 2 +6x-1
-5x 2 +5x
x-1
x-1
I
x -1
6x 2 - Sx + 1
0
27
; h11111111f'M JII':I,Y.IIIii'<IT )\(~JICIII15I:
+ 6x -1 = (x -1) (6x 2- 5x + 1) .
JJnl!;~eM t<Oplili KBa,z:t:paTHOro TpeX'-IJieHa 6x 2 - 5x + 1. fiOJIJ"--HM:
1
1
x1 =2, x2 =3.
G.r:1 -11x
2
6x 3 ~11x 2 +6x-1
1 1
.
1; 2; 3'
CJie,l(oBaTeJihHO, MHoro'-IJieH
HMeeT TPH Kopm:c
1 ; 1 ; 1.
3 2
Omeem:
llpHMep 5.
YnpocTHTe BhrpaiKeHHe
a-b
a 2+b 2+a
b2+b+ab+a
2a-b- 2a 2+ab-b 2
7
72
( 4-b4
-:-+-4-'-ab-;:-2 ""'+'"-:a2~)""":('--2b -+a.....,.)
b +1
Peuteuue.
BhiiiOJIHHM nocJie,l(oBaTeJihHO ,l(eHCTBH5I c MHOfO'-IJieHa-
MH, rrpe,l(BapHTeJibHO paCKJia,z:t:hiBa5I HX Ha MHOJKHTeJIH:
1) 2a 2 +ab-b 2 =a 2 +ab+a 2-b 2 =a(a+b )+(a-b )(a+b) =
=(a+b)(2a-b);
) a-b
a2+b 2 +a _ (a-b)(a+b)-(a 2 +b 2 +a) _
2
2a-b (a+b)(2a-b)(a+b)(2a-b)
2
=a 2-b 2 -a 2-b 2-a= -2b 2-a _ -(2b +a) .
(a+b)(2a-b)
(a+b)(2a-b)- (a+b)(2a-b)'
2
4b4+4ab2+a2 (2b2+a)
2
3)
=
=
2b +a.
1
2
2b-+a
2b +a
2
-(2b +a)
-(2b 2 +a)
4) ..,.-'-;-:--:-:,---:...,...,.. · ( 2b 2 + a ) =
=
(a +b )(2a-b) ·
(a +b) (2a -b)( 2b 2 +a)
-1
.
= (a+b)(2a-b)'
5) b2 +h+ab+a = b(b+1)+a(b+1) =(b+1)(b+a);
6
)
-(b+1)(b+a)
(a+b)(2a-b)(b+1)
.
1
OmBem. b _ a .
2
28
=
1
1
- 2a-b = b-2a ·
IIpHMep 6.
Ynpocnue Bbipa:IKeHHe
~216x 3 (5+2.J6) ..,)3-JfX -2.J3X.
Peweuue.
BhiTIOJIHHM CJieJJ:yiOIIJHe .n:eiiCTBH5I:
1) -h.J2X-2.J3X
=~(3../2X-2.f3X) 2 =~30x-12x.J6 =
=~6x(5-2.J6);
2)
~216x 3 (5+2.J6)~6x(5-2.J6) =~(6x) 4 (5+2.J6)(5-2../6) =
=~(6x) 4 (25-24) =~(6x) 4 =6x.
Omeem: 6x.
IIpHMep 7.
YnpocTHTe Bbipa:IKemre
f?j 2~
ao
a33 a2
if;;?!"
~
4
Peweuue.
I1ocJie.l(OBaTeJihHO npHMeH5I5I <J:>opMyJihi
2.1 0, 2.5, 2.3,
2.4, TIOJiy<IHM:
2.1 s.1.1 -1.1.1
~a2~ r.r _ a 3a 4 3a 4 3
va1
11
3·- 2·-·~a3~
a 4a 3 4
Omeem:O.
IIpHMep 8.
YnpocTHTe Bbipa:IKeHHe
~
4x
W
~ - 'VX
3
3x -2 x
Peweuue.
I1oJiara.H
¥X =y
2
_2 .
