x0x1x2x3
x16x17x19 x4x25x1ax1bx1cxax26x0x27x28x1cx29xb
x25 x26 10–1
1,x6fx70(x2 +ax?b)(x2?1)+(x2?ax+b)(x2 +1).
x29,2x4?2ax+2b.
2,x6fx70x49x7ex7fx3 +2x2 +3x?1x1c3x2 +2x+4x18x8x9.
x29,3x5 +8x4 +17x3 +11x2 +10x?4.
3,xd
f(x) = 3x2?5x+3;
g(x) = ax(x?1)+b(x+2)(x?1)+cx(x+2);
x3cx66x58a;b;c,x4ef(x) = g(x).
x29,x51x =?2,x4fa = 256 ;x51x = 0,x4fb =? 32,x51x = 1,x4fc = 13,
4,xdf(x),g(x)x75h(x)x32x21x2x73x79x49x7ex7f,x5ex5f,x34x35
f2(x) = xg2(x)+xh2(x);
x5bx23
f(x) = g(x) = h(x) = 0:
x27x28,x34f(x) 6= 0,xcx1ax7fx18x1ax79x2ex1bx79,x4ax1cx7fx18x1ax79x2ex1dx79,x1ex1f,x33f(x) = 0,x49x4a
g2(x)+h2(x) = 0:
x43,g(x);h(x)x20x2ex2x73x79x49x7ex7f,x49x4ag2(x);h2(x)x18x21x7ex73x79x32x21x7ax7ax79,x4ax6bx40x76x79x74x75x2ex7b,x33
g(x);h(x)x18x21x7ex73x79x32x21x7b,x49x4ag(x) = h(x) = 0.
x25 x26 10–2
1,x58g(x)x22f(x),x4cx23q(x)x1cx6dx7fr(x):
(1) f(x) = x4 +4x2?x+6,g(x) = x2 +x+1;
(2) f(x) = x3 +3x2?x?1,g(x) = 3x2?2x+1.
x29,(1) q(x) = x2?x+4,r(x) =?4x+2.
(2) q(x) = 19 (3x+11),r(x) = 109 (x?2).
2,m,p,qx24x25x22x23x46x47x53,x24
(1) x2 +mx+1 j x3 +px+q;
(2) x2 +mx+1 j x4 +px2 +q.
x29,(1) p = 1?m2,q =?m.
(2)
( m = 0
p = 1+q x6b
( p =?m2 +2
q = 1
￠ 1 ￠
3,x58x26x25x22x71x4cx23q(x)x2ex6dx7fr(x):
(1) f(x) = x4?2x3 +4x2?6x+8,g(x) = x?2;
(2) f(x) = 2x5?5x3?8x,g(x) = x+2.
x29,(1) q(x) = x3 +4x+2,r(x) = 12.
(2) q(x) = 2x4?4x3 +3x2?6x+4,r(x) =?8.
4,x58x26x25x22x71x72f(x)x2ex?x0x18x3ex27:
(1) f(x) = x4?2x3 +3x2?2x+1,x0 = 2;
(2) f(x) = x4?2x2 +3,x0 =?2;
(3) f(x) = x4 +2ix3?(1+i)x2?3x+1?2i,x0 =?i.
x29,(1) f(x) = (x?2)4 +6(x?2)3 +15(x?2)2 +18(x?2)+9.
(2) f(x) = (x+2)4?8(x+2)3 +22(x+2)2?24(x+2)+11.
(3) f(x) = (x+i)4?2i(x+i)3?(1+i)(x+i)2?5(x+i)+(1+2i).
5,x47hxi0 = 1,hxik = x(x?1)(x?2)￠￠￠(x?k +1),(k > 1),x3cx28f(x)x72x2e
c0 +c1hxi+c2hxi2 +￠￠￠
x18x70x7f:
(1) f(x) = x4?2x3 +x2?1;
(2) f(x) = x5.
x29,(1) 1 1?2 1 0?1
1?1 0
2 1?1 0 0
2 2
3 1 1 2
3
1 4
x1fx6cf(x) =?1+2hxi2 +4hxi3 +hxi4.
(2) f(x) = hxi+15hxi2 +25hxi3 +10hxi4 +hxi5.
6,kx21x57x9x79,x5ex5f,x j fk(x)x50x46x61x50x j f(x);
x27x28,xdf(x)x18x60x79x7ex2ea,xcfk(x)x18x60x79x7ex2eak,x1fx6cx j fk(x) () ak = 0 () a = 0 ()
x j f(x).
7,xda;bx2ex40x76x60x18x1ex18x60x79,x5ex5f,x49x7ex7ff(x)x29(x?a)(x?b)x22x10x4fx6dx7fx2e
f(a)?f(b)
a?b x+
af(b)?bf(a)
a?b,
x27x28,xdf(x) = (x?a)(x?b)q(x)+Ax+B,xc
f(a) = aA+B; f(b) = bA+B;
x1bx6cx4f
A = f(a)?f(b)a?b ; B = af(b)?bf(a)a?b,
x1fx6cx38x39x3ax3b.
8,xdf1(x),f2(x),g1(x),g2(x)x32x21x79x1Kx7x18x49x7ex7f,x2dx2af1(x) 6= 0.
x5ex5f,x34x35g1(x)g2(x) j f1(x)f2(x),f1(x) j g1(x),xcg2(x) j f2(x).
￠ 2 ￠
x27x28,xdf1(x)f2(x) = g1(x)g2(x)q1(x),g1(x) = f1(x)q2(x),xcf1(x)f2(x) = f1(x)q2(x)g2(x)q1(x),
x1bx15f1(x) 6= 0,x40x4ff2(x) = g2(x)q2(x)q1(x),x4bg2(x) j f2(x).
9,x5ex5f,xd?1 j xn?1x50x46x61x50d j n.
x27x28,())x11n = dq,xc
xn?1 = (xd?1)(xd(q?1) +xd(q?2) +￠￠￠+xd +1):
x1fx6cxd?1 j xn?1.
(()xdn = dq +r,0 6 r < d,x1bx7x5e,xdq?1 · 0 (mod xd?1),x4b
xdq · 1 (mod xd?1);
xn · xdq+r · xdq ￠xr · xr (mod xd?1);
xn?1 · xr?1 (mod xd?1):
x4axd?1 j xr?1,r = 0,x1fx6cxd?1 j xn?1,r = 0,d j n.
x25 x26 10–3
1,x4cx5ex7cx5x1fx7f(f(x);g(x)):
(1) f(x) = x4 +x3?3x2?4x?1,g(x) = x3 +x2?x?1;
(2) f(x) = x5 +x4?x3?2x?1,g(x) = 3x4 +2x3 +x2?2;
(3) f(x) = x4?x3?4x2 +4x+1,g(x) = x2?x?1.
x29,(1) x+1.
(2) 1.
(3) 1.
2,x4cu(x),v(x),x4eu(x)f(x)+v(x)g(x) = (f(x);g(x)):
(1) f(x) = x4 +2x3?x2?4x?2,g(x) = x4 +x3?x2?2x?2;
(2) f(x) = 4x4?2x3?16x2 +5x+9,g(x) = 2x3?x2?5x+4;
(3) f(x) = 2x4 +3x3?3x2?5x+2,g(x) = 2x3 +x2?x?1.
x29,(1) u(x) =?x?1,v(x) = x+2,d(x) = x2?2.
(2) u(x) =? 13 (x?1),v(x) = 13 (2x2?2x?3),d(x) = x?1.
(3) u(x) =? 16 (2x2 +3x),v(x) = 16 (2x3 +5x2?6),d(x) = 1.
3,x5ex5f,x34x35d(x) j f(x),d(x) j g(x),x46d(x)x2ef(x)x1cg(x)x18x66x76x5dx25,x5bx23d(x)x21f(x)x1cg(x)
x18x66x76x5ex7cx5x1fx7f.
x27x28,xdd(x) = u(x)f(x) + v(x)g(x),xcx14x34x35x18h(x) 2 K[x],x34h(x) j f(x),h(x) j g(x),xc
h(x) j d(x).
x43,d(x)x2ef(x)x1cg(x)x18x66x76x5x1fx7f,x33d(x)x21f(x)x1cg(x)x18x66x76x5ex7cx5x1fx7f.
4,x5ex5f,x34x35h(x)x2ex21x66x49x7ex7f,xc
(f(x)h(x);g(x)h(x)) = (f(x);g(x))h(x):
x27x28,xdd(x) = (f(x);g(x)) 6= 0,xcx44x71u(x);v(x)x4e
d(x) = u(x)f(x)+v(x)g(x):
￠ 3 ￠
x10x16
d(x)h(x) = u(x)f(x)h(x)+v(x)g(x)h(x):
x43x1fd(x)h(x) j f(x)h(x),d(x)h(x) j g(x)h(x),x10x16d(x)h(x)x21f(x)h(x)x1cg(x)h(x)x18x66x76x5ex7cx5x1fx7f.
x43x1fd(x);h(x)x32x21x21x66x49x7ex7f,x33d(x)h(x)x3bx21x21x66x49x7ex7f,x49x4a
(f(x)h(x);g(x)h(x)) = d(x)h(x) = (f(x);g(x))h(x):
x43x34d(x) = 0,xcf(x) = g(x) = 0,x72x1ex7fx2ax13x3ax3b.
5,x5ex5f,x34x35f(x),g(x)x60x79x2ex7b,xc
f(x)
(f(x);g(x));
g(x)
(f(x);g(x))
= 1:
x27x28,x1ff(x),g(x)x60x79x2ex7b,x33(f(x);g(x)) 6= 0,x10x16
(f(x);g(x)) =
f(x)
(f(x);g(x)) (f(x);g(x));
g(x)
(f(x);g(x)) (f(x);g(x))
=
f(x)
(f(x);g(x)) ;
g(x)
(f(x);g(x))
(f(x);g(x))
(x1bx4ex4f4)x40x52x2bx2c(f(x);g(x)),x4f
f(x)
(f(x);g(x)) ;
g(x)
(f(x);g(x))
= 1:
6,x5ex5f,x34x35f(x),g(x)x60x79x2ex7b,x46
u(x)f(x)+v(x)g(x) = (f(x);g(x));
xc(u(x);v(x)) = 1.
x27x28,x1ff(x),g(x)x60x79x2ex7b,x33(f(x);g(x)) 6= 0,x1fx6c
u(x) f(x)(f(x);g(x)) +v(x) g(x)(f(x);g(x)) = 1;
(u(x);v(x)) = 1:
7,x5ex5f,x34x35(f(x);g(x)) = 1,(f(x);h(x)) = 1,x5bx23
(f(x);g(x)h(x)) = 1:
x27x28,x44x71u(x);v(x);s(x);t(x),x4e
u(x)f(x)+v(x)g(x) = 1;
s(x)f(x)+t(x)h(x) = 1;
x10x16
f(x)(u(x)s(x)f(x)+u(x)t(x)h(x)+s(x)v(x)g(x))+v(x)t(x)g(x)h(x) = 1;
(f(x);g(x)h(x)) = 1:
8,xd f1(x);￠￠￠ ;fm(x),g1(x);￠￠￠ ;gn(x) x32x21x49x7ex7f,x46 (fi(x);gj(x)) = 1 (i = 1;￠￠￠ ;m;j =
1;￠￠￠ ;n),x5ex5f:
(f1(x)f2(x)￠￠￠fm(x);g1(x)g2(x)￠￠￠gn(x)) = 1:
x27x28,x1b(fi(x);gj(x)) = 1,x40x4f(fi(x);g1(x)g2(x)) = 1,:::,(fi(x);g1(x)g2(x)￠￠￠gn(x)) = 1,x49x4a
(f1(x)f2(x);g1(x)￠￠￠gn(x)) = 1,(f1(x)f2(x)f3(x);g1(x)￠￠￠gn(x)) = 1,:::,
(f1(x)f2(x)￠￠￠fm(x);g1(x)￠￠￠gn(x)) = 1.
9,x5ex5f,x34x35(f(x);g(x)) = 1,x5bx23(f(x)+g(x);f(x)g(x)) = 1.
￠ 4 ￠
x27x28,x1bx15(f(x);g(x)) = 1,x10x16
(f(x)+g(x);g(x)) = (f(x);g(x)) = 1;
(f(x)+g(x);f(x)) = (g(x);f(x)) = 1;
x1fx6c
(f(x)+g(x);f(x)g(x)) = 1:
10,xdf1(x) = af(x)+bg(x),g1(x) = cf(x)+dg(x),x46ad?bc 6= 0,x5ex5f:
(f(x);g(x)) = (f1(x);g1(x)):
x27x28,x1bx4fxdx40x4f(f(x);g(x)) j (f1(x);g1(x)),x43
f(x) = dad?bc f1(x)? bad?bc g1(x);
g(x) =?cad?bc f1(x)+ aad?bc g1(x);
x10x16
(f1(x);g1(x)) j (f(x);g(x)):
x43x1f(f1(x);g1(x))x1c(f(x);g(x))x18x21x7ex73x79x18x61,x33(f(x);g(x)) = (f1(x);g1(x)).
11,x5ex5f,x34x35f(x)x1cg(x)x1xb,x5bx23f(xm)x1cg(xm)x3bx1xb.
x27x28,x1bx4fxd,x44x71x49x7ex7fu(x);v(x)x4e
u(x)f(x)+v(x)g(x) = 1:
x10x16
u(xm)f(xm)+v(xm)g(xm) = 1:
x33(f(xm);g(xm)) = 1.
