PolynomialGCD[f[x],g[x],h[x]]
-1+x
- , ,
. (2.1)
(2.1) .
, , - . , , .
3.3.
.
d[x_]=PolynomialGCD[f[x],g[x]]
1+x
g[x]- , u[x] ; f[x] , v[x] .
A=Table[a[n-1],{n,1,7}];
d1[x_]=f[x]*u[x]+g[x]*v[x];
T1=Table[Coefficient[d1[x],x,n-1],{n,1,7}];
T2=Table[Coefficient[d[x],x,n-1],{n,1,7}];
T=Table[T1[[n]]T2[[n]],{n,1,7}];
R=Solve[T,A]
a[6]=0;
u[x_]=u[x]/.R[[1]]
v[x_]=v[x]/.R[[1]]
Expand[f[x]*u[x]+g[x]*v[x]]
X
.
- , . . , , . ,
:
,
, Mathematica.
a0[x_]=f[x];a1[x_]=g[x];u0[x_]=1;v0[x_]=0;u1[x_]=0;v1[x_]=1;
q[x_]=PolynomialQuotient[a0[x],a1[x],x];
a2[x_]=a0[x]-a1[x]*q[x];u2[x_]=u0[x]-u1[x]*q[x];v2[x_]=v0[x]-v1[x]*q[x];
u1[x] v1[x] a1[x]
While[Length[CoefficientList[a2[x],x]]>0,a0[x_]=a1[x];
a1[x_]=a2[x];q[x_]=PolynomialQuotient[a0[x],a1[x],x];
u0[x_]=u1[x];u1[x_]=u2[x];v0[x_]=v1[x];v1[x_]=v2[x];
a2[x_]=Expand[a0[x]-a1[x]*q[x]];
u2[x_]=Expand[u0[x]-u1[x]*q[x]];
v2[x_]=Expand[v0[x]-v1[x]*q[x]]]
u1[x]
X
v1[x]
a1[x]
Simplify[f[x]*u1[x]+g[x]*v1[x]]
.
, , , , .
,
|
|
, (3.3.1)
,
, - .
, . , .
, , , .
, , .
. ; - , , - , .. , . , . , , , .
. (3.3.2)
(3.3.2) . .
. ,
.
.
.
Mathematica .
3.3.1.
.
f[x_]=x^8+2*x^7+5*x^6+6*x^5+
8*x^4+6*x^3+5*x^2+2*x+1;
g[x]
g[x_]=f[x];
g[x_]=PolynomialGCD[g[x],D[g[x],x]];
d=Append[d,g[x]];
While[Length[CoefficientList[g[x],x]]>1,g[x_]=
PolynomialGCD[g[x],D[g[x],x]];
d=Append[d,g[x]]];