3
X
'if3 =a , 3armrneM .n:aHHoe Bbipa:IKe-
H
HHe B BH.l(e:
a2y5 _ 4y3
ay 2 - 2y
-->"-.,--~-- 2y2
y2(ay-2)(ay+2) -2y2
ay-2
'ifX =y
. ~-4x
31
2
"\13X
y(ay- 2)
2y2
=
=y2 (ay+2)-2y2 =y2(ay+2-2) =ay3.
Y'1HThiBM no.n:cTaHOBKY
npeo6pa3oBaHH5I.
= y3(a2y2 _ 4)
-2'lfX
-
H
¥3 =a ,
3anHrneM pe3yJihTaT
2 =v3T<>
-:z
3.JX .
X
3
Omeem: ¥3x.
29
llpuMep 9. Yrrpocnue Bbrpa)Kemre
J.. .!. 2
m+n
am -an +22am-n
0
(aLa;)~+~) (a:•: J.
Peweuue. Ha ocHoBamm cBoikTB cTerreHeii 2.1, 2.3 H 2.11 BbiiiOJIHHM cJieJJ:yrorn:ne npeo6pa3oBaHH5l:
(m+n)
1) a(m-n)
JO
=1;
[
E:!!!!.
..!!!...+..!!_
.!.+_!_
.!. ..!..
2) amn =amn mn=an m=an.am;
m+1 n+1 m+1
n+1
3) ~am+ 1 +?</an+1 =a---;;;- +a--;;- =r;;;; +a;;-;;=
1+.1..
1+.!.
..!..
.!.
( ..!..
.!. )
=a m +a n =a·am +a·an =a am +an .
.!. 2
..!..
lloJiarM
..!..
.!.
am= x H an= y, 3arrnrneM A=
..!.. .!.
(x- )2 +4x
y
y
a ( x 2 - y 2 ) ( x + y)
.
I1pHMeH5l5l cpopMyJibl COKpall(eHHOfO yMHO)KeHH5l, IIOJiyqnM:
2
2
2
A=x -2xy+y +4xy =
(x+y)
2
2
a(x -y )(x+y) a(x-y)(x+y)(x+y)
1
yql1TbiBM, qyo
1
am =X l1 an
=y' 3aiiHllieM:
1
llpnMep 10. YnpocTHTe Bbrpa)KeHHe
30
1
a(x- y) ·
1
A= ( rnr nr)
a ~a-~a
sthV54 tsms
+
-=:-?=.;,.;;,;:;....;....::::i~;,;;;;;=
3V4~ +3V9tff62 .
.
Peme1-1Ue.
Ha ocHoBaHHH CBOHCTB CTeneHeil BhiiTOJIHHM npeo6pa-
3oBaHH51 t.£IICJIHTeJI51 H 3HaMeHaTeJI51 i'1,p06H:
1)
sth~h-33 +1SV2' = 5~7 .2k ·3+15·2~ =5
A~
1
4
2k (21+15·4) =
1
= s\J2 3 ·34 = s.i12 ·3 = 15-ii2;
'1~
2
5
2
1
1
2) 3'\J"l.~'\1"2" +3~3 2 {/2.3 4 =3-i3.ii2+3·33.ii2.33 =
13
1
1
1
= 3.212 +9· i12 = 212 · (6+9) = 1S-ii2.
1
3anmueM:
?
15 · 2
=1.
15-212
Omeem: 1.
IIpnMep 11. Haili'J,HTe .Ja --Ja+12, ecJIH .Ja +Ja+12 =130.
Pemenue. IloJiara51 .Ja- J a+ 12 =x , HaHi'J,eM rrpoH3Bei'J,eHH51 rrpaBhiX II JieBhiX t.£aCTeH paBeHCTB .Ja +.Ja+12 = 130 H .Ja -Ja+12 =X.
I10JIYt.£IIM: (.JQ +Ja+12)(.JQ --Ja+12) = 130x. 0TKyi'J,a:
a-(a+12)=130x, a-a-12=130x, x=- 6 .
65
6
.
Omeem: 65
llpUMCp 12. llpoBepbTe CnpaBei'J,JIIIBOCTb paBeHCTBa
_1-.Ji -310-7-Ji
.Ji+1- 10+7-Ji.
Pemenue.