12,x5ex5f,x14x34x35x18x57x9x79n,x32x24
(f(x);g(x))n = (fn(x);gn(x)):
x27x28,xd(f(x);g(x)) = d(x),f(x) = d(x)f1(x),g(x) = d(x)g1(x),xc(f1(x);g1(x)) = 1.
x1bx4ex4f8x40x4f
(fn1 (x);gn1(x)) = 1:
x15x21
(fn(x);gn(x)) = (dn(x)fn1 (x);dn(x)gn1(x))
= dn(x)(fn1 (x);gn1(x)) = dn(x)
= (f(x);g(x))n:
13,x3cx4cxm?1x1cxn?1x18x5ex7cx5x1fx7f.
x29,x53d = (m;n),xcx78x2dx4ex4f10–2.9,xd?1 j xm?1,xd?1 j xn?1.
xdh(x)x21xm?1x1cxn?1x18x5x1fx7f,xcx24
xm?1 · 0 (mod h(x));xn?1 · 0 (mod h(x)) =) xm · 1 (mod h(x));xn · 1 (mod h(x)):
x1bx15d = (m;n),x1fx6cx44x71u;v 2Zx4ex4fd = um+vn.
xd = xum+vn · 1 (mod h(x)) =) xd?1 · 0 (mod h(x)):
￠ 5 ￠
x43xdd = ms?nt,s;t > 0,xcd+nt = ms,x15x21
xms?1 = xd+nr?1 = (xd?1)xnr +xnr?1:
x11f(x) 2 K[x]x12xff(x) j xm?1,f(x) j xn?1,xc(f(x);x) = 1,x46f(x) j xms?1,f(x) j xnt?1,x15x21
f(x) j (xd?1)xnr,x1bf(x)x1cxx1xbx40x4ff(x) j xd?1,x1fx6c(xm?1;xn?1) = xd?1,x2dx2ad = (m;n).
14,x5ex5f,x6cx45 f(x)
(f(x);g(x)),
g(x)
(f(x);g(x)) x18x1ax79x32x7cx15x7b,x54x40x16x24x50x2ex2fx24x25x1ex7f
u(x)f(x)+v(x)g(x) = (f(x);g(x))
x18u(x)x1cv(x),x4e
degu(x) < deg
g(x)
(f(x);g(x))
; degv(x) < deg
f(x)
(f(x);g(x))
:
x27x28,x44x71x49x7ex7fs(x);t(x) 2 K[x]x4e
s(x)f(x)+t(x)g(x) = (f(x);g(x)):
xc
s(x) f(x)(f(x);g(x)) +t(x) g(x)(f(x);g(x)) = 1,(*)
x53
s(x) = g(x)(f(x);g(x)) q(x)+u(x);
x2dx2au(x) = 0x6bdegu(x) < deg g(x)(f(x);g(x)), x47v(x) = f(x)(f(x);g(x)) q(x)+t(x),xcx1b(*)x75,
u(x) f(x)(f(x);g(x)) +v(x) g(x)(f(x);g(x)) = 1,(**)
x1bx30xd,f(x)(f(x);g(x)) x1c g(x)(f(x);g(x)) x18x1ax79x32x7cx15x7b,x10x16u(x);v(x)x32x60x21x7bx49x7ex7f,x15x21
degu(x) < deg g(x)(f(x);g(x)),
x1b(**)x75
deg
u(x) f(x)(f(x);g(x))
= deg
v(x) g(x)(f(x);g(x))
;
x49x4a
degv(x) < deg f(x)(f(x);g(x)),
x25 x26 10–4
1,xd(f(x);m(x)) = 1,x5ex5f,x14x34x4cx18x49x7ex7fg(x),x32x44x71x49x7ex7fh(x),x4e
h(x)f(x) · g(x) (mod m(x)):
x27x28,x1bx30xd,x44x71u(x);v(x) 2 K[x],x4e
u(x)f(x)+v(x)m(x) = 1:
x10x16
g(x)u(x)f(x)+g(x)v(x)m(x) = g(x):
x15x21
g(x)u(x)f(x) · g(x) (mod m(x)):
￠ 6 ￠
x53h(x) = g(x)u(x),xc
h(x)f(x) · g(x) (mod m(x)):
2,xdm1(x);￠￠￠ ;ms(x)x2ex66x5dx40x40x1xbx18x49x7ex7f,x5ex5f,x14x34x4cx18x49x7ex7ff1(x);￠￠￠ ;fs(x),x32x44x71
x49x7ex7fF(x),x4e
F(x) · fi(x) (mod mi(x)); i = 1;￠￠￠ ;s:
x27x28,x53M(x) = m1(x)m2(x)￠￠￠ms(x),Ri(x) = M(x)m
i(x)
,xc(Ri(x);mi(x)) = 1,mj(x) j Ri(x),
i 6= j,x44x71hi(x)x4e(x4ex4f1)
hi(x)Ri(x) · fi(x) (mod mi(x))
x53
F(x) =
sX
i=1
hi(x)Ri(x);
xc
F(x) ·
sX
i=1
hi(x)Ri(x) (mod mk(x))
· hk(x)Rk(x) (mod mk(x))
· fk(x) (mod mk(x)):
3,xdm(x)x2ex3x73x79x49x7ex7f,x46m(0) 6= 0,x5ex5f,x44x71x3x73x79x49x7ex7ff(x),x4e
f2(x) · x (mod m(x)):
x27x28,(a)x21x7bx5ex5fx14x34x35x18a 6= 0,x61x6dx7f
f2(x) · x (mod (x?a)m)
x24x6,xdpax21ax18x34x35x66x76x2fx3ex78,xc
(x?a)m = ((px?pa)(px+pa))m = (px?pa)m(px+pa)m
= (h(x)px?g(x))(h(x)px+g(x)) = h2(x)x?g2(x):
x15x21
g2(x) · h2(x)x (mod (x?a)m)
x4ah(a)pa + g(a) = (pa + pa)m 6= 0,x4ah(a)pa? g(a) = (pa?pa)m = 0,x1fx6cg(a)h(a) 6= 0,x49x4a
(h(x);(x?a)m) = 1,x44x71h1(x) 2 K[x]x4eh1(x)h(x) · 1 (mod (x?a)m),x15x21
(h1(x)g(x))2 · x (mod (x?a)m)
x51f(x) = h1(x)g(x),xcx24
f2(x) · x (mod (x?a)m):
(b)xdm(x) = (x?a1)m1(x?a2)m2 ￠￠￠(x?as)ms,ai 6= ajx14i 6= j,xc(x?a1)m1;￠￠￠ ;(x?as)ms
x40x40x1xb,x1b(a),x44x71fi(x) 2 K[x],x4e
f2i (x) · x (mod (x?ai)mi):
x1bx4ex4f2,x44x71f(x)x4e
f(x) · fi(x) (mod (x?ai)mi)
x15x21
f2(x) · x (mod (x?ai)mi)
￠ 7 ￠
x1b(x?a1)m1;￠￠￠ ;(x?as)msx40x40x1xbx40x4f
f2(x) · x (mod m(x)):
x25 x26 10–5
1,x5ex5f,gm(x) j fm(x) () g(x) j f(x).
x27x28,xd
f(x) = apl11 (x)pl22 (x)￠￠￠plss (x);
g(x) = bpk11 (x)pk22 (x)￠￠￠pkss (x);
x2dx2aa;b 2 K,p1(x);￠￠￠ ;ps(x)x21x40x40x1xbx18x60x40x8x49x7ex7f,x46li;ki > 0,i = 1;￠￠￠ ;s,xc
g(x) j f(x) () ki 6 li; i = 1;￠￠￠ ;s
() mki 6 mli; i = 1;￠￠￠ ;s
() gm(x) j fm(x):
2,xdf(x);g(x) 2 K[x],x46x24x13x6x7f
f(x) = apr11 (x)pr22 (x)￠￠￠prss (x); ri > 0; i = 1;￠￠￠ ;s;
g(x) = bpt11 (x)pt22 (x)￠￠￠ptss (x); ti > 0; i = 1;￠￠￠ ;s;
x2dx2ap1(x);￠￠￠ ;ps(x)x21x60x61x18x21x66x60x40x8x49x7ex7f,x5ex5f:
[f(x);g(x)] = pmax(r1;t1)1 (x)pmax(r2;t2)2 (x)￠￠￠pmax(rs;ts)s (x):
x27x28,x53mi = max(ri;ti),i = 1;￠￠￠ ;s.
m(x) = pm11 (x)pm22 (x)￠￠￠pmss (x);
xcx1fri 6 mi,ti 6 mi,x1fx6c
f(x) j m(x); g(x) j m(x) =) [f(x);g(x)] j m(x):
xds(x) 2 K[x]x21f(x);g(x)x18x5x7x7f,xcx24
s(x) = pl11 (x)pl22 (x)￠￠￠plss (x)h(x); li 6 ri; li 6 ti; (h(x);pi(x)) = 1; i = 1;￠￠￠ ;s:
x15x21
li > max(ri;ti); i = 1;￠￠￠ ;s; =) m(x) j s(x):
x1fx6c
[f(x);g(x)] = pm11 (x)pm22 (x)￠￠￠pmss (x):
3,xdf(x);g(x) 2 K[x]x32x21x21x66x49x7ex7f,x5ex5f:
[f(x);g(x)] = f(x)g(x)(f(x);g(x)):
x27x28,xd
f(x) = pr11 (x)pr22 (x)￠￠￠prss (x); ri > 0; i = 1;￠￠￠ ;s;
g(x) = pt11 (x)pt22 (x)￠￠￠ptss (x); ti > 0; i = 1;￠￠￠ ;s;
x2dx2ap1(x);￠￠￠ ;ps(x)x21x60x61x18x21x66x60x40x8x49x7ex7f,x53
mi = max(ri;ti); li = min(ri;ti); i = 1;￠￠￠ ;s:
￠ 8 ￠
xc
f(x)g(x) = pr1+t11 (x)pr2+t22 (x)￠￠￠prs+tss (x);
(f(x);g(x)) = pl11 (x)pl22 (x)￠￠￠plss (x);
x1bx15ri +ti?li = mi,i = 1;￠￠￠ ;s,x1fx6c
f(x)g(x)
(f(x);g(x)) = p
m1
1 (x)p
m2
2 (x)￠￠￠p
ms
s (x) = [f(x);g(x)]:
4,x4cx17x2cx49x7ex7fx18x5ex22x5x7x7f:
(1) f(x) = x4?4x3 +1,g(x) = x3?3x2 +1;
(2) f(x) = x4?x?1+i,g(x) = x2 +1.
x29,(1)x1bx15(f(x);g(x)) = 1,[f(x);g(x)] = f(x)g(x) = x7?7x6 +12x5 +x4?3x3?3x2 +1.
(2)x1bx15(f(x);g(x)) = x?i,[f(x);g(x)] = f(x)(x+i) = x5 +ix4?x2?x?(1+i).
5,xdp(x)x21x1ax79x7cx15x7bx18x49x7ex7f,x5ex5f,x34x35x14x15x34x4cx49x7ex7ff(x);g(x),x1bp(x) j f(x)g(x)x40x16
x31x1ep(x) j f(x)x6bx32p(x) j g(x),xcp(x)x21x60x40x8x49x7ex7f.
x27x28,x11p(x)x40x8,xcx44x71x1ax79x22x15p(x)x18x7ax60x79x49x7ex7ff(x);g(x)x4ep(x) = f(x)g(x),x49x4a
p(x) j f(x)g(x),x41x1f
degf(x) < degp(x); degg(x) < degp(x);
p(x)-f(x),p(x)-g(x),x1cx30xdx1ex1f,x1fx6cp(x)x60x40x8.
6,x5ex5f,x1ax79x7cx150x18x21x66x49x7ex7ff(x)x21x33x66x60x40x8x49x7ex7fx18x3ex27x18x43x13x44x45x46x47x21,x14x34x35x18
x49x7ex7fg(x)x44x24(f(x);g(x)) = 1,x6bx32x14x33x66x57x9x79m,f(x) j gm(x).
x27x28,())xdf(x) = pm(x),x2dx2ap(x)x60x40x8,xcx11g(x) 2 K[x]x12xfp(x) j g(x),x24
f(x) = pm(x) j gm(x):
x34p(x)-g(x),xc(p(x);g(x)) = 1,x49x4a(pm(x);g(x)) = 1,x4b(f(x);g(x)) = 1.
(() xdp(x)x21f(x)x18x66x76x21x66x60x40x8x1fx20,xc(p(x);f(x)) = p(x),x49x4ax44x71x33x76x57x9x79m,x4e
f(x) j pm(x),x6bx1ax5fp(x)x21f(x)x18x34x66x60x40x8x1fx20,x10x16f(x) = cpr(x),x43x1ff(x);p(x)x18x21x7ex73x79x32
x211,x33c = 1,x49x4af(x) = pr(x).
7,x5ex5f,x1ax79x7cx150x18x21x66x49x7ex7ff(x)x21x33x66x60x40x8x49x7ex7fx18x3ex27x18x43x13x44x45x46x47x21,x14x34x35x18
x49x7ex7fg(x);h(x),x1bf(x) j g(x)h(x)x40x16x31x1ef(x) j g(x),x6bx32x14x33x66x57x9x79m,f(x) j hm(x).
x27x28,()) xdf(x) = pm(x),x2dx2ap(x)x21x21x66x60x40x8x49x7ex7f,xcx1bf(x) j g(x)h(x),x40x4fp(x) j
g(x)h(x),x49x4ap(x) j g(x)x6bp(x) j h(x),x15x21f(x) = pm(x) j gm(x)x6bf(x) = pm(x) j hm(x).
(() xdp(x)x21f(x)x18x66x76x21x66x60x40x8x1fx20,xcf(x) = p(x)f1(x),x49x4af(x) j p(x)f1(x),x4a
f(x) - f1(x),x49x4ax44x71x33x76x57x9x79m,x4ef(x) j pm(x),x6bx1ax5fp(x)x21f(x)x18x34x66x60x40x8x1fx20,x10x16
f(x) = cpr(x),x43x1ff(x);p(x)x18x21x7ex73x79x32x211,x33c = 1,x49x4af(x) = pr(x).
x25 x26 10–6
1,x15x7dx17x2cx24x0x73x79x49x7ex7fx24xax77x1fx7f,x11x24,xcx4cx1ex77x1fx7f:
(1) f(x) = x5?10x3?20x2?15x?4;
(2) f(x) = x4?4x3 +16x?16;
(3) f(x) = x5?6x4 +16x3?24x2 +20x?8;
(4) f(x) = x6?15x4 +8x3 +51x2?72x+27.
￠ 9 ￠
x29,(1) x+1,4x77.
(2) x?2,3x77.
(3) x2?2x+2,2x77.
(4) x+3,2x77,x?1,3x77.
2,a;bx35x12xfx22x23x46x47,x17x2cx49x7ex7fx24x77x1fx7f?
(1) f(x) = x3 +3ax+b; (2) f(x) = x4 +4ax+b.
x29,(1)x50a = b = 0x243x77x1fx7fx,x504a3 =?b2x46a 6= 0,x242x77x1fx7f2ax+b.
