1. P[x]. ˳ P[x].
2. .
3. .
1. P[x].
1: d(x) f(x) g(x), f(x) g(x).
2: f(x) g(x), , f(x) g(x) (f,g).
! d(x) (f(x) g(x), d(x), , , .
! .
3: f(x),g(x) P[x] -, ( )
1: - f(x) g(x) P[x] d(x), d(x)
d(x)=f(x)u(x)+g(x)v(x),
u(x) v(x) P[x].
.
g(x) i f(x), f(x) g(x). , :
f(x)=g(x) (x)+ (x),
g(x)= (x) (x)+ (x),
(x)= (x) (x)+ (x), (1)
(x)= (x) (x)+ (x),
(x)= (x).
ϳ (x)=0.
deg (x)<deg g(x), deg (x)< deg (x) .. deg (x)<deg (x). , (x) , , 1 0.
deg (x)=0, =0, - .
(x) m-1, m g(x), (1) m.
1. f(x)= -3 +3x-1, g(x)=-1. .
0 (f,g)=x-1
˳
d(x)= (x), (1)
d= - . (2)
-
- (3)
...
- . (4)
ϳ (2) (3) :
d= - ( - ( - ) )
d= + (1+ )
,, d i , . , i .
d=fu+gv (5)
2: 1
d= =x-1
(x)= (x)+f(x)= (x)
g(x)= (x)
4. f(x), g(x) P[x] - S(x) P[x] , S(x): f(x) S(x): g(x). () f(x) g(x) f(x) g(x), - . f(x) g(x) [f,g].
3. - f(x) g(x)
|
|
r2(x) = g(x) r1(x) S2(x)
r2(x) = g(x) (f(x) g(x) S1(x)) S2(x)
r2(x) = g(x) f(x) S2(x) + g (S1(x) S2(x)) = g(x) (1 + S1(x) S2(x)) f(x) S2(x)
r2(x) = (x3 1) (1 + (- x - ) (x3 3x2 + 3x + 1) (- x - )
V(x) = ( - x), U(x) = ( x - )
= -4()
2. .
1. f(x) P[x] , , c f(x), .
2. f(x) P[x] ,
deg f ≥1 g(x), S(x) P[x], f(x) = g(x) S(x) P[x], deg g ≥1, deg S ≥1.
, .
3. (2 + 1) . ³ .
4. (2 - 2) . ³ + b (a, b Q): (2 - 2) = ( - ) ( + )
2. .
3.
3. f(x) :
f(x) = p1(x) p2(x) pl(x), (6)
pk(x) . (6) pk(x).
. :
f(x) = [p1(x)]k1 [p2(x)]k2 [pm(x)]km (7)
(7) f(x) .
3. pi(x) (7) kj, , pj(x) kj f(x). .
5. f(x) = 3 + 2 5 + 3 :
f(x) = 3 + 2 5 + 3 = ( 1) ( 1) ( +3) = ( 1)2( + 3)
p1(x) = x 1 2.
P2(x) = x + 3 1.
- , 1- .
6. f(x) = 4 2 + 4 = (2 2) .
f(x) = ( - ) ( + ) . 2.
4. f(x) g(x) , (f,g) , f(x) g(x). , (f,g) = 1.
7. f(x) = ( 1)2( + 3), g(x) = ( 1)3. (f,g) = ( 1)2, (f,g) = 2 2 +1
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3
1. . .
2. .
3. . ³.
4. . .