BhiiTOJIHIIM rrpeo6pasoBaHH51 npaBOH qacTII paBeHCTBa,
YMHO%a51 t.£IICJIIITeJib II 3HaMeHaTeJib i'1,po6II Ha Bbipa%eHIIe conp51%eHHOe Bhipa%eHIIIO, 3aiTIICaHHOMY B 3HaMeHaTeJie i'1,p06II:
2
.,f10=7.J2 -3
(10-7.Ji)
-3100-140.Ji+98V10+1J2- (to+ 7.Ji)(1o-7.Ji)1oo-9s
=3 2(99~70-Ji) =~99-70-Ji.
AHaJioriit.£HhiM o6pa30M BhiiiOJIHIIM npeo6pasoBaHII51 JieBoH: qacTII
2
(.J2 -1 )
- 2- 2.J2 + 1 - 3- 2-Ji
.J2+ 1 - (.J2+ 1)(.Ji- 1)- 2 _ 1 -
. .J2 -1 paBeHCTBa.
31
ITocKOJihKY rrpaBa5I qacTh paBeHCTBa rrpeJJ:cTaBJieHa KOpHeM TpeTheH: cTerrenH, 3aiiHilleM:
3-2../i =~(3~2v'2) = <!27 -54../i + 72-16../i =<199-70../i.
3
Omeem: PaBeHCTBO cnpaBeJJ:JIHBO.
IlpHMep 13.
0CB060JJ:HTeCh OT HppaQHOHaJihHOCTH B 3HaMeHaTeJie
3 +J2 +J6.
3-J2-J6
Pew-e11ue. 3arri1IlleM 3HaMenaTeJih JJ:po6H B BHJJ:e 3-( .J2 +J6).
JJ:p06H
3+.J2+J6
In
,-;:;.
HOJKI1B qlfCJIHTeJih l1 3HaMeHaTeJih JJ:PO 611
3+( .J2 +J6),
corrpH:>KeHHOe Bhipa:>Kemno
Ha Bhipa:>KeHHe
3-v2-v6
3-( .J2 +J6),
H rrpuMeH5I5I
cpopMyJJhi coKpaiiJeHHoro yMHOJKeHHH, IIOJJyquM:
3+.J2 +J6 (3+.J2 +J6)2
=
3-( J2 +J6)- (3-( .J2 +J6))·(3+( .J2 +J6))
=
9+2+6+6v'2+6J6 +4.J3 17 +6.J2 +6J6 +4.J3
9-(.J2+J6/
=
9-2-4.J3-6
=
(17 +6v'i +6J6 +4.J3)( 1+4.J3)
=
(1-4.J3)(1+4.J3)
=
+68.J3 +24./6 + 72./i +48
= 17 +6v'i +6J6 +4.J3 1-48
=
=
65+ 78.J2 + 72.J3 +30J6
47
65+ 78.J2 + 72.J3 +30J6
47
IlpHMep 14. HaiiJJ:HTe qucno, 125% KOToporo paBHhi qucny.
Omeem:
(~+~):~(5·~)2.
Pewe11ue.
HaiiJJ:eM 3HaqenHe Bhipa:>KeHH5I
(~+~):~(5·~)
1
.
llonyqHM.
32
1
_(33·5)3+(23·5)3
A-
1
11
- -·54.534
2
=A.
_3·5~+2·5~ _5·5~-
-
1
1
-+5412
YM-
--1--5.
53-
3Ha5I, '-ITO 125
% IICKOMOrO
l.JHCJia paBHbl 5, 3aTIHIIICM H peiiiHM
nponop1IIIIO: x-100%.0n<yl(a 125·x=100·5 x=4.
5-125%
'
Omaem: 4.
llpHMep
15.
YnpocTHTe BbrpaJKeHHe
~.J2 + 1: ~(7 -5.J2)- 1 .