(2)x50a = b = 0x244x77x1fx7fx,x5027a4 = b3x46a 6= 0,x242x77x1fx7f3ax+b.
3,xdp(x)x21f0(x)x18kx77x1fx7f,x55x17x1ap(x)x21f(x)x18k +1x77x1fx7f,x2ex22x23?
x29,x60x55.x1fx2ex43x40x55f0(x)x34x66x77x1fx7fx32x60x21f(x)x18x1fx7f,x3fx34f(x) = x4?1,f0(x) = 4x3.
4,x5ex5f,x34x35(f0(x);f00(x)) = 1,x5bx23,f(x)x18x77x1fx7fx32x21f(x)x18x19x77x1fx7f.
x27x28,x1bx15(f0(x);f00(x)) = 1,f0(x)x18x34x66x1fx7fx32x60x21f00(x)x18x1fx7f,xdp(x)x21f(x)x18x77x1fx7f,xc
p(x) j f0(x),x15x21p(x)-f00(x),x1ax5fp(x)x21f0(x)x18x7fx1fx7f,x33p(x)x21f(x)x18x19x77x1fx7f.
5,x5ex5f,K[x]x2ax60x40x8x49x7ex7fp(x)x21f(x) 2 K[x]x18k (k > 1)x77x1fx7fx18x43x13x44x45x46x47x21p(x)x21
f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f.
x27x28,())x14kx58x36x37x71,x50k = 1x53x38x39x38x13x3ax3b,x39xdx38x39x14k?1x3ax3b,xdp(x)x21f(x)x18k
x77x1fx7f,xcf(x) = pk(x)g(x),x2dx2a(p(x);g(x)) = 1,xc
f0(x) = kpk?1(x)g(x)+pk(x)g0(x) = pk?1(x)(kg(x)+p(x)g0(x)):
x1b(p(x);g(x)) = 1x40x4f(p(x);kg(x)+p(x)g0(x)) = 1,x1fx6cp(x)x21f0(x)x18k?1x77x1fx7f,x78x2dx36x37x30xd,
p(x)x21f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f,x4ap(x)x21f(x)x18x1fx7fx21x74x75x18.
(() x34p(x)x21f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f,xcp(x)x21f(k?1)(x)x18x66
x77x1fx7f,x3ax4a,p(x)x21f(k?2)(x)x18x19x77x1fx7f,x3bx1ax3cx31,x40x75p(x)x21f(x)x18kx77x1fx7f.
6,x3cx4cx49x7ex7fx1999 +1x22x16(x?1)2x10x4fx6dx7f.
x29,xdx1999 +1 = (x?1)2q(x)+ax+b,xcx40x52x4cx3dx2x4f
1999x1998 = 2(x?1)q(x)+(x?1)2q(x)+a:
x16x = 1x62xax7x40x7f,x4f
a = 1999; b =?1997:
x33x10x4cx6dx7fx2e1999x?1997.
x25 x26 10–7
1,x4cx17x2cx49x7ex7fx18x5x6x78:
(1) f(x) = x4 +2x2 +9,g(x) = x4?4x3 +4x2?9;
(2) f(x) = x3 +2x2 +2x+1,g(x) = x4 +x3 +2x2 +x+1.
x29,(1) 1+p2i,1?p2i.
(2)?1+
p3i
2,
1?p3i
2,
2,x34x35(x?1)2 j Ax4 +Bx2 +1,x4cA;B.
x29,A = 1,B =?2.
3,x74x75x4?3x3 +6x2 +ax+bx55x29x2?1x9x22,x4ca;b.
￠ 10 ￠
x29,a = 3,b =?7.
4,x5ex5f,x34x35f(x) j f(xn),x5bx23f(x)x18x78x6cx55x21x7bx6bx7fx0x78.
x27x28,xdax21f(x)x18x66x76x78,xcf(a) = 0,x15x21f(an) = 0,x43x40x4fx3ef((an)n) = f(an2) = 0,:::,
f(ann) = 0,x1fx4aa;an;an2;￠￠￠ ;ann x32x21f(x)x18x78,x41f(x)x18x60x61x78x61x24x24x3fx49x76,x33x44x24k < lx4e
ank = anl,x4b
ank(anl?nk?1) = 0:
x15x21a = 0x6banl?nk = 1,x33ax2e0x6bx7fx0x78.
5,x5ex5f,sinxx60x21x49x7ex7f.
x27x28,sinxx24xax3fx49x76x60x61x18x78k…,k 2Z,x4ax49x7ex7fx6cx24x24x3fx49x76x78,x1fx6csinxx60x21x49x7ex7f.
6,x74x75x49x7ex7ff(x) = x5?10x2 +15x?6x24x77x78,x3cx4cx1fx18x10x24x78x30x66x58x78x18x77x79.
x29,?3+
p15i
2,
3?p15i
2,1,1,1.
7,x4ctx18x52,x4ef(x) = x3?3x2 +tx?1x24x77x78.
x29,t = 3x53,1x2e3x77x78; t =? 154 x53,? 12 x2e2x77x78.
8,x4cx49x7ex7ff(x) = x3 +px+qx24x77x78x18x46x47.
x29,4p3 +27q2 = 0.
9,x5ex5f,x17x2cx49x7ex7fx77x24x77x78:
(1) f(x) = 1+x+ x22! +￠￠￠+ xnn! ;
(2) f(x) = 1+2x+3x2 +￠￠￠+(n+1)xn.
x27x28,(1)
(f(x);f0(x)) =
1+x+ x
2
2! +￠￠￠+
xn
n! ;1+x+
x2
2! +￠￠￠+
xn?1
(n?1)!
=
xn
n! ;1+x+
x2
2! +￠￠￠+
xn?1
(n?1)!
= 1:
x10x16f(x)xax77x78.
(2)xd
g(x) = (1?x)2(1+2x+3x2 +￠￠￠+(n+1)xn) = 1?(n+2)xn+1 +(n+1)xn+2;
g0(x) = (n+2)(n+1)xn+1?(n+2)(n+1)xn;
(g(x);g0(x)) = x?1:
x10x16g(x)x61x24x18x77x78x21x = 1,x43f(x)x18x77x78x38x13x32x21g(x)x18x77x78,x4ax = 1x60x21f(x)x18x78,x33f(x)xa
x77x78.
10,x5ex5f,f(x) = xn +axn?m +b (n > 2;n > m > 0)x60x55x24x7ax7bx18x77x79x7cx152x18x78.
x27x28,f0(x) = xn?m?1[nxm +(n?m)a].
(a)x50a 6= 0x53,nxm +(n?m)ax18x78x32x21x7fx78,x10x16f(x)x18x77x79x7cx152x18x78x6cx40x55x21x = 0.
(b)x50a = 0x53,f0(x)x18x61x24x18x77x78x2ex = 0,x33f(x)x18x77x79x7cx152x18x78x6cx40x55x21x = 0.
11,x34x35ax21f000(x)x18x66x76kx77x78,x5ex5f,ax21
g(x) = x?a2 [f0(x)+f0(a)]?f(x)+f(a)
x18x66x76k +3x77x78.
￠ 11 ￠
x27x28:
g(x) = x?a2 [f0(x)+f0(a)]?f(x)+f(a);
g0(x) = 12 [f0(a)?f0(x)]+ x?a2 f00(x);
g00(x) = x?a2 f000(x);
x38x13ax21g(x);g0(x);g00(x)x18x78,x43ax21f000(x)x18kx77x78,x1fx6cax21g00(x)x18k +1x77x78,x21g(x)x18k +3x77
x78.
12,x5ex5f,x0 x21f(x)x18k x77x78x18x43x13x44x45x46x47x21 f(x0) = f0(x0) = ￠￠￠ = f(k?1)(x0) = 0x4a
f(k)(x0) 6= 0.
x27x28,x0x21f(x)x18kx77x78 () x?x0x21f(x)x18kx77x1fx7f
() x?x0x21f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f
() f(x0) = f0(x0) = ￠￠￠ = f(k?1)(x) = 0;f(k)(x0) 6= 0.
13,x5ex5f,x34x35f0(x) j f(x),xcf(x)x24nx77x78,x2dx2an = degf(x).
x27x28,x1bx30xd,f(x)(f(x);f0(x)) = c(x? a),x49x4ax? ax2ef(x)x61x24x18x60x40x8x1fx7f(x31x396.4),x10x16
f(x) = c(x?a)n,f(x)x24nx77x78.
14,x3cx40x17x72x10x41x18x79x52,x4cx1ax79x5ex42x18x49x7ex7f:
x 1 2 3 4
y 2 1 4 3
x29,f(x) =? 43 x3 +10x2? 653 x+15.
15,x35x58x43x44x45x71xcx3dx1ex44x46x47x48x49x52x5x7f.
x27x28,xdx10x4cx49x7ex7fx2e
f(x) = c0 +c1x+￠￠￠+cn?1xn?1;
x2dx2acix4ax58.x28ai;bix62xax7x7fx40x52,x4fc0;c1;￠￠￠ ;cn?1x18x69x28x3ex6ax5d:
8>
>>><
>>>>
:
c0 +c1a1 +￠￠￠+cn?1an?11 = b1
c0 +c1a2 +￠￠￠+cn?1an?12 = b2
:::::::::::::::::::::::::
c0 +c1an +￠￠￠+cn?1an?1n = bn
x6cx69x28x3ex6ax5dx18x73x79x4bx3fAx21x14x4bx4cx4bx4bx3f:
A =
0
BB
B@
1 a1 a21 ￠￠￠ an?11
1 a2 a22 ￠￠￠ an?12
...,..,..,..,..
1 an a2n ￠￠￠ an?1n
1
CC
CA;
jAj =
Y
16i<j6n
(aj?ai):
x1bx15aix1x60x18x61,x33jAj6= 0,x10x16x69x28x3ex6ax5dx24x34x66x6.0
B@
c0
...
cn?1
1
CA = A?1
0
B@
b1
...
bn
1
CA:
￠ 12 ￠
x33x10x4cx18x34x66x1ax79x60x4dx4en?1x18x49x7ex7f
f(x) = (1 x ￠￠￠ xn?1)
0
B@
c0
...
cn?1
1
CA
= (1 x ￠￠￠ xn?1)A?1
0
B@
b1
...
bn
1
CA
= 1jAj (1 x ￠￠￠ xn?1)A?
0
B@
b1
...
bn
1
CA
= 1jAj
nX
k=1
(?1)n+kbk
flfl
flfl
flfl
flfl
flfl
flfl
1 ￠￠￠ 1 1 ￠￠￠ 1 1
a1 ￠￠￠ ak?1 ak+1 ￠￠￠ an x
a21 ￠￠￠ a2k?1 a2k+1 ￠￠￠ a2n x2
...,..,..,..,..,..,..
an?11 ￠￠￠ an?1k?1 an?1k+1 ￠￠￠ an?1n xn?1
flfl
flfl
flfl
flfl
flfl
flfl
= 1jAj
nX
k=1
(?1)n+kbk
nY
i=1i6=k
(x?ai)
Y
16i<j6n
i;j6=k
(aj?ai)
=
nX
k=1
(?1)n+k bkF(x)(x?a
k)(an?ak)￠￠￠(ak+1?ak)(ak?ak?1)￠￠￠(ak?a1)
=
nX
k=1
bkF(x)
(x?ak)F0(ak),
x6bx4fF(x) = (x?a1)(x?a2)￠￠￠(x?an).
16,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x79,F(x) = (x?a1)(x?a2)￠￠￠(x?an).
x5ex5f,x34x4cx49x7ex7ff(x)x58F(x)x22x10x4fx18x6dx7fx2e
nX
i=1
f(ai)F(x)
(x?ai)F0(ai):
x27x28,x36x37
1
F(x) =
A1
x?a1 +
A2
x?a2 +￠￠￠+
An
x?an,
x40x52x61x8x16x?ai,x50x53x = ai,x40x4f
Ai = 1F0(a
i)
:
x1fx6cx40x4fx51x1ex7f
1
F(x) =
1
(x?a1)F0(a1) +
1
(x?a2)F0(a2) +￠￠￠+
1
(x?an)F0(an),
x49x4a
1 =
nX
i=1
F(x)
(x?ai)F0(ai),
x53
f(x) = (x?ai)fi(x)+f(ai);
￠ 13 ￠
xc
f(x) =
nX
i=1
[(x?ai)fi(x)+f(ai)] F(x)(x?a
i)F0(ai)
=
nX
i=1
fi(x)F(x)
F0(ai) +
nX
i=1
f(ai)F(x)
(x?ai)F0(ai)
= F(x)
nX
i=1
fi(x)
F0(ai)
!
+
nX
i=1
f(ai)F(x)
(x?ai)F0(ai),
x1bx15
nP
i=1
f(ai)F(x)
(x?ai)F0(ai) 2 K[x],x46deg
nP
i=1
f(ai)F(x)
(x?ai)F0(ai) 6 n?1,x10x16x58F(x)x22f(x)x10x4fx18x6dx7fx2e
nP
i=1
f(ai)F(x)
(x?ai)F0(ai),
17,x74x75a1;￠￠￠ ;an; b1;￠￠￠ ;bnx2ex1x60x18x61x18x79,x4cx6x17x2cx3ex6ax5d:
8>
>>>>
<
>>>>
>:
1
b1?a1 x1 +
1
b1?a2 x2 +￠￠￠+
1
b1?an xn =?1;
1
b2?a1 x1 +
1
b2?a2 x2 +￠￠￠+
1
b2?an xn =?1;
:::::::::::::::::::::::::::::::::::::::::::::::::
1
bn?a1 x1 +
1
bn?a2 x2 +￠￠￠+
1
bn?an xn =?1:x29,xdx
1;￠￠￠ ;xnx21x6cx3ex6ax5dx18x34x66x6,x36x37x24x0x13x7f
F(x) = 1+ x1x?a
1
+ x2x?a
2
+￠￠￠+ xnx?a
n; (*)
xcF(bi) = 0,i = 1;￠￠￠ ;n.
x53F(x) = g(x)(x?a
1)(x?a2)￠￠￠(x?an)
,xcdegg(x) = n,x46g(x)x18x21x7ex2exn,x1bx15F(bi) = 0,x33
g(bi) = 0,i = 1;￠￠￠ ;n,x10x16
g(x) = (x?b1)(x?b2)￠￠￠(x?bn):
F(x) = (x?b1)(x?b2)￠￠￠(x?bn)(x?a
1)(x?a2)￠￠￠(x?an)
:
x53
f(x) = (x?a1)(x?a2)￠￠￠(x?an):
x36x37h(x) = g(x)?f(x),xcdegh(x) 6 n?1.
x1bx15h(ai) = g(ai),x1bx44x46x47x48x5x7f,
h(x) =
nX
i=1
g(ai)f(x)
(x?ai)f0(ai) ;
g(x) = f(x)+
nX
i=1
g(ai)f(x)
(x?ai)f0(ai) ;
F(x) = g(x)f(x) = 1+
nX
i=1
1
(x?ai) ￠
g(ai)
f0(ai) ;
x1c(*)x50x51,x4bx4f
x1 = g(a1)f0(a
1);x2 = g(a2)f0(a
2);￠￠￠ ;xn = g(an)f0(a
n)
:
x25 x26 10–8
1,x13x7dx4cx49x7ex7ff(x) = x5?3x4 +4x3?4x2 +3x?1x71x3x79x1x75x2x79x1x7x18x6fx5x13x6x7f.
