1. f(x) , f(x) ( f(x)=0).
1. f (x)=x5+2x3-3x.
1 f(x), 2 f(x), f(1)=15+2∙13-3∙1=0, f(2)=25+2∙23-3∙2=42≠0.
, .
f(x), f() -.
2. - f(), f() -. k, f() (-)k, (-)k+1. k f(), - k- . k=1, .
k f() :
k- f(), k>1 (k-1)- ; k=1, f '().
. k- f() k- .
2. , 2 f()=4-43+16-16. .
. 2 f(), 24-4∙23+16∙2-16=0.
f '(x)=4x3-12x2+16, f '(2)=4∙23-12∙22+16=0;
f ''(x)=12x2-24x, f ''(2)=12∙22-24∙2=0;
f '''(x)=24x-24, f '''(2)=24∙2-24≠0.
2 f'''(), 2 f().
f() n≥1 1: f()=n+a1xn-1++an-1x+an α1,...,αn . , :
a1= -(α1+...+αn),
a2=α1α2+...+αn-1αn,
a3= -(α1α2α3+...+αn-2αn-1αn),
...........................
an=(-1)nα1α2...αn.
.
3. , 2; 3 1.
. :
1= (2+311)=-3,
2=23+2(1)+2(1)+3(1)+3(1)+(1)(1)= 3,
3= (2(1)(1)+3(1)(1)+32(1)+32(1))= 7,
4=32(1)(1)=6.
433327+6.
3. f()Ì[x] n , φ() ψ() [x], n:
f(x)=φ(x)ψ(x). (1)
f(x)ÎP[x] , [x] 0, n.
:
f() [x] [x] .
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, f()Î[x] n, n≥1, :
, (2)
- P[x] , . .
, , . k (2), k- f(). () , f().
(2) , :
, (3)
1(),,r(x) . k1,,kr . (3) :
, (4)
F1(x) , - .. (3). (3) m- , .
F1(x),,Fs(x) f(x) , , ( ) :
, f(x) .
f(x) F1(x),,Fs(x) (4), , .
4. f(x)=5-4-53+2+8+4.
. f(x) f '(x)=5x4-4x3-15x2+2x+8.
d1(x)=[ f(x), f '(x)]=x3-3x-2.
d2(x)=(d1(x), d1'(x)).
d2(x)=x+1.
d3(x)=(d2(x), d2'(x)), d2'(x)=1,
d3(x)=1.
v1(x), v2(x), v3(x).
( ).
v1(x)=x2-x-2.
( ).
F3(x)=v3(x)=x+1,
, f(x) f(x)=(-2)2(+1)3. (3) f(x) , -2 +1.
1. , f(x) ( f(x)=F1(x)).
2. .
1
1. , 34-53+32+4-2 1+i. .
2. 5+54-53-452+108.
3. , : 5, i, i+3.
2
1. 0=2 f(x)=x5-74+123+162-64+48? .
2. 5-64+163-242+20-8.
3. x3+px+q=0, 1, 2, 3, .
3
1. 0=4 4-73+92+8+16? .
2. 6-25-4-23+52+4+4.
3. λ , : x3-7x+λ=0.
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4
1. , =3 f(x)=4-63+102-6+9. .
2. 5+64+133+142+12+8.
3. 23-2-7+λ=0 1. λ.
5
1. , 0=-2 4+3-182-52-40. .
2. f(x)=5-54-53+452-108.
3. 1, 2, 3, 1+i.
6
1. , 5+4+b , .
2. 6+154-83+512-72+27.
3. 0n+a1xn-1++an 1, 2,, n. : 1) a0xn -a1xn-1+a2xn-2++(-1)nan;
2) anxn+an-1xn-1++a0?
7
1. , =-2 45+244+473+262-12-8. .
2. 4+3-32-5-2.
3. 23-22-4-1.
8
1. , =1 6-5-44+63+2-5+2. . .
2. 5-34+43-42+3-1.
3. . f()=3-72+14+λ.
9
1. , 5+103+5b+ , .
2. 7-36+55-74+73-52+3-1.
3. 3-62+q+2=0, , .
10
1. , =3 f(x)=4-123+532-102+72. , .
2. 6-44-162+16.
3. 1, 2+i, 3.
11
1. , =2 5-64+133-142+12-8. .
2. 4+3-32-5-2.
3. , 1=2, x2=1-i, x3=3.
12
1. , =-1 4+3-32-5-2. .
2. 5-34+43-42+3-1.
3. , 1=i, x2=2+i, x3=x4=2.
13
1. 0=4 4-73+92+8+16? .
2. 6-25-4-23+52+4+4.
3. λ , 3-7+λ=0 .