PeweHUe. IJpel(CTaBHM KOpHH BTOpOH I1 TpeTbCH CTCTICHH B BIIJ(e
KOpHeH IIICCTOH CTCTICHH I1 npHMCHHM <lJopMyJihl COKpall(eHHOrO
YMHOJKCHIUL Ho TIOCKOJJbKY 7- 5.J2 =
..J4§- -J50 < 0 '
TO 3aTIHIIICM
7- 5.J2 = - ( 5.J2- 7) H TIOJIY'-IHM:
_24(.J2+1)3
=
.3~(5.J2-7)2 =
-~( 2.J2 + 6 + 3.J2 + t) (5.J2- 7/
=
-((7 + 5_.J2_)_(5-.Ji---7~) =
2
=-~( 5.J2 + 7)( 5.J2- 7)( 5.J2- 7) = -~(50-49)(5.J2- 7) =
=-~5.J2-7 =-~2.J2-6+3.J2-1 =-~(.Ji-1) 3 =-~.Ji-1·
Omaem:
-~ .J2 -1 .
r
fipHMep 16. I-!a(Il(HTC 3Hal.JCHHC BbipaJKCHHH
(1(-J2-ir -~(t-.J2)'
PeweHue. YnpocnrM BbipaJKCHI1e
flocKOJibKY
.J2- ~ == .J2-
TO, 3HaH, '-ITO
~( .J2 -~
.Ji =.J2- g
2
(a-b) n =(b-aln,
,n;eM HMCTb
<0 ,
3aTIHIIICM (
Tor,n;a corJiaCifO CBOikTBY 2.14 TIOJIY'-IHM:
YrrpocTHM BhrpaJKeHHe
r.
~(1-.J2) 5
.J2 -~r =(~-.J2r
~(~-J2r =~-.J2.
. CorJiacno CBOHCTBY 2.15
6y-
~(1-.J2) 5 =1-.J'i.
33
Hail:,n:eM 3HaqeHne ncxo,n:Horo Bbipa:>KeHn.a:
4
)-4 = (21 )-
4
3
)- = ( 12-1
1
3
)- = ( 2-~-1+~
(2-~-(1-~)
4
=16.
Omeem: 16.
3aAaYII1 AJUI caMocTosnenbHOro peweHIIIH
YnpocTHTe Bbipa:>KeHH.a H Haii,n:nTe nx 3HaqeHH.a B cJiyqae, ecJIHH3BeCTHbi qifCJIOBbie 3HaqeHH.H napaMeTpOB ( 1-70):
1. (1+x+x2 1-x+x2 +2J-1: (5-2x2)-1;
2x+x 2
2x-x 2
3a 2+2ax-x 2 10ax-30x2
2
9x 2 -a 2
· (3x+a)(a+x)
°-
1
-hh(
2
2
2
2
.JX =~3,92~.
·
J·
2a-2b-2c
1_a -b -c
•
-1 +-12bc
·
22bc
'
0
b+c
a=50- 1; c=1,07; b=-11,05.
3
b-1
' (abc+a+c)
1
2
1 .
1.
a+-- 2a+2bb+c-1
2
11x-2
-2x 2 -x-2 J-1
; x=7,(3).
5. (x +2x+ _ x_ )· x+1+
x+
3 1 (
3 1
4
6.
1+(a+xr1 [ 1-(a2+x2))·
1. 1
2
'
1-(a+xr
ax
-3
X
1.
=(a-1)3 .
2
(ab- 1 +a-1b+1): (a- 1 -b- 1)7
' a2b-2 +a-2b2-(ab-1 +a- 1b)
1
x 2 (-2x-1)
S. 16+4x-2 +x-4 . 4-4x- 1 +x-2 + 0,25-0,Sx ·
x-6 -64
9. 2a 4 +a 3 +4a 2 +a+2
2a 3 -a 2 +a-2
34
35
25
2p-G
- 2-
12
--2
3
--2
. x2p +6p : x9-p . x3p-p .
26 •
n
2nm
m(m-n~
m
2 'J
ym-n : y
m -n1
2
1!22
-4x q
27.
1 1]2 +4x E.~1
xq -xP
(
q
1
29 •
1
___.,._(_1--.1-1---::]2-----:(-m+-n),...---3 xm +3xn
30.
-36x (mn)
(3
_I~ +·.E_+ 2 - 2Ja -a ifi
3 1. ~a ~
affa-aW ·
2
1 1
-a
32 (a+2
· -ffa + ffa +2
33.
36
3U) , x=64.