￠ 14 ￠
x29,f(x) = (x2 +1)(x?1)3 = (x+i)(x?i)(x?1)3.
2,x13x7dx4cx49x7ex7ff(x) = xn?1x71x3x79x1x75x2x79x1x7x18x6fx5x13x6x7f.
x29,x71x3x79x1x7x18x13x6x7f:
f(x) =
n?1Y
k=0
x?cos 2k…n?isin 2k…n
;
x71x2x79x1x7x18x13x6x7f:
f(x) =
8>
>><
>>>
:
(x?1)
n?1
2Q
k=1
x2?2cos 2k…n x+1
·; nx2ex1dx79;
(x?1)(x+1)
n?2
2Q
k=1
x2?2cos 2k…n x+1
·; nx2ex1bx79.
3,x74x75m;n;px2ex7ax7ax9x79,x5ex5f,x3m +x3n+1 +x3p+2x55x29x2 +x+1x9x22.
x27x28,x1fx2e
x3m +x3n+1 +x3p+2 = x3m?1+x3n+1?x+x3p+2?x2 +x2 +x+1
= (x3m?1)+x(x3n?1)+x2(x3p?1)+x2 +x+1:
x1bx15x3?1 j x3m?1,x3?1 j x3n?1,x3?1 j x3p?1,x10x16x2 +x+1 j x3m?1+x(x3n?1)+x2(x3p?
1)+(x2 +x+1).
x52x27:xd"1;"2x2ex2+x+1x18x78,xc"31 = "32 = 1,x10x16f("1) = "3m1 +"3n+11 +"3p+21 = 1+"1+"21 = 0.
x61x0f("2) = 0,x10x16x2 +x+1 j (x).
4,x5ex5f,x34x35x2 +x+1 j f1(x3)+xf2(x3),x5bx23f1(1) = f2(1) = 0.
x27x28,xd" =?1+
p3i
2,xc";"x32x21x
2 +x+1x18x78,x1bx15x2 +x+1 j f1(x3)+xf2(x3),x10x16
f1(1)+"f2(1) = 0; f1(1)+"f2(1) = 0:
x1bx6cx4ff1(1) = f2(1) = 0.
5,x5ex5f,x34x35x?1 j f(xn),x5bx23xn?1 j f(xn).
x27x28,x1bx15x?1 j f(xn),x10x16f(1) = 0,x49x4ax14x34x35x18nx1ax7fx0x78",
f("n) = f(1) = 0;
x10x16xn?1 j f(xn).
6,x74x75x49x7ex7ff(x) = x3 +ix2 +(1?i)x?10?2ix24x2x78,x3cx4cf(x)x18x79x3x78.
x29,2,?1+?1+
p17
2 i,?1+
1?p17
2 i.
7,x5ex5f,x2x73x79x49x7ex7ff(x)x40x72x2ex40x76x2x73x79x49x7ex7fx18x2fx3ex75x18x43x13x44x45x46x47x21x14x34x4cx18x2x79
a,x32x24f(a) > 0.
x27x28,x44x45x28x38x13,x17x5ex43x13x28.
xd
f(x) = c(x?a1)l1(x?a2)l2 ￠￠￠(x?at)lt(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks;
x6bx4fa1 < a2 < ￠￠￠ < at,p2i? 4qi < 0,li > 0,ki > 0,x1bx46x47x75,c > 0,x34x51b;cx4ear?1 < b < ar,
ar < c < ar+1,xcf(b)x18x53xex2e(?1)lr+￠￠￠+lt,f(c)x18x53xex2e(?1)lr+1+￠￠￠+lt,x43x1ff(b) > 0,f(c) > 0,x33
(?1)lr > 0,lrx21x1bx79,r = 1;￠￠￠ ;t,x49x4a
f(x) = g2(x)(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks:
￠ 15 ￠
xd
x2 +pix+qi = (x?fii)(x?fii); fii 2C;
xc
(x?fi1)(x?fi2)￠￠￠(x?fis) = u(x)+iv(x); u(x);v(x) 2R[x];
(x?fi1)(x?fi2)￠￠￠(x?fis) = u(x)?iv(x):
x49x4a
(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks = u2(x)+v2(x):
f(x) = g2(x)(u2(x)+v2(x)) = (g(x)u(x))2 +(g(x)v(x))2:
8,x3cx58x54x33x55x58x0x56x57x17x2cx49x7ex7fx18x2x78:
(1) x3?3x?1; (2) x3 +x2?2x?1;
(3) x4 +x?1; (4) x4 +4x3?12x+9.
x29,(1)x54x33x55x2cx2e,x3?3x?1,2?1,2+1,1,x5bxex79x34x17x72:
1?2?1 0 1 2 +1
f0(x) + + +
f1(x) + + 0? 0 + +
f2(x) + + + +
f3(x) + + + + + + +
V(x) 3 3 2 1 1 0 0
x1bx6cx75,f(x)x243x76x2x78,x2x78x14x78x21(?2;?1),(?1;0),(1;2).
(2)x54x33x55x2cx2e,x3 +x2?2x?1,3x2 +2x?2,2x+1,1.
x2x78x79x2e3,x2x78x14x78x21(?2;?1),(?1;0),(1;2).
(3)x54x33x55x2cx2e,x4 +x?1,4x3 +1,?3x?4,?1.
x2x78x79x2e2,x2x78x14x78x21(?2;?1),(0;1).
(4)x54x33x55x2cx2e,x4 +4x3?12x+9,x3 +3x2?3,x2 +3x?4,?4x?3,1,xax2x78.
x25 x26 10–9
1,x3cx4cx17x2cx49x7ex7fx18x24x0x78:
(1) x5?7x3?12x2 +6x+36; (2) 6x4 +19x3?7x2?26x+12;
(3) 10x4?13x3 +15x2?18x?15;
(4) x6?6x5 +11x4?x3?18x2 +20x?8.
x29,(1) 3,?2.
(2)?3,12,
(3)? 12,
(4) 2,2,2.
2,x5ex5fx17x2cx49x7ex7fx71x24x0x79x1x7x60x40x8:
(1) x4?8x3 +12x2?6x+2; (2) x5?12x3 +36x?12;
(3) x4?x3 +2x+1; (4) x4 +4kx+1,kx2ex9x79
(5) xp +px+1,px2ex1dxbx79; (6) x4 +5x3?3x2?5x+1.
x27x28,(1)x51p = 2,x1bx58x59x5ax5bx1fx15x7dx71x75,f(x)x60x40x8.
(2)x51p = 3,x1bx58x59x5ax5bx1fx15x7dx71x75,f(x)x60x40x8.
￠ 16 ￠
(3) f(y +1) = y4 +3y3 +3y2 +3y +3,x51p = 3,x1bx58x59x5ax5bx1fx15x7dx71x75,f(y +1)x60x40x8,x33f(x)
x60x40x8.
(4) f(y +1) = y4 +4y3 +6y2 +4(k +1)y +2(2k +1),x51p = 2,x1bx58x59x5ax5bx1fx15x7dx71x75,f(y +1)
x60x40x8,x33f(x)x60x40x8.
(5)
f(y?1) = (y?1)p +p(y?1)+1 =
pX
k=0
Ckp(?1)kyp?k +p(y?1)+1
=
p?1X
k=0
Ckp(?1)kyp?k +py?p
= yp?pyp?1 + p(p?1)2 yp?2 +￠￠￠+ p(p?1)2 y2 +2py?p:
x1bx58x59x5ax5bx1fx15x7dx71x75,f(y?1)x60x40x8,x33f(x)x60x40x8.
(6)x1fx2ef(x)xax24x0x78,x33x11f(x)x40x8,xcx44x24
f(x) = (x2 +ax+1)(x2 +bx+1) x6b f(x) = (x2 +ax?1)(x2 +bx?1):
x14x15x1ax7f,x6fx70x2d3x1ax7ex2e1x1ax7ex73x79,x4fa+b = 5,a+b =?5,x60x40x55.
x14x1cx7f,x53x = 1,x4fab =?1,x43a+b = 5,x3bx60x40x55.
x33f(x)x60x40x8.
3,x3cx28x17x2cx13x7fx18x13x5cx24x0x42:
(1) 11+ 3p2+2 3p4 ; (2) 11? 4p2+p2 ;
(3) 11+p2?p3 ;
(4) a2?3a?1a2 +2a+1,x2dx2a,ax2ex3ex6ax3 +x2 +3x+4 = 0x18x78.
x29,(1)x36x37f(x) = 2x2 +x+1x2eg(x) = x3?2,x76x75(f(x);g(x)) = 1,x5dx6fx70x75
(2x2 +x+1) x
2 +7x?3
23?(x
3?2)?2x+13
23 = 1:
x10x16
1
1+ 3p2+2 3p4 =
1
23 (?3+7
3p2? 3p4):
(2) 11? 4p2+p2 = 17 (1+3 4p2+2p2? 4p8).
(3) 11+p2?p3 = 14 (2+p2+p6).
(4) a2?3a?1a2 +2a+1 = 17a2?3a+55.
4,xdf(x)x21x66x76x9x73x79x49x7ex7f,x5ex5f,x34x35f(0)x75f(1)x32x21x1dx79,xcf(x)xax9x79x78.
x27x28,x3ex5e,x34f(x)x24x9x79x78a,xcf(x) = (x?a)g(x),x2dx2ag(x)x2ex9x73x79x49x7ex7f,xc0?ax1c1?a
x2ax5ex5fx24x66x76x21x1bx79,x49x4af(0);f(1)x2ax5ex5fx24x66x76x2ex1bx79,x1ex1f.
5,xdf(x) = x3 +bx2 +cx+dx21x66x76x9x73x79x49x7ex7f,x5ex5f,x34x35bd+cdx2ex1dx79,xcf(x)x71x24x0x79
x1x7x60x40x8.
x27x28,x1bx4fxd,dx1cb+cx32x21x1dx79,x49x4af(0) = dx16x2ef(1) = 1+b+c+dx60x2ex1dx79,x33f(x)xax9
x79x78,x43x1ff(x)x18x21x7ex73x79x2e1,x46degf(x) = 3,x10x16f(x)x60x40x8.
6,x74x75x9x73x79x49x7ex7ff(x) = a0xn +a1xn?1 +￠￠￠+anxax24x0x78,x5ex5f,x34x35x24xbx79p,x4e
￠ 17 ￠
(1) p-a0;
(2) p j ai;i = 2;3;￠￠￠ ;an;
(3) p2 -an
xcf(x)x71Qx7x60x40x8.
x27x28,x34p j a1,xcx1bx58x59x5ax5bx1fx15x7dx71x75f(x)x71Qx7x60x40x8.
x16x17xdp-a1,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],x1bx15f(x)xax24x0x78,x1fx6c2 6 degg(x) 6 n?2,
2 6 degh(x) 6 n?2,xd
g(x) = b0xk +b1xk?1 +￠￠￠+bk; k > 2;m > 2;
h(x) = c0xm +c1xm?1 +￠￠￠+cm; k +m = n:
x1bx15bkcm = an,p j an,p2 -an,x40xdp j bk,p-cm,x43x1fp-b0,xdblx21x49x61x62x63x5ex7bx66x76x60x55x29px9x22
x18x73x79,xc
p-am+l = cmbl +cm?1bl+1 +￠￠￠
x41x1fm+l > 2,p j am+l,x1ex1f,x1fx6cf(x)x71Qx7x60x40x8.
7,x3cx66x58x10x24x18x9x79m,x4ex5 +mx?1x71x24x0x79x1x7x40x8.
x27x28,(a)x34m = 0,xcx5?1x38x13x40x8.
(b)x34f(x)x24x66x1ax1fx7f,xc1+m?1 = 0x6b?1?m?1 = 0,x49x4am = 0x6b?2.
(c)x11f(x)x60x64x66x1ax1fx7f,x41x40x8,xcx40xd
x5 +mx?1 = (x2 +ax§1)(x3 +bx2 +cxcurrency11):
x50x51x40x52x73x79,x4f
a+b = 0; ab+c§1 = 0; ac§bcurrency11 = 0; currency1(a?c) = m:
x33b =?a,8
><
>:
a2 +c = currency11
accurrency1a = §1
m = currency1(a?c)
x71x65x66x66x7dx70x17,c = 0,a =?1,m = 1;x71x65x19x66x7dx70x17,c = 2,a(c+2) =?1,x60x40x55.x10x16mx18x40x55
x51x52x2e0,1,?2,x71x6c3x66x7dx67x17x5 +mx?1x32x40x8.
8,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x9x79,x5ex5f,x49x7ex7f
f(x) = (x?a1)(x?a2)￠￠￠(x?an)?1
x71Qx7x60x40x8.
x27x28,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],degg(x);degh(x) < degf(x),xcf(ai) = g(ai)h(ai) =
1,x33g(ai) =?h(ai) = §1,x49x4ag(ai)+h(ai) = 0,x15x21x49x7ex7f
F(x) = g(x)+h(x)
x24nx76x60x61x18x78,x41degF(x) < n,x6cx55F(x) = 0,g(x) =?h(x),f(x) =?g2(x),x4ax50xx43x13x7cx53,x24
f(x) > 0,?g2(x) 6 0,x1ex1f,x1fx6c
9,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x9x79,x5ex5f,x49x7ex7f
f(x) = (x?a1)2(x?a2)2￠￠￠(x?an)2 +1
x71Qx7x60x40x8.
