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c , n 1,2,3,, n k.
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, (1) 䳿 α, . (1) α. f(α) f() =α.
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4
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1. . . .
. f(x)=nanxn-1+(n-1)an-1xn-2++2a2x +a1(*).
, - .
.
.
deg f=degf-1.
( ) , , :
[f(x)+g(x)]=f(x)+g(x) (1)
[f(x)*g(x)]=f(x)g(x)+f(x)g(x) (2)
, ,
[cf(x)]=cf(x) (c - ) (3)
, , .
(1). R[x]:
f(x)=anxn +an-1xn-1++a1x+a0,
g(x) = bmxm +bm-1xm-1++b1x+b0, (n≥m),
f(x)+g(x)= anxn +an-1xn-1++(am+bm)xm++(a1+b1)x+(a0+b0).
(*), , (1) :
nanxn-1+(n-1)an-1xn-2++m(am+bm)xm-1++(a1+b1)= [nanxn-1+(n-1)an-1xn-2++mamxm-1++a1] +[mbmxm-1++b1](4)
f(x), f(x), f(x),fk(x). fk(x) f(k+1)x, fk(x)=0, k>n, fn(x)=n!an(an f(x)).
2. .
, :
f(x)=[p1(x)]k1[p2(x)]k2[pi(x)]ki (5)
(5), , , , .
, , , (5).
(5) pi(x), ki 1, 1(x):
1(x)=pi1(x) pi2(x) pis(x)
pj(x) kj =2, , (5) 2.
2(x)=pj1(x) pj2(x) pjs(x)
2(x) , , [2(x)]2.
3(x) 3 ..
(5) :
f(x)= 1(x)[ 2(x)]2[ 3(x)]3[ m(x)]m, (6)
f= 1 22 33... mm.
k<m (6) k=1.
(6) , (5) . , f(x)= 1 (x).
1.
f(x)= x13-5x12+6x11+4x10-9x9+5x8-6x7-4x6+8x5=x5(x-2)3(x2+1)(x+1)2(x-1).
1=(x2+1)(x-1)=x3-x2+x-1, 2=x+1, 3=x-2, 4=1, 5=5.
f(x)= 1(x)*[ 2(x)]2*[ 3(x)]3*[ 4(x)]4*[ 5(x)]5.
f(x) k(x) , , 쳺 1-3 .
3