5(y2 -x2)(~ +Vi/)
3/5 3r-2f 3i"T2 3rs-5 </XY+'\JY
-'I X~ - '\J X y HJ X y + '\J y
)i2-a2
a-:g;; a+2 ·
2
(Ja __
1 J2 :(Ja-1+ Ja +1 J-1
2 2Ja
Ja+1 1-Ja
34.
b
35 . ..fi (a-x)
a- 2x
3
2
a= 52' x
((
--JX )2
J2X +.fa
-(ili +.fa_J-1]~;
-2Ta
2
=52.
36. ( ~36mn 2p +.J3mn +.J3np )( ~36mn
2
p -.J3mn -.j2np).
J3 (a- b2 ) + 2J3b 2
37
.
J2a _J2C
~2(b 2 -a) 2 +(-2bffa) 2 . ...[3.[;;1 -J3J?.
_v;zs. (H + 2+"¥ri
'fn2 J-(vn + 2lfii ) 22 _v;;z
¥n -2 ~ +2"¥ri.
38 23
. 2+lfii .
39
.Q
40.
m'lfm-33n'lfm
·(1-3~)-~
+3~ +32RJnZ.
nm
m .
(.JX+2)(~-1 )-(.JX-2)(~+1 )-~
(2-.Jx+2){
41.
:[~ +{(211
1
1-.J'ii
1-1(2t)
3
1+1(2tr
.J2t
4rn::
1-"\/.a
2
~~+1- 2 ~)
..fit].
2
x3 + 2R/XY + 22y3
42
· (x¥X- 23 y<!X): R/XY
43.
(x-R +2-2fx)
x-2+x-
(2-~xy- 1 )
2
(1 +fx)
1
44. (2.Jab- 2ab( a+.Jab)45, ( lJs- x'+3
1
)· ( 2(
x:~~:':x~
2
·
r--1 ,_--.,---3 1
1
'\1 x -v4x- +4+x.
.Jab-b)~( ab )-2 )-
3
} ({ ,J2-
1
.
1
v'XT +( .JX +,nt)
37
(~a2+a~ -~a2-a~t
58.
59.
a.J(Jj
:
2
(Jb-Ja)2
.JOjj
, a >b>0.
211+0,25( J;;:t -Jat
I
.
\}1 +0,25( J;;:t -Jat -o,5( /;;:t -Ja)
x+__1£_
60.
JX+4 + -4-+.Jx+4.
2-.Jx+4
.Jx+4
2
6 1. (1+.JX +
.J1+x
62.
2a
4 .3
1
2
(2x + 2x..J x 2 -1
4
t
i2
x(x+~) -x
64.
~(x-2)2 .
.JX -2:.JX
65.
2~4x(11+4.J6) -~4ffx -2.J3x -~50x 2.
~-x-~·~1-x~
6
~(x2-1)3
6 .
.
(~( x2 +4 )..J1 +4x-2 -~(x 2 -4 )..J1-4x-2
67.
68 .
x2-
..J 4
x -16
~(abc+4)a- 1 +4~;
...raiJC +2
3
69
.
22 +2 3 a- 1 +2 2 a-2
(a+1)3
63.
2
.Jf+x ) -(1-.JX + .J1+x )
-1-.JX
.Jf+x .JX-1
a=
f
5_2 •
3
. (2a+1)2 +(2a-1)2.
J4a+2~(2a) 2 -1
70. 4a2-b2. Ja4-(2ab)2+(2b2)2 .a4+2a2b2+4b4;
a6 -8b 6 \j
4a 2 +4ab+b 2
3 b
1
a=4, =4·
39
BbiiiOJIHIIB J'Ka3aHHyro ITOil,CTaHOBKJ', yrrpocTIITe Bbip<PKeHim
76. 0,(3)x 3 -x+0,3;
x=<f.J3 -.J2 +<f.J3 +.J2.
BhP-IHCJIIITe 3Ha'-!eHIIJI BbipaJKeHIIH
77.
(77 -85):
30</4m92 +42</18~
3/
~96m+48rns
30
4
6
+
,-;;):(2.J3+10).
78. ( ,-;; + ,-;:;
v3 -1 v3 -2 3-v3
79. ~ +~64W:S -<fsM/2.