￠ 18 ￠
x27x28,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],x46
0 < degg(x) < 2n; 0 < degh(x) < 2n:
x43x1fdegg(x) + degh(x) = 2n,x33g(x);h(x)x2ax5ex5fx24x66x76x18x1ax796 n,x60x68xddegh(x) 6 n,x43xd
g(x);h(x)x60x2ex21x66x49x7ex7f.
x1bx15f(x)x71x2x79x7x69x6ax51x57x52,x1fx6cf(x)xax2x78,g(x);h(x)x6bxax2x78,x15x21g(x);h(x)x71x2x79x7
x69x6ax51x57x52.x43x1ff(ai) = 1,x33h(ai) = g(ai) = 1,h(x)?1x24nx76x60x61x18x78a1;￠￠￠ ;an,x10x16
h(x) = (x?a1)￠￠￠(x?an)+1:
x49x4adegg(x) = n,x3ax4a
g(x) = (x?a1)￠￠￠(x?an)+1:
x15x21
g(x)h(x) = [(x?a1)￠￠￠(x?an)+1]2
= (x?a1)2￠￠￠(x?an)2 +2(x?a1)￠￠￠(x?an)+1 6= f(x);
x1ex1f,x1fx6cf(x)x60x40x8.
10,xdx7ex72x49x7ex7ff(x)x71x24x0x79x1x7x60x40x8,x5ex5f,f(x2)x71x24x0x79x1x7x40x8x18x43x13x44x45x46x47x21
x44x71x9x79c 6= 0x2ex9x73x79x49x7ex7fg(x);h(x),x4e
cf(x) = g2(x)?xh2(x):
x27x28,x43x13x28x38x13,x16x17x5ex44x45x28.
xdg(x)x2ef(x2)x18x34x66x60x40x8x1fx7f,xcx1bg(x) j f(x2)x40x4fg(?x) j f(x2),x38x13g(?x)x3bx60x40x8.
g(x)x1cg(?x)x18x29x73x61x24x16x173x66x40x55:
(a) g(x) = g(?x); (b) g(x) =?g(?x); (3) (g(x);g(?x)) = 1.
(a) x34 g(x) = g(?x),xc g(x) = h(x2),x1b h(x2) j f(x2) x4f h(x) j f(x),x4a f(x) x60x40x8,x10x16
h(x) = cf(x),g(x) = cf(x2),x1cf(x2)x40x8x1ex1f,x1fx6cg(x) 6= g(?x).
(b)x34g(x) =?g(?x),xcg(x) =?xh(x2),xh(x2) j f(x2),x33x j f(x),x15x21§f(x) =?x = 02?x￠12,
x38x39x3ax3b.
(c)x34(g(x);g(?x)) = 1,xcg(x)g(?x) j f(x2),xdg(x) = u(x2)+xv(x2),xc
g(x)g(?x) = u2(x2)?x2v2(x2):
x4au2(x2)?x2v2(x2) j f(x2),x1fx6c
u2(x)?xv2(x) j f(x):
x33x44x71c 6= 0x4ecf(x) = u2(x)?xv2(x),x5ex6c.
11,x5ex5f,x14x10x24x18x57x9x79n,f(x) = x2n?x2n?1 +1x71x24x0x79x1x7x60x40x8,(x6dx6e,x6fnx70x71x72x73
x74x75x70x76x210)
x27x28,x21x7bx45x77x4ex4f10x18x38x39x3cx78x2e,x50f(x)x21x7ex72x49x7ex7fx53,x40x51c = 1,x2ex5ex6bx66x38x39,x36x37
f(x2) = c?1(g2(x2)?x2h2(x2)) = c?1(g(x2)+xh(x2))(g(x2)?xh(x2));
x79x35x3ex11g(x2)+xh(x2) = r(g1(x2)+xh1(x2)),x2dx2ag1(x2)+xh1(x2)x21x7ex72x49x7ex7f,xcg1(x2)?xh1(x2)
x3bx21x7ex72x49x7ex7f,x15x21
f(x2) = c?1r2(g1(x2)+xh1(x2))(g1(x2)?xh1(x2)) = c?1r2(g21(x2)?x2h21(x2));
￠ 19 ￠
x78x2dx7ax5ax7bx0,c?1r2 = 1,x15x21f(x) = g21(x)?xh21(x).
x14nx58x36x37x71,x30x35x58x3cx78x7cx18x4ex4f10.
x50n = 1x53,x76x75x2?x+1x71x24x0x79x1x7x60x40x8.
x39xdx2n?x2n?1 +1x71x24x0x79x1x7x60x40x8,x4ax2n+1?x2n +1x71x24x0x79x1x7x40x8,xcx78x2dx3cx78x18x4e
x4f10,x44x71g(x);h(x) 2Z[x],x4e
x2n?x2n?1 +1 = g2(x)?xh2(x);
x40x52x4cx3dx4f
2nx2n?1?2n?1x2n?1?1 = 2g(x)g0(x)?h2(x)?2xh(x)h0(x):
xc2 j h2(x),2 j h(x),x10x16
x2n?x2n?1 +1 = g2(x)+4p(x):
x53
g(x) = x2n?1?x2n?2 +1+k(x)+2l(x);
x2dx2ak(x)x18x16x7ex73x79x32x210x6b1,xc
x2n?x2n?1 +1 = x2n?x2n?1 +1+4x2n?1?2x2n?2?2x3￠2n?2 +k2(x)+4p2(x):
x1fx6c2 j k(x),4 j k2(x),x3ax4a
x2n?x2n?1 +1 = x2n?x2n?1 +1?2x2n?2?2x3￠2n?2 +4p3(x);
4p3(x) = 2(x2n?2 +x3￠2n?2;
x6bx60x40x55,x49x4ax75x2n+1?x2n +1x71x24x0x79x1x7x60x40x8.
￠ 20 ￠
x16x17x19 x4x25x1ax1bx1cxax26x0x27x28x1cx29xb
x25 x26 10–1
1,x6fx70(x2 +ax?b)(x2?1)+(x2?ax+b)(x2 +1).
x29,2x4?2ax+2b.
2,x6fx70x49x7ex7fx3 +2x2 +3x?1x1c3x2 +2x+4x18x8x9.
x29,3x5 +8x4 +17x3 +11x2 +10x?4.
3,xd
f(x) = 3x2?5x+3;
g(x) = ax(x?1)+b(x+2)(x?1)+cx(x+2);
x3cx66x58a;b;c,x4ef(x) = g(x).
x29,x51x =?2,x4fa = 256 ;x51x = 0,x4fb =? 32,x51x = 1,x4fc = 13,
4,xdf(x),g(x)x75h(x)x32x21x2x73x79x49x7ex7f,x5ex5f,x34x35
f2(x) = xg2(x)+xh2(x);
x5bx23
f(x) = g(x) = h(x) = 0:
x27x28,x34f(x) 6= 0,xcx1ax7fx18x1ax79x2ex1bx79,x4ax1cx7fx18x1ax79x2ex1dx79,x1ex1f,x33f(x) = 0,x49x4a
g2(x)+h2(x) = 0:
x43,g(x);h(x)x20x2ex2x73x79x49x7ex7f,x49x4ag2(x);h2(x)x18x21x7ex73x79x32x21x7ax7ax79,x4ax6bx40x76x79x74x75x2ex7b,x33
g(x);h(x)x18x21x7ex73x79x32x21x7b,x49x4ag(x) = h(x) = 0.
x25 x26 10–2
1,x58g(x)x22f(x),x4cx23q(x)x1cx6dx7fr(x):
(1) f(x) = x4 +4x2?x+6,g(x) = x2 +x+1;
(2) f(x) = x3 +3x2?x?1,g(x) = 3x2?2x+1.
x29,(1) q(x) = x2?x+4,r(x) =?4x+2.
(2) q(x) = 19 (3x+11),r(x) = 109 (x?2).
2,m,p,qx24x25x22x23x46x47x53,x24
(1) x2 +mx+1 j x3 +px+q;
(2) x2 +mx+1 j x4 +px2 +q.
x29,(1) p = 1?m2,q =?m.
(2)
( m = 0
p = 1+q x6b
( p =?m2 +2
q = 1
￠ 1 ￠
3,x58x26x25x22x71x4cx23q(x)x2ex6dx7fr(x):
(1) f(x) = x4?2x3 +4x2?6x+8,g(x) = x?2;
(2) f(x) = 2x5?5x3?8x,g(x) = x+2.
x29,(1) q(x) = x3 +4x+2,r(x) = 12.
(2) q(x) = 2x4?4x3 +3x2?6x+4,r(x) =?8.
4,x58x26x25x22x71x72f(x)x2ex?x0x18x3ex27:
(1) f(x) = x4?2x3 +3x2?2x+1,x0 = 2;
(2) f(x) = x4?2x2 +3,x0 =?2;
(3) f(x) = x4 +2ix3?(1+i)x2?3x+1?2i,x0 =?i.
x29,(1) f(x) = (x?2)4 +6(x?2)3 +15(x?2)2 +18(x?2)+9.
(2) f(x) = (x+2)4?8(x+2)3 +22(x+2)2?24(x+2)+11.
(3) f(x) = (x+i)4?2i(x+i)3?(1+i)(x+i)2?5(x+i)+(1+2i).
5,x47hxi0 = 1,hxik = x(x?1)(x?2)￠￠￠(x?k +1),(k > 1),x3cx28f(x)x72x2e
c0 +c1hxi+c2hxi2 +￠￠￠
x18x70x7f:
(1) f(x) = x4?2x3 +x2?1;
(2) f(x) = x5.
x29,(1) 1 1?2 1 0?1
1?1 0
2 1?1 0 0
2 2
3 1 1 2
3
1 4
x1fx6cf(x) =?1+2hxi2 +4hxi3 +hxi4.
(2) f(x) = hxi+15hxi2 +25hxi3 +10hxi4 +hxi5.
6,kx21x57x9x79,x5ex5f,x j fk(x)x50x46x61x50x j f(x);
x27x28,xdf(x)x18x60x79x7ex2ea,xcfk(x)x18x60x79x7ex2eak,x1fx6cx j fk(x) () ak = 0 () a = 0 ()
x j f(x).
7,xda;bx2ex40x76x60x18x1ex18x60x79,x5ex5f,x49x7ex7ff(x)x29(x?a)(x?b)x22x10x4fx6dx7fx2e
f(a)?f(b)
a?b x+
af(b)?bf(a)
a?b,
x27x28,xdf(x) = (x?a)(x?b)q(x)+Ax+B,xc
f(a) = aA+B; f(b) = bA+B;
x1bx6cx4f
A = f(a)?f(b)a?b ; B = af(b)?bf(a)a?b,
x1fx6cx38x39x3ax3b.
8,xdf1(x),f2(x),g1(x),g2(x)x32x21x79x1Kx7x18x49x7ex7f,x2dx2af1(x) 6= 0.
x5ex5f,x34x35g1(x)g2(x) j f1(x)f2(x),f1(x) j g1(x),xcg2(x) j f2(x).
￠ 2 ￠
x27x28,xdf1(x)f2(x) = g1(x)g2(x)q1(x),g1(x) = f1(x)q2(x),xcf1(x)f2(x) = f1(x)q2(x)g2(x)q1(x),
x1bx15f1(x) 6= 0,x40x4ff2(x) = g2(x)q2(x)q1(x),x4bg2(x) j f2(x).
9,x5ex5f,xd?1 j xn?1x50x46x61x50d j n.
x27x28,())x11n = dq,xc
xn?1 = (xd?1)(xd(q?1) +xd(q?2) +￠￠￠+xd +1):
x1fx6cxd?1 j xn?1.
(()xdn = dq +r,0 6 r < d,x1bx7x5e,xdq?1 · 0 (mod xd?1),x4b
xdq · 1 (mod xd?1);
xn · xdq+r · xdq ￠xr · xr (mod xd?1);
xn?1 · xr?1 (mod xd?1):
x4axd?1 j xr?1,r = 0,x1fx6cxd?1 j xn?1,r = 0,d j n.
x25 x26 10–3
1,x4cx5ex7cx5x1fx7f(f(x);g(x)):
(1) f(x) = x4 +x3?3x2?4x?1,g(x) = x3 +x2?x?1;
(2) f(x) = x5 +x4?x3?2x?1,g(x) = 3x4 +2x3 +x2?2;
(3) f(x) = x4?x3?4x2 +4x+1,g(x) = x2?x?1.
x29,(1) x+1.
(2) 1.
(3) 1.
2,x4cu(x),v(x),x4eu(x)f(x)+v(x)g(x) = (f(x);g(x)):
(1) f(x) = x4 +2x3?x2?4x?2,g(x) = x4 +x3?x2?2x?2;
(2) f(x) = 4x4?2x3?16x2 +5x+9,g(x) = 2x3?x2?5x+4;
(3) f(x) = 2x4 +3x3?3x2?5x+2,g(x) = 2x3 +x2?x?1.
x29,(1) u(x) =?x?1,v(x) = x+2,d(x) = x2?2.
(2) u(x) =? 13 (x?1),v(x) = 13 (2x2?2x?3),d(x) = x?1.
(3) u(x) =? 16 (2x2 +3x),v(x) = 16 (2x3 +5x2?6),d(x) = 1.
3,x5ex5f,x34x35d(x) j f(x),d(x) j g(x),x46d(x)x2ef(x)x1cg(x)x18x66x76x5dx25,x5bx23d(x)x21f(x)x1cg(x)
x18x66x76x5ex7cx5x1fx7f.
x27x28,xdd(x) = u(x)f(x) + v(x)g(x),xcx14x34x35x18h(x) 2 K[x],x34h(x) j f(x),h(x) j g(x),xc
h(x) j d(x).
x43,d(x)x2ef(x)x1cg(x)x18x66x76x5x1fx7f,x33d(x)x21f(x)x1cg(x)x18x66x76x5ex7cx5x1fx7f.
