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1: f()= 2-3+1, g()=2+1. г 3-3+1=(2+1)+(-4+1) , f() ÷ g() . S()=, r() =-4+1. g( ) f() : 2+1=0 *( 3-3+1)+2+1. S1()=0 , r1()=2+1 .
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1: f() [] - g() , 0 , [] .
: [] S() r() - f()= g() [] g()≠0. f()= nn+an-1 xn-1++a1x+a0, g() = bmxm + bm-1xm-1++b1x+b0.
f()=0, S()=0, r()=0. n=deg f()<deg g(x)=m S(x)=0, r(x)=f(x). n≥m.
n, n=0. n=0, m=0, f(x)=a0, g(x)=b0≠0 S(x)= a0/ b0, r(x)=0. S(x) P(x), a0/b0 .
, , deg f <n n.
()=f(x)=an/bm xn-m g(x) an≠0, bm≠0.
n/bm xn-m g(x)=anxn f(x). deg P<n () g(x);
P(x)= g(x)S1(x) + r1(x), S1(x), r1(x) P[x], r1(x)=0 deg r1<deg g.
f(x)- an/bm xn-m g(x)= g(x)S1(x) + r1(x),
f(x)=g(x) S(x)+r(x), r(x)=r1(x), S(x)=S1(x)+ an/bm xn-m, S(x) r(x) P[x] r(x)=0, deg r= deg r1<deg g.
f(x) g(x) . S(x) r(x). , :
f(x)= g(x)S(x)+ r(x), deg r<deg g;
f(x)= g(x)S^(x)+r^(x), deg r^< deg g;
³ , :
g(x) [S(x)- S^(x)]=r(x)- r^(x). (2)
g(x)≠0
, r(x) ≠ r^(x), S(x) ≠ S^(x) ( [x] ). . (2) deg g (1). (2) r(x)= r^ (x) S(x)= S^(x), , . .
4. ij . . . . , , 1.
1: f(x)=x4-2x3+x-1, a g(x)=x2-2
: g1(x)=f(x)- an/bm xn-m g(x):
g1(x)=x4-2x3+x-1-x2 (x2-2)=-2x3+2x2+x+1 g1(x) f(x);
g2(x)=-2x3+2x2+x-1+2x(x2-2)=2x2-3x-1;
g3(x)=2x2-3x-1-2(x2-2)=-3x+3
deg g3(x)<deg g(x) S(x)=x2-2x+2, r(x)= -3x+3.
X4-2x3+x-1=(x2-2)(x2-2x+2)+ (-3x+3)
:
X4-2x3+x-1 x2-2
X4-2x3+x-1=(x2-2)S(x)+r(x) (3)
c S(x)≤n-m=2 r(x)<m, deg r(x)=1.
S(x)=A2x2+A1x+A0 r(x)=B1x+B0
ϳ (3) :
X4-2x3+x-1=(x2-2)(A2x2+A1x+A0)+(B1x+B0) . :
A2=1, A1=-2, -2A2+A0=0, -2A1+B1= 1, -2A0+B0=-1,
A2=1, A1=-2, A0=2, B1=-3, B0=3
S(x)=x2-2x+2, r(x)=-3x+3
S(x) n-m, a r(x)- m-1
g(x)= -2 .
anxn+an-1xn-1++a1x+a0=(x- 2)(An-1xn-1+An-2xn-2++A1x+A0)+r. r-const. (4)
(4) :
n=An-1,
an-1=An-2- α An-1
..
a1=A0- α A1,
a0= r- α A0,
An-1= an
An-2= an-1+ α An-1
An-3=an-2+ α A
..
A0= a1+ α A1
r= a0+ α A0
(5) , - α - :
n | an-1 | an-2 | an-3 | a1 | a0 | ||
α | an An-1 | αAn-1+an-1 An-2 | αAn-2+an-2 An-3 | αAn-3+an-3 An-4 | αA1+a1 A0 | αA0+a0 r |
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, -1 r α .
2: 4-32+2-1 -2 4=1, 3=0, 2=-3, 1=2, 0= -1, α=2.
-3 | -1 | ||||
2*2-3=1 | 2*1+2=4 | 2*4-1=7 |
S(x)= x3+2x2+x+4, r=7
-3 | -1 | ||||
: f(x) - α = α f(x). .
. - α f(x) P[x] - α= f(x).
: (1) :
f(x)=(- α)S(x)+r, (6)
r , , - α. , (6) , - . = α, f(α)=r, .
. f(x) - α.
f(x)- n- , α- . ij - α : f(x)=(- α) f1()+0, (7)
f1()- (n-1)- [x], a C0- . n˃1 :
f1(x)=(x- α) f2(x) + C1 (8)
f2(x)=(x- α)f3(x)+C2
fn-1(x)= (x- α)fn(x)+Cn-1
fn(x) fn(x)=Cn
(7) (8) fn-1(x),fn-2(x),,f2(x), f1(x) :
f(x)=Cn(x- α)n+Cn-1(x-α)n-1++C1(x-α)+C0 (9)
, f(x) , =- α. 0, 1, n α 0, 1, ,n f(). , 0 f() f1(x) - α .. Cn .
4: f(x)=5-33+2-2+1 -1.
-3 | -2 | |||||
-2 | -1 | -3 | -2 | |||
-1 | -4 | |||||
5=1, 4=5, 3=7, 2=2, 1=-4, 0=-2
f()= (-1)5+5(-1)4+7(-1)3+2(-1)2-4(-1)-2.
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