80.
5~48~ +J32</2,25 -11<!12-/8.
81. .J160.JT2 +.Ji.5J48 --t/1200 -.J240J27.
82.
<!7 +JSO +<!7 -.JSO.
83. J3+.J3 -<f-1o-Jf08.
84. </26+$75 ·(2+.J3t.
85.
40
,J5-.J24 (5+2.J6)(49-10.J24)
J2j -Jf62 +.JfOS -JS
.
(71-76):
0CB060,ll,HTeCb OT HppaqHOHaJibHOCTH B 3HaMeHaTeJie ,11,p06H
86.
4fn
5
(86-88):
4r::> .
'\/8-'\/3
1-x
87. 3r r·
-vx -vx
88
23
. -J2 +.J3 -.J6.
89.
Hali,ll,HTe
.J'"'""(3---a..,..)-,-(2_+_a_), ecJIH -J3-a +-J2+a =15.
90.
Hali,ll,HTe
Jtoo-x 2 +J5b-x 2
2
, ecJIH H3BecTHo, qTO
2
Jtoo-x -J5o-x =10.
Om6embt: 1. 0,04. 2. 1. 3. 10. 4. -1. 5. 20. 6.
a
3
1
(a-t). 7. ab ·
2
a 2 +1
+(ij
2
8. (2x + 1) 2. 9. - . 10. 0,2. 11. 0. 12. x+1. 13. !tfX
JC . 14. 16a .
4r
a- 1
'\IX -~y
15.
22.
1
12
f""?."""
va 2b
;b.
'
1
.16. x -1.17. x- 7 .18. a +b .19.1.20. -b .21.
a
2(
.JP +J(i)
(p-q)
1
(Jm-Jn(
23.
1
24. y
10
.
25. xP-
2
1
3
.
4
1
26. y. 27. xP-x(j.
1
xm+3xn
1
1-a
. 30. 80. 31. 1. 32. ,./a +.J'i.. 33. ,./a .
3
34. -4. 35. 1. 36. -3n ( m + p) . 37. -JaG . 38. 2. 39. 0. 40. 2. 41. 1.
28.
qr-
2'\/X.
29.
a2b
42. -1. 43. x(x+1)(x+2). 44. -b. 45.
a47. m 2 . 48. -1. 49.
53. Vx+y -Vx-y. 54.
.
ecJIH aE ( -1,
.
(%r+b.
(
~
3
'¥8-x
-J2 . 46.
2
(a-b) 2 .
50. -1. 51. q(p+q). 52. 1-x 2 .
2
4 a-x
) .
55. 1-a, ecJIH
.
aE
(-oo; -1); a-1,
O)u(O, 1)u(1, +oo). 56. -1. 57. -25. 58.
a+ 1
1 ( 16xJX
59. - . 60 . -4. 6.
2)
a
1- x ( x -1)
.
(Ja +,Jlj)2
a-
b
.
2~
1
62. . 63. !?-:"
a
" x 2 -1
41
-JX, ecJIH
64.
ecJIH
XE (0; 2);
.JX, eCJIH
xe(-oo; -1)u(1; +oo);
XE
firr\
;;;L.,eCJIH
20x. 66. -lfi,
2¥X
xe(-1;1).67. -.68.5.
(2; +oo). 65.
X
69. 4a-..f4a 2-1. 70. ~. 71. 0. 72. 0,2. 73. -0,5.J6. 74. a+b. 75. 1.
76. 20~+ 9 . 77.31. 78.o,5. 79.
4
m .8o. 2m .81.0.82.2.83 . ..13+1.
1
4
84. 1. 85. 1. 86. (-tls+ti3)(2.J2+.J3). 87.
(J.X+¥X)(x+\lx2+¥X)
X
88. 7.J2+5.J3+.J6+12.89.110.90.5.91.1.
KoHTpOilbHbllil TeeT
N!!
BapnaHTbi OTBeToo
3a,ll;aHHH
1)-3a2b4.J:ilj;
2) 2a 2b4 .J-2b ;
Pe3yJihTaT BbiHeceHH.H MHOJKHTeJia H3-IIO,ll;
1
3HaKa KOpHa
4 9
.J-8a b
paBeH
3) 3ab 4 J3ab;
4) 3a 2b4 .fb;
5) -a 2b4 .J27b .