4,x5ex5f,x34x35h(x)x2ex21x66x49x7ex7f,xc
(f(x)h(x);g(x)h(x)) = (f(x);g(x))h(x):
x27x28,xdd(x) = (f(x);g(x)) 6= 0,xcx44x71u(x);v(x)x4e
d(x) = u(x)f(x)+v(x)g(x):
￠ 3 ￠
x10x16
d(x)h(x) = u(x)f(x)h(x)+v(x)g(x)h(x):
x43x1fd(x)h(x) j f(x)h(x),d(x)h(x) j g(x)h(x),x10x16d(x)h(x)x21f(x)h(x)x1cg(x)h(x)x18x66x76x5ex7cx5x1fx7f.
x43x1fd(x);h(x)x32x21x21x66x49x7ex7f,x33d(x)h(x)x3bx21x21x66x49x7ex7f,x49x4a
(f(x)h(x);g(x)h(x)) = d(x)h(x) = (f(x);g(x))h(x):
x43x34d(x) = 0,xcf(x) = g(x) = 0,x72x1ex7fx2ax13x3ax3b.
5,x5ex5f,x34x35f(x),g(x)x60x79x2ex7b,xc
f(x)
(f(x);g(x));
g(x)
(f(x);g(x))
= 1:
x27x28,x1ff(x),g(x)x60x79x2ex7b,x33(f(x);g(x)) 6= 0,x10x16
(f(x);g(x)) =
f(x)
(f(x);g(x)) (f(x);g(x));
g(x)
(f(x);g(x)) (f(x);g(x))
=
f(x)
(f(x);g(x)) ;
g(x)
(f(x);g(x))
(f(x);g(x))
(x1bx4ex4f4)x40x52x2bx2c(f(x);g(x)),x4f
f(x)
(f(x);g(x)) ;
g(x)
(f(x);g(x))
= 1:
6,x5ex5f,x34x35f(x),g(x)x60x79x2ex7b,x46
u(x)f(x)+v(x)g(x) = (f(x);g(x));
xc(u(x);v(x)) = 1.
x27x28,x1ff(x),g(x)x60x79x2ex7b,x33(f(x);g(x)) 6= 0,x1fx6c
u(x) f(x)(f(x);g(x)) +v(x) g(x)(f(x);g(x)) = 1;
(u(x);v(x)) = 1:
7,x5ex5f,x34x35(f(x);g(x)) = 1,(f(x);h(x)) = 1,x5bx23
(f(x);g(x)h(x)) = 1:
x27x28,x44x71u(x);v(x);s(x);t(x),x4e
u(x)f(x)+v(x)g(x) = 1;
s(x)f(x)+t(x)h(x) = 1;
x10x16
f(x)(u(x)s(x)f(x)+u(x)t(x)h(x)+s(x)v(x)g(x))+v(x)t(x)g(x)h(x) = 1;
(f(x);g(x)h(x)) = 1:
8,xd f1(x);￠￠￠ ;fm(x),g1(x);￠￠￠ ;gn(x) x32x21x49x7ex7f,x46 (fi(x);gj(x)) = 1 (i = 1;￠￠￠ ;m;j =
1;￠￠￠ ;n),x5ex5f:
(f1(x)f2(x)￠￠￠fm(x);g1(x)g2(x)￠￠￠gn(x)) = 1:
x27x28,x1b(fi(x);gj(x)) = 1,x40x4f(fi(x);g1(x)g2(x)) = 1,:::,(fi(x);g1(x)g2(x)￠￠￠gn(x)) = 1,x49x4a
(f1(x)f2(x);g1(x)￠￠￠gn(x)) = 1,(f1(x)f2(x)f3(x);g1(x)￠￠￠gn(x)) = 1,:::,
(f1(x)f2(x)￠￠￠fm(x);g1(x)￠￠￠gn(x)) = 1.
9,x5ex5f,x34x35(f(x);g(x)) = 1,x5bx23(f(x)+g(x);f(x)g(x)) = 1.
￠ 4 ￠
x27x28,x1bx15(f(x);g(x)) = 1,x10x16
(f(x)+g(x);g(x)) = (f(x);g(x)) = 1;
(f(x)+g(x);f(x)) = (g(x);f(x)) = 1;
x1fx6c
(f(x)+g(x);f(x)g(x)) = 1:
10,xdf1(x) = af(x)+bg(x),g1(x) = cf(x)+dg(x),x46ad?bc 6= 0,x5ex5f:
(f(x);g(x)) = (f1(x);g1(x)):
x27x28,x1bx4fxdx40x4f(f(x);g(x)) j (f1(x);g1(x)),x43
f(x) = dad?bc f1(x)? bad?bc g1(x);
g(x) =?cad?bc f1(x)+ aad?bc g1(x);
x10x16
(f1(x);g1(x)) j (f(x);g(x)):
x43x1f(f1(x);g1(x))x1c(f(x);g(x))x18x21x7ex73x79x18x61,x33(f(x);g(x)) = (f1(x);g1(x)).
11,x5ex5f,x34x35f(x)x1cg(x)x1xb,x5bx23f(xm)x1cg(xm)x3bx1xb.
x27x28,x1bx4fxd,x44x71x49x7ex7fu(x);v(x)x4e
u(x)f(x)+v(x)g(x) = 1:
x10x16
u(xm)f(xm)+v(xm)g(xm) = 1:
x33(f(xm);g(xm)) = 1.
12,x5ex5f,x14x34x35x18x57x9x79n,x32x24
(f(x);g(x))n = (fn(x);gn(x)):
x27x28,xd(f(x);g(x)) = d(x),f(x) = d(x)f1(x),g(x) = d(x)g1(x),xc(f1(x);g1(x)) = 1.
x1bx4ex4f8x40x4f
(fn1 (x);gn1(x)) = 1:
x15x21
(fn(x);gn(x)) = (dn(x)fn1 (x);dn(x)gn1(x))
= dn(x)(fn1 (x);gn1(x)) = dn(x)
= (f(x);g(x))n:
13,x3cx4cxm?1x1cxn?1x18x5ex7cx5x1fx7f.
x29,x53d = (m;n),xcx78x2dx4ex4f10–2.9,xd?1 j xm?1,xd?1 j xn?1.
xdh(x)x21xm?1x1cxn?1x18x5x1fx7f,xcx24
xm?1 · 0 (mod h(x));xn?1 · 0 (mod h(x)) =) xm · 1 (mod h(x));xn · 1 (mod h(x)):
x1bx15d = (m;n),x1fx6cx44x71u;v 2Zx4ex4fd = um+vn.
xd = xum+vn · 1 (mod h(x)) =) xd?1 · 0 (mod h(x)):
￠ 5 ￠
x43xdd = ms?nt,s;t > 0,xcd+nt = ms,x15x21
xms?1 = xd+nr?1 = (xd?1)xnr +xnr?1:
x11f(x) 2 K[x]x12xff(x) j xm?1,f(x) j xn?1,xc(f(x);x) = 1,x46f(x) j xms?1,f(x) j xnt?1,x15x21
f(x) j (xd?1)xnr,x1bf(x)x1cxx1xbx40x4ff(x) j xd?1,x1fx6c(xm?1;xn?1) = xd?1,x2dx2ad = (m;n).
14,x5ex5f,x6cx45 f(x)
(f(x);g(x)),
g(x)
(f(x);g(x)) x18x1ax79x32x7cx15x7b,x54x40x16x24x50x2ex2fx24x25x1ex7f
u(x)f(x)+v(x)g(x) = (f(x);g(x))
x18u(x)x1cv(x),x4e
degu(x) < deg
g(x)
(f(x);g(x))
; degv(x) < deg
f(x)
(f(x);g(x))
:
x27x28,x44x71x49x7ex7fs(x);t(x) 2 K[x]x4e
s(x)f(x)+t(x)g(x) = (f(x);g(x)):
xc
s(x) f(x)(f(x);g(x)) +t(x) g(x)(f(x);g(x)) = 1,(*)
x53
s(x) = g(x)(f(x);g(x)) q(x)+u(x);
x2dx2au(x) = 0x6bdegu(x) < deg g(x)(f(x);g(x)), x47v(x) = f(x)(f(x);g(x)) q(x)+t(x),xcx1b(*)x75,
u(x) f(x)(f(x);g(x)) +v(x) g(x)(f(x);g(x)) = 1,(**)
x1bx30xd,f(x)(f(x);g(x)) x1c g(x)(f(x);g(x)) x18x1ax79x32x7cx15x7b,x10x16u(x);v(x)x32x60x21x7bx49x7ex7f,x15x21
degu(x) < deg g(x)(f(x);g(x)),
x1b(**)x75
deg
u(x) f(x)(f(x);g(x))
= deg
v(x) g(x)(f(x);g(x))
;
x49x4a
degv(x) < deg f(x)(f(x);g(x)),
x25 x26 10–4
1,xd(f(x);m(x)) = 1,x5ex5f,x14x34x4cx18x49x7ex7fg(x),x32x44x71x49x7ex7fh(x),x4e
h(x)f(x) · g(x) (mod m(x)):
x27x28,x1bx30xd,x44x71u(x);v(x) 2 K[x],x4e
u(x)f(x)+v(x)m(x) = 1:
x10x16
g(x)u(x)f(x)+g(x)v(x)m(x) = g(x):
x15x21
g(x)u(x)f(x) · g(x) (mod m(x)):
￠ 6 ￠
x53h(x) = g(x)u(x),xc
h(x)f(x) · g(x) (mod m(x)):
2,xdm1(x);￠￠￠ ;ms(x)x2ex66x5dx40x40x1xbx18x49x7ex7f,x5ex5f,x14x34x4cx18x49x7ex7ff1(x);￠￠￠ ;fs(x),x32x44x71
x49x7ex7fF(x),x4e
F(x) · fi(x) (mod mi(x)); i = 1;￠￠￠ ;s:
x27x28,x53M(x) = m1(x)m2(x)￠￠￠ms(x),Ri(x) = M(x)m
i(x)
,xc(Ri(x);mi(x)) = 1,mj(x) j Ri(x),
i 6= j,x44x71hi(x)x4e(x4ex4f1)
hi(x)Ri(x) · fi(x) (mod mi(x))
x53
F(x) =
sX
i=1
hi(x)Ri(x);
xc
F(x) ·
sX
i=1
hi(x)Ri(x) (mod mk(x))
· hk(x)Rk(x) (mod mk(x))
· fk(x) (mod mk(x)):
3,xdm(x)x2ex3x73x79x49x7ex7f,x46m(0) 6= 0,x5ex5f,x44x71x3x73x79x49x7ex7ff(x),x4e
f2(x) · x (mod m(x)):
x27x28,(a)x21x7bx5ex5fx14x34x35x18a 6= 0,x61x6dx7f
f2(x) · x (mod (x?a)m)
x24x6,xdpax21ax18x34x35x66x76x2fx3ex78,xc
(x?a)m = ((px?pa)(px+pa))m = (px?pa)m(px+pa)m
= (h(x)px?g(x))(h(x)px+g(x)) = h2(x)x?g2(x):
x15x21
g2(x) · h2(x)x (mod (x?a)m)
x4ah(a)pa + g(a) = (pa + pa)m 6= 0,x4ah(a)pa? g(a) = (pa?pa)m = 0,x1fx6cg(a)h(a) 6= 0,x49x4a
(h(x);(x?a)m) = 1,x44x71h1(x) 2 K[x]x4eh1(x)h(x) · 1 (mod (x?a)m),x15x21
(h1(x)g(x))2 · x (mod (x?a)m)
x51f(x) = h1(x)g(x),xcx24
f2(x) · x (mod (x?a)m):
(b)xdm(x) = (x?a1)m1(x?a2)m2 ￠￠￠(x?as)ms,ai 6= ajx14i 6= j,xc(x?a1)m1;￠￠￠ ;(x?as)ms
x40x40x1xb,x1b(a),x44x71fi(x) 2 K[x],x4e
f2i (x) · x (mod (x?ai)mi):
x1bx4ex4f2,x44x71f(x)x4e
f(x) · fi(x) (mod (x?ai)mi)
x15x21
f2(x) · x (mod (x?ai)mi)
￠ 7 ￠
x1b(x?a1)m1;￠￠￠ ;(x?as)msx40x40x1xbx40x4f
f2(x) · x (mod m(x)):
x25 x26 10–5
1,x5ex5f,gm(x) j fm(x) () g(x) j f(x).
x27x28,xd
f(x) = apl11 (x)pl22 (x)￠￠￠plss (x);
g(x) = bpk11 (x)pk22 (x)￠￠￠pkss (x);
x2dx2aa;b 2 K,p1(x);￠￠￠ ;ps(x)x21x40x40x1xbx18x60x40x8x49x7ex7f,x46li;ki > 0,i = 1;￠￠￠ ;s,xc
g(x) j f(x) () ki 6 li; i = 1;￠￠￠ ;s
() mki 6 mli; i = 1;￠￠￠ ;s
() gm(x) j fm(x):
2,xdf(x);g(x) 2 K[x],x46x24x13x6x7f
f(x) = apr11 (x)pr22 (x)￠￠￠prss (x); ri > 0; i = 1;￠￠￠ ;s;
g(x) = bpt11 (x)pt22 (x)￠￠￠ptss (x); ti > 0; i = 1;￠￠￠ ;s;
x2dx2ap1(x);￠￠￠ ;ps(x)x21x60x61x18x21x66x60x40x8x49x7ex7f,x5ex5f:
[f(x);g(x)] = pmax(r1;t1)1 (x)pmax(r2;t2)2 (x)￠￠￠pmax(rs;ts)s (x):
x27x28,x53mi = max(ri;ti),i = 1;￠￠￠ ;s.
m(x) = pm11 (x)pm22 (x)￠￠￠pmss (x);
xcx1fri 6 mi,ti 6 mi,x1fx6c
f(x) j m(x); g(x) j m(x) =) [f(x);g(x)] j m(x):
xds(x) 2 K[x]x21f(x);g(x)x18x5x7x7f,xcx24
s(x) = pl11 (x)pl22 (x)￠￠￠plss (x)h(x); li 6 ri; li 6 ti; (h(x);pi(x)) = 1; i = 1;￠￠￠ ;s:
x15x21
li > max(ri;ti); i = 1;￠￠￠ ;s; =) m(x) j s(x):
x1fx6c
[f(x);g(x)] = pm11 (x)pm22 (x)￠￠￠pmss (x):
3,xdf(x);g(x) 2 K[x]x32x21x21x66x49x7ex7f,x5ex5f:
[f(x);g(x)] = f(x)g(x)(f(x);g(x)):
x27x28,xd
f(x) = pr11 (x)pr22 (x)￠￠￠prss (x); ri > 0; i = 1;￠￠￠ ;s;
g(x) = pt11 (x)pt22 (x)￠￠￠ptss (x); ti > 0; i = 1;￠￠￠ ;s;
x2dx2ap1(x);￠￠￠ ;ps(x)x21x60x61x18x21x66x60x40x8x49x7ex7f,x53
mi = max(ri;ti); li = min(ri;ti); i = 1;￠￠￠ ;s:
￠ 8 ￠
xc
f(x)g(x) = pr1+t11 (x)pr2+t22 (x)￠￠￠prs+tss (x);
(f(x);g(x)) = pl11 (x)pl22 (x)￠￠￠plss (x);
x1bx15ri +ti?li = mi,i = 1;￠￠￠ ;s,x1fx6c
f(x)g(x)
(f(x);g(x)) = p
m1
1 (x)p
m2
2 (x)￠￠￠p
ms
s (x) = [f(x);g(x)]:
4,x4cx17x2cx49x7ex7fx18x5ex22x5x7x7f:
(1) f(x) = x4?4x3 +1,g(x) = x3?3x2 +1;
(2) f(x) = x4?x?1+i,g(x) = x2 +1.
x29,(1)x1bx15(f(x);g(x)) = 1,[f(x);g(x)] = f(x)g(x) = x7?7x6 +12x5 +x4?3x3?3x2 +1.
(2)x1bx15(f(x);g(x)) = x?i,[f(x);g(x)] = f(x)(x+i) = x5 +ix4?x2?x?(1+i).
5,xdp(x)x21x1ax79x7cx15x7bx18x49x7ex7f,x5ex5f,x34x35x14x15x34x4cx49x7ex7ff(x);g(x),x1bp(x) j f(x)g(x)x40x16
x31x1ep(x) j f(x)x6bx32p(x) j g(x),xcp(x)x21x60x40x8x49x7ex7f.
x27x28,x11p(x)x40x8,xcx44x71x1ax79x22x15p(x)x18x7ax60x79x49x7ex7ff(x);g(x)x4ep(x) = f(x)g(x),x49x4a
p(x) j f(x)g(x),x41x1f
degf(x) < degp(x); degg(x) < degp(x);
p(x)-f(x),p(x)-g(x),x1cx30xdx1ex1f,x1fx6cp(x)x60x40x8.
6,x5ex5f,x1ax79x7cx150x18x21x66x49x7ex7ff(x)x21x33x66x60x40x8x49x7ex7fx18x3ex27x18x43x13x44x45x46x47x21,x14x34x35x18
x49x7ex7fg(x)x44x24(f(x);g(x)) = 1,x6bx32x14x33x66x57x9x79m,f(x) j gm(x).
x27x28,())xdf(x) = pm(x),x2dx2ap(x)x60x40x8,xcx11g(x) 2 K[x]x12xfp(x) j g(x),x24
f(x) = pm(x) j gm(x):
x34p(x)-g(x),xc(p(x);g(x)) = 1,x49x4a(pm(x);g(x)) = 1,x4b(f(x);g(x)) = 1.
(() xdp(x)x21f(x)x18x66x76x21x66x60x40x8x1fx20,xc(p(x);f(x)) = p(x),x49x4ax44x71x33x76x57x9x79m,x4e
f(x) j pm(x),x6bx1ax5fp(x)x21f(x)x18x34x66x60x40x8x1fx20,x10x16f(x) = cpr(x),x43x1ff(x);p(x)x18x21x7ex73x79x32
x211,x33c = 1,x49x4af(x) = pr(x).
7,x5ex5f,x1ax79x7cx150x18x21x66x49x7ex7ff(x)x21x33x66x60x40x8x49x7ex7fx18x3ex27x18x43x13x44x45x46x47x21,x14x34x35x18
x49x7ex7fg(x);h(x),x1bf(x) j g(x)h(x)x40x16x31x1ef(x) j g(x),x6bx32x14x33x66x57x9x79m,f(x) j hm(x).
x27x28,()) xdf(x) = pm(x),x2dx2ap(x)x21x21x66x60x40x8x49x7ex7f,xcx1bf(x) j g(x)h(x),x40x4fp(x) j
g(x)h(x),x49x4ap(x) j g(x)x6bp(x) j h(x),x15x21f(x) = pm(x) j gm(x)x6bf(x) = pm(x) j hm(x).
(() xdp(x)x21f(x)x18x66x76x21x66x60x40x8x1fx20,xcf(x) = p(x)f1(x),x49x4af(x) j p(x)f1(x),x4a
f(x) - f1(x),x49x4ax44x71x33x76x57x9x79m,x4ef(x) j pm(x),x6bx1ax5fp(x)x21f(x)x18x34x66x60x40x8x1fx20,x10x16
f(x) = cpr(x),x43x1ff(x);p(x)x18x21x7ex73x79x32x211,x33c = 1,x49x4af(x) = pr(x).
x25 x26 10–6
1,x15x7dx17x2cx24x0x73x79x49x7ex7fx24xax77x1fx7f,x11x24,xcx4cx1ex77x1fx7f:
(1) f(x) = x5?10x3?20x2?15x?4;
(2) f(x) = x4?4x3 +16x?16;
(3) f(x) = x5?6x4 +16x3?24x2 +20x?8;
(4) f(x) = x6?15x4 +8x3 +51x2?72x+27.
￠ 9 ￠
x29,(1) x+1,4x77.
(2) x?2,3x77.
(3) x2?2x+2,2x77.
(4) x+3,2x77,x?1,3x77.
2,a;bx35x12xfx22x23x46x47,x17x2cx49x7ex7fx24x77x1fx7f?
(1) f(x) = x3 +3ax+b; (2) f(x) = x4 +4ax+b.
x29,(1)x50a = b = 0x243x77x1fx7fx,x504a3 =?b2x46a 6= 0,x242x77x1fx7f2ax+b.
(2)x50a = b = 0x244x77x1fx7fx,x5027a4 = b3x46a 6= 0,x242x77x1fx7f3ax+b.
3,xdp(x)x21f0(x)x18kx77x1fx7f,x55x17x1ap(x)x21f(x)x18k +1x77x1fx7f,x2ex22x23?
x29,x60x55.x1fx2ex43x40x55f0(x)x34x66x77x1fx7fx32x60x21f(x)x18x1fx7f,x3fx34f(x) = x4?1,f0(x) = 4x3.
4,x5ex5f,x34x35(f0(x);f00(x)) = 1,x5bx23,f(x)x18x77x1fx7fx32x21f(x)x18x19x77x1fx7f.
x27x28,x1bx15(f0(x);f00(x)) = 1,f0(x)x18x34x66x1fx7fx32x60x21f00(x)x18x1fx7f,xdp(x)x21f(x)x18x77x1fx7f,xc
p(x) j f0(x),x15x21p(x)-f00(x),x1ax5fp(x)x21f0(x)x18x7fx1fx7f,x33p(x)x21f(x)x18x19x77x1fx7f.
5,x5ex5f,K[x]x2ax60x40x8x49x7ex7fp(x)x21f(x) 2 K[x]x18k (k > 1)x77x1fx7fx18x43x13x44x45x46x47x21p(x)x21
f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f.
x27x28,())x14kx58x36x37x71,x50k = 1x53x38x39x38x13x3ax3b,x39xdx38x39x14k?1x3ax3b,xdp(x)x21f(x)x18k
x77x1fx7f,xcf(x) = pk(x)g(x),x2dx2a(p(x);g(x)) = 1,xc
f0(x) = kpk?1(x)g(x)+pk(x)g0(x) = pk?1(x)(kg(x)+p(x)g0(x)):
x1b(p(x);g(x)) = 1x40x4f(p(x);kg(x)+p(x)g0(x)) = 1,x1fx6cp(x)x21f0(x)x18k?1x77x1fx7f,x78x2dx36x37x30xd,
p(x)x21f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f,x4ap(x)x21f(x)x18x1fx7fx21x74x75x18.
(() x34p(x)x21f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f,xcp(x)x21f(k?1)(x)x18x66
x77x1fx7f,x3ax4a,p(x)x21f(k?2)(x)x18x19x77x1fx7f,x3bx1ax3cx31,x40x75p(x)x21f(x)x18kx77x1fx7f.
6,x3cx4cx49x7ex7fx1999 +1x22x16(x?1)2x10x4fx6dx7f.
x29,xdx1999 +1 = (x?1)2q(x)+ax+b,xcx40x52x4cx3dx2x4f
1999x1998 = 2(x?1)q(x)+(x?1)2q(x)+a:
x16x = 1x62xax7x40x7f,x4f
a = 1999; b =?1997:
x33x10x4cx6dx7fx2e1999x?1997.
x25 x26 10–7
1,x4cx17x2cx49x7ex7fx18x5x6x78:
(1) f(x) = x4 +2x2 +9,g(x) = x4?4x3 +4x2?9;
(2) f(x) = x3 +2x2 +2x+1,g(x) = x4 +x3 +2x2 +x+1.
x29,(1) 1+p2i,1?p2i.
(2)?1+
p3i
2,
1?p3i
2,
2,x34x35(x?1)2 j Ax4 +Bx2 +1,x4cA;B.
x29,A = 1,B =?2.
3,x74x75x4?3x3 +6x2 +ax+bx55x29x2?1x9x22,x4ca;b.
￠ 10 ￠
x29,a = 3,b =?7.
4,x5ex5f,x34x35f(x) j f(xn),x5bx23f(x)x18x78x6cx55x21x7bx6bx7fx0x78.
x27x28,xdax21f(x)x18x66x76x78,xcf(a) = 0,x15x21f(an) = 0,x43x40x4fx3ef((an)n) = f(an2) = 0,:::,
f(ann) = 0,x1fx4aa;an;an2;￠￠￠ ;ann x32x21f(x)x18x78,x41f(x)x18x60x61x78x61x24x24x3fx49x76,x33x44x24k < lx4e
ank = anl,x4b
ank(anl?nk?1) = 0:
x15x21a = 0x6banl?nk = 1,x33ax2e0x6bx7fx0x78.
5,x5ex5f,sinxx60x21x49x7ex7f.
x27x28,sinxx24xax3fx49x76x60x61x18x78k…,k 2Z,x4ax49x7ex7fx6cx24x24x3fx49x76x78,x1fx6csinxx60x21x49x7ex7f.
6,x74x75x49x7ex7ff(x) = x5?10x2 +15x?6x24x77x78,x3cx4cx1fx18x10x24x78x30x66x58x78x18x77x79.
x29,?3+
p15i
2,
3?p15i
2,1,1,1.
7,x4ctx18x52,x4ef(x) = x3?3x2 +tx?1x24x77x78.
x29,t = 3x53,1x2e3x77x78; t =? 154 x53,? 12 x2e2x77x78.
8,x4cx49x7ex7ff(x) = x3 +px+qx24x77x78x18x46x47.
x29,4p3 +27q2 = 0.
9,x5ex5f,x17x2cx49x7ex7fx77x24x77x78:
(1) f(x) = 1+x+ x22! +￠￠￠+ xnn! ;
(2) f(x) = 1+2x+3x2 +￠￠￠+(n+1)xn.
x27x28,(1)
(f(x);f0(x)) =
1+x+ x
2
2! +￠￠￠+
xn
n! ;1+x+
x2
2! +￠￠￠+
xn?1
(n?1)!
=
xn
n! ;1+x+
x2
2! +￠￠￠+
xn?1
(n?1)!
= 1:
x10x16f(x)xax77x78.
(2)xd
g(x) = (1?x)2(1+2x+3x2 +￠￠￠+(n+1)xn) = 1?(n+2)xn+1 +(n+1)xn+2;
g0(x) = (n+2)(n+1)xn+1?(n+2)(n+1)xn;
(g(x);g0(x)) = x?1:
x10x16g(x)x61x24x18x77x78x21x = 1,x43f(x)x18x77x78x38x13x32x21g(x)x18x77x78,x4ax = 1x60x21f(x)x18x78,x33f(x)xa
x77x78.
10,x5ex5f,f(x) = xn +axn?m +b (n > 2;n > m > 0)x60x55x24x7ax7bx18x77x79x7cx152x18x78.
x27x28,f0(x) = xn?m?1[nxm +(n?m)a].
(a)x50a 6= 0x53,nxm +(n?m)ax18x78x32x21x7fx78,x10x16f(x)x18x77x79x7cx152x18x78x6cx40x55x21x = 0.
(b)x50a = 0x53,f0(x)x18x61x24x18x77x78x2ex = 0,x33f(x)x18x77x79x7cx152x18x78x6cx40x55x21x = 0.
11,x34x35ax21f000(x)x18x66x76kx77x78,x5ex5f,ax21
g(x) = x?a2 [f0(x)+f0(a)]?f(x)+f(a)
x18x66x76k +3x77x78.
￠ 11 ￠
x27x28:
g(x) = x?a2 [f0(x)+f0(a)]?f(x)+f(a);
g0(x) = 12 [f0(a)?f0(x)]+ x?a2 f00(x);
g00(x) = x?a2 f000(x);
x38x13ax21g(x);g0(x);g00(x)x18x78,x43ax21f000(x)x18kx77x78,x1fx6cax21g00(x)x18k +1x77x78,x21g(x)x18k +3x77
x78.
12,x5ex5f,x0 x21f(x)x18k x77x78x18x43x13x44x45x46x47x21 f(x0) = f0(x0) = ￠￠￠ = f(k?1)(x0) = 0x4a
f(k)(x0) 6= 0.
x27x28,x0x21f(x)x18kx77x78 () x?x0x21f(x)x18kx77x1fx7f
() x?x0x21f(x);f0(x);￠￠￠ ;f(k?1)(x)x18x1fx7f,x41x60x21f(k)(x)x18x1fx7f
() f(x0) = f0(x0) = ￠￠￠ = f(k?1)(x) = 0;f(k)(x0) 6= 0.