EcJIH
2
HH.H
a= 68, b =4,
r
TO 3HaqeHHe BhipaJKe-
(, 'J[ , ,
a4 +b4 · ba -4 +b4a-l
paBHO
Pe3yJihTaT yrrpom.eHH.H BbipaJKeHH.H
3
[(~+M) (~+~ax')-1
lfa-VX
paBeH
42
~)'
1) 17;
2) 72;
3) 0,75;
4) 64;
5) 1.
1) a2;
X
a2
3) 4 ;
2)
r;
4) a 2x 4 ;
X
5)
(~
R[;J_\Q".
.
JlapHaHTbl OTBeTOB
2) 1;
4
5
.J8- X - .J3- X =8 , TO 3Ha'!eHHe
Bbrpa:>KeHH5I .Js~x +.J3-x paBHo
EcJIH
Ecmr
Fa+ Jb =S, ab =4 , TO 3Ha'Iemre
Bbrpa:>Kemm
a.Ja +bJb
4)
1) 20;
2) 263;
3) 9S;
4) 400;
5) 40.
paBHO
1) 10;
x + y =4 , y + z =8 , x + z =6 ,
3) 40;
6
TO 3Ha'!eHIIe Bbipa:>KeHH5I X- y + 2z paBHO
5) 2.
Ecmr
7
Hmr6oJibiiiee 3Ha'!eHHe Bbrpa:>KeHH5I
-4b· (5a +b) -(Sa- 2)( Sa+ 2)
5.
4'
paBHO
1) 4;
3) 40;
2) 18;
4) 8;
2) 14;
4) 20;
5) 5.
1) x-2;
8
OcTaToK OT .n:eJieHH5I MHoroqJieHOB
x 3 -5x 2 +3x-1 H x+2 paBeH
Pe3yJihTaT Bhi'IHCJieHH5I Bhrpa:>KeHH5I
9
3
.
(0,25f2 +3·0,0081
_.!4
0 75
+(0,0625f' · paBeH
B pe3yJihTaTe
npeo6pa3oBaHH5I Bbrpa:>KeHH5I
JO 5Rl6J32 -Rl243Jf62 -11VfB+Rl600.J50
2) x 2 +1;
3) 0;
4) 3S;
5) -35.
1) 25;
2) 26;
3) -1;
4) 260;
5) 3..
1) 0;
3) 1;
5) lf5.
2) R./6;
4)Vf8;
1) 4;
3) 2;
5) J6.
2) 14;
4) f/6;
TIOJIY'IHM
Pe3yJihTaT Bhi'IHCJieHH5I Bhrpa:>KeHH5I
11
4../S-.JU
43
N2
12
13
3a,~~;aHH.H
BapnaHTbiOTBCTOB
1) 5-2J5;
2) 0;
3) 1;
Pe3yJihTaT ynpo:r.u:emm: Bbipa)KeHIHI
(V9+J80 +:if2+.J5)·:V2-J5
EcJIII
paBeH
4) 4-2J5;
5) -2.
ab- 1 = 2-2 , TO 3Ha'-!eHIIe Bbipa)KeHII5I
..fa+2b -../b-3a
../4a+3b +..f13a-b
1)
2
7;
2)
3.
3) 4'
5) 12.
paBHO
7.
2'
4) 4;
1) 2a-3.
2~ '
Pe3yJihTaT ynpon~eHIUI Bbrpa)KeHII5I
1
1
T
-1
14 )4
-+---2
+ ~--+-+2
1
1
5
2) ~;
2
a
4a-
yc.JIOBIIII, 'ITO
4aa
0 <a< 4 , paBeH
npii
5 .
3) 2~'
4) 2~;
5) 5~.
OmBem'bt
HoMep 3a,namui
1
2
3
4
5
6
7
HoMep
npaBHJILHoro
OTBeTa
2
1
3
3
3
4
1
HoMep 3a)l:aHH5!
8
9
10
11
12
13
14
HoMep
npaBHJILHoro
OTBeTa
5
2
1
1
5
1
3
44