13,x5ex5f,x34x35f0(x) j f(x),xcf(x)x24nx77x78,x2dx2an = degf(x).
x27x28,x1bx30xd,f(x)(f(x);f0(x)) = c(x? a),x49x4ax? ax2ef(x)x61x24x18x60x40x8x1fx7f(x31x396.4),x10x16
f(x) = c(x?a)n,f(x)x24nx77x78.
14,x3cx40x17x72x10x41x18x79x52,x4cx1ax79x5ex42x18x49x7ex7f:
x 1 2 3 4
y 2 1 4 3
x29,f(x) =? 43 x3 +10x2? 653 x+15.
15,x35x58x43x44x45x71xcx3dx1ex44x46x47x48x49x52x5x7f.
x27x28,xdx10x4cx49x7ex7fx2e
f(x) = c0 +c1x+￠￠￠+cn?1xn?1;
x2dx2acix4ax58.x28ai;bix62xax7x7fx40x52,x4fc0;c1;￠￠￠ ;cn?1x18x69x28x3ex6ax5d:
8>
>>><
>>>>
:
c0 +c1a1 +￠￠￠+cn?1an?11 = b1
c0 +c1a2 +￠￠￠+cn?1an?12 = b2
:::::::::::::::::::::::::
c0 +c1an +￠￠￠+cn?1an?1n = bn
x6cx69x28x3ex6ax5dx18x73x79x4bx3fAx21x14x4bx4cx4bx4bx3f:
A =
0
BB
B@
1 a1 a21 ￠￠￠ an?11
1 a2 a22 ￠￠￠ an?12
...,..,..,..,..
1 an a2n ￠￠￠ an?1n
1
CC
CA;
jAj =
Y
16i<j6n
(aj?ai):
x1bx15aix1x60x18x61,x33jAj6= 0,x10x16x69x28x3ex6ax5dx24x34x66x6.0
B@
c0
...
cn?1
1
CA = A?1
0
B@
b1
...
bn
1
CA:
￠ 12 ￠
x33x10x4cx18x34x66x1ax79x60x4dx4en?1x18x49x7ex7f
f(x) = (1 x ￠￠￠ xn?1)
0
B@
c0
...
cn?1
1
CA
= (1 x ￠￠￠ xn?1)A?1
0
B@
b1
...
bn
1
CA
= 1jAj (1 x ￠￠￠ xn?1)A?
0
B@
b1
...
bn
1
CA
= 1jAj
nX
k=1
(?1)n+kbk
flfl
flfl
flfl
flfl
flfl
flfl
1 ￠￠￠ 1 1 ￠￠￠ 1 1
a1 ￠￠￠ ak?1 ak+1 ￠￠￠ an x
a21 ￠￠￠ a2k?1 a2k+1 ￠￠￠ a2n x2
...,..,..,..,..,..,..
an?11 ￠￠￠ an?1k?1 an?1k+1 ￠￠￠ an?1n xn?1
flfl
flfl
flfl
flfl
flfl
flfl
= 1jAj
nX
k=1
(?1)n+kbk
nY
i=1i6=k
(x?ai)
Y
16i<j6n
i;j6=k
(aj?ai)
=
nX
k=1
(?1)n+k bkF(x)(x?a
k)(an?ak)￠￠￠(ak+1?ak)(ak?ak?1)￠￠￠(ak?a1)
=
nX
k=1
bkF(x)
(x?ak)F0(ak),
x6bx4fF(x) = (x?a1)(x?a2)￠￠￠(x?an).
16,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x79,F(x) = (x?a1)(x?a2)￠￠￠(x?an).
x5ex5f,x34x4cx49x7ex7ff(x)x58F(x)x22x10x4fx18x6dx7fx2e
nX
i=1
f(ai)F(x)
(x?ai)F0(ai):
x27x28,x36x37
1
F(x) =
A1
x?a1 +
A2
x?a2 +￠￠￠+
An
x?an,
x40x52x61x8x16x?ai,x50x53x = ai,x40x4f
Ai = 1F0(a
i)
:
x1fx6cx40x4fx51x1ex7f
1
F(x) =
1
(x?a1)F0(a1) +
1
(x?a2)F0(a2) +￠￠￠+
1
(x?an)F0(an),
x49x4a
1 =
nX
i=1
F(x)
(x?ai)F0(ai),
x53
f(x) = (x?ai)fi(x)+f(ai);
￠ 13 ￠
xc
f(x) =
nX
i=1
[(x?ai)fi(x)+f(ai)] F(x)(x?a
i)F0(ai)
=
nX
i=1
fi(x)F(x)
F0(ai) +
nX
i=1
f(ai)F(x)
(x?ai)F0(ai)
= F(x)
nX
i=1
fi(x)
F0(ai)
!
+
nX
i=1
f(ai)F(x)
(x?ai)F0(ai),
x1bx15
nP
i=1
f(ai)F(x)
(x?ai)F0(ai) 2 K[x],x46deg
nP
i=1
f(ai)F(x)
(x?ai)F0(ai) 6 n?1,x10x16x58F(x)x22f(x)x10x4fx18x6dx7fx2e
nP
i=1
f(ai)F(x)
(x?ai)F0(ai),
17,x74x75a1;￠￠￠ ;an; b1;￠￠￠ ;bnx2ex1x60x18x61x18x79,x4cx6x17x2cx3ex6ax5d:
8>
>>>>
<
>>>>
>:
1
b1?a1 x1 +
1
b1?a2 x2 +￠￠￠+
1
b1?an xn =?1;
1
b2?a1 x1 +
1
b2?a2 x2 +￠￠￠+
1
b2?an xn =?1;
:::::::::::::::::::::::::::::::::::::::::::::::::
1
bn?a1 x1 +
1
bn?a2 x2 +￠￠￠+
1
bn?an xn =?1:x29,xdx
1;￠￠￠ ;xnx21x6cx3ex6ax5dx18x34x66x6,x36x37x24x0x13x7f
F(x) = 1+ x1x?a
1
+ x2x?a
2
+￠￠￠+ xnx?a
n; (*)
xcF(bi) = 0,i = 1;￠￠￠ ;n.
x53F(x) = g(x)(x?a
1)(x?a2)￠￠￠(x?an)
,xcdegg(x) = n,x46g(x)x18x21x7ex2exn,x1bx15F(bi) = 0,x33
g(bi) = 0,i = 1;￠￠￠ ;n,x10x16
g(x) = (x?b1)(x?b2)￠￠￠(x?bn):
F(x) = (x?b1)(x?b2)￠￠￠(x?bn)(x?a
1)(x?a2)￠￠￠(x?an)
:
x53
f(x) = (x?a1)(x?a2)￠￠￠(x?an):
x36x37h(x) = g(x)?f(x),xcdegh(x) 6 n?1.
x1bx15h(ai) = g(ai),x1bx44x46x47x48x5x7f,
h(x) =
nX
i=1
g(ai)f(x)
(x?ai)f0(ai) ;
g(x) = f(x)+
nX
i=1
g(ai)f(x)
(x?ai)f0(ai) ;
F(x) = g(x)f(x) = 1+
nX
i=1
1
(x?ai) ￠
g(ai)
f0(ai) ;
x1c(*)x50x51,x4bx4f
x1 = g(a1)f0(a
1);x2 = g(a2)f0(a
2);￠￠￠ ;xn = g(an)f0(a
n)
:
x25 x26 10–8
1,x13x7dx4cx49x7ex7ff(x) = x5?3x4 +4x3?4x2 +3x?1x71x3x79x1x75x2x79x1x7x18x6fx5x13x6x7f.
￠ 14 ￠
x29,f(x) = (x2 +1)(x?1)3 = (x+i)(x?i)(x?1)3.
2,x13x7dx4cx49x7ex7ff(x) = xn?1x71x3x79x1x75x2x79x1x7x18x6fx5x13x6x7f.
x29,x71x3x79x1x7x18x13x6x7f:
f(x) =
n?1Y
k=0
x?cos 2k…n?isin 2k…n
;
x71x2x79x1x7x18x13x6x7f:
f(x) =
8>
>><
>>>
:
(x?1)
n?1
2Q
k=1
x2?2cos 2k…n x+1
·; nx2ex1dx79;
(x?1)(x+1)
n?2
2Q
k=1
x2?2cos 2k…n x+1
·; nx2ex1bx79.
3,x74x75m;n;px2ex7ax7ax9x79,x5ex5f,x3m +x3n+1 +x3p+2x55x29x2 +x+1x9x22.
x27x28,x1fx2e
x3m +x3n+1 +x3p+2 = x3m?1+x3n+1?x+x3p+2?x2 +x2 +x+1
= (x3m?1)+x(x3n?1)+x2(x3p?1)+x2 +x+1:
x1bx15x3?1 j x3m?1,x3?1 j x3n?1,x3?1 j x3p?1,x10x16x2 +x+1 j x3m?1+x(x3n?1)+x2(x3p?
1)+(x2 +x+1).
x52x27:xd"1;"2x2ex2+x+1x18x78,xc"31 = "32 = 1,x10x16f("1) = "3m1 +"3n+11 +"3p+21 = 1+"1+"21 = 0.
x61x0f("2) = 0,x10x16x2 +x+1 j (x).
4,x5ex5f,x34x35x2 +x+1 j f1(x3)+xf2(x3),x5bx23f1(1) = f2(1) = 0.
x27x28,xd" =?1+
p3i
2,xc";"x32x21x
2 +x+1x18x78,x1bx15x2 +x+1 j f1(x3)+xf2(x3),x10x16
f1(1)+"f2(1) = 0; f1(1)+"f2(1) = 0:
x1bx6cx4ff1(1) = f2(1) = 0.
5,x5ex5f,x34x35x?1 j f(xn),x5bx23xn?1 j f(xn).
x27x28,x1bx15x?1 j f(xn),x10x16f(1) = 0,x49x4ax14x34x35x18nx1ax7fx0x78",
f("n) = f(1) = 0;
x10x16xn?1 j f(xn).
6,x74x75x49x7ex7ff(x) = x3 +ix2 +(1?i)x?10?2ix24x2x78,x3cx4cf(x)x18x79x3x78.
x29,2,?1+?1+
p17
2 i,?1+
1?p17
2 i.
7,x5ex5f,x2x73x79x49x7ex7ff(x)x40x72x2ex40x76x2x73x79x49x7ex7fx18x2fx3ex75x18x43x13x44x45x46x47x21x14x34x4cx18x2x79
a,x32x24f(a) > 0.
x27x28,x44x45x28x38x13,x17x5ex43x13x28.
xd
f(x) = c(x?a1)l1(x?a2)l2 ￠￠￠(x?at)lt(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks;
x6bx4fa1 < a2 < ￠￠￠ < at,p2i? 4qi < 0,li > 0,ki > 0,x1bx46x47x75,c > 0,x34x51b;cx4ear?1 < b < ar,
ar < c < ar+1,xcf(b)x18x53xex2e(?1)lr+￠￠￠+lt,f(c)x18x53xex2e(?1)lr+1+￠￠￠+lt,x43x1ff(b) > 0,f(c) > 0,x33
(?1)lr > 0,lrx21x1bx79,r = 1;￠￠￠ ;t,x49x4a
f(x) = g2(x)(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks:
￠ 15 ￠
xd
x2 +pix+qi = (x?fii)(x?fii); fii 2C;
xc
(x?fi1)(x?fi2)￠￠￠(x?fis) = u(x)+iv(x); u(x);v(x) 2R[x];
(x?fi1)(x?fi2)￠￠￠(x?fis) = u(x)?iv(x):
x49x4a
(x2 +p1x+q1)k1 ￠￠￠(x2 +psx+qs)ks = u2(x)+v2(x):
f(x) = g2(x)(u2(x)+v2(x)) = (g(x)u(x))2 +(g(x)v(x))2:
8,x3cx58x54x33x55x58x0x56x57x17x2cx49x7ex7fx18x2x78:
(1) x3?3x?1; (2) x3 +x2?2x?1;
(3) x4 +x?1; (4) x4 +4x3?12x+9.
x29,(1)x54x33x55x2cx2e,x3?3x?1,2?1,2+1,1,x5bxex79x34x17x72:
1?2?1 0 1 2 +1
f0(x) + + +
f1(x) + + 0? 0 + +
f2(x) + + + +
f3(x) + + + + + + +
V(x) 3 3 2 1 1 0 0
x1bx6cx75,f(x)x243x76x2x78,x2x78x14x78x21(?2;?1),(?1;0),(1;2).
(2)x54x33x55x2cx2e,x3 +x2?2x?1,3x2 +2x?2,2x+1,1.
x2x78x79x2e3,x2x78x14x78x21(?2;?1),(?1;0),(1;2).
(3)x54x33x55x2cx2e,x4 +x?1,4x3 +1,?3x?4,?1.
x2x78x79x2e2,x2x78x14x78x21(?2;?1),(0;1).
(4)x54x33x55x2cx2e,x4 +4x3?12x+9,x3 +3x2?3,x2 +3x?4,?4x?3,1,xax2x78.
x25 x26 10–9
1,x3cx4cx17x2cx49x7ex7fx18x24x0x78:
(1) x5?7x3?12x2 +6x+36; (2) 6x4 +19x3?7x2?26x+12;
(3) 10x4?13x3 +15x2?18x?15;
(4) x6?6x5 +11x4?x3?18x2 +20x?8.
x29,(1) 3,?2.
(2)?3,12,
(3)? 12,
(4) 2,2,2.
2,x5ex5fx17x2cx49x7ex7fx71x24x0x79x1x7x60x40x8:
(1) x4?8x3 +12x2?6x+2; (2) x5?12x3 +36x?12;
(3) x4?x3 +2x+1; (4) x4 +4kx+1,kx2ex9x79
(5) xp +px+1,px2ex1dxbx79; (6) x4 +5x3?3x2?5x+1.
x27x28,(1)x51p = 2,x1bx58x59x5ax5bx1fx15x7dx71x75,f(x)x60x40x8.
(2)x51p = 3,x1bx58x59x5ax5bx1fx15x7dx71x75,f(x)x60x40x8.
￠ 16 ￠
(3) f(y +1) = y4 +3y3 +3y2 +3y +3,x51p = 3,x1bx58x59x5ax5bx1fx15x7dx71x75,f(y +1)x60x40x8,x33f(x)
x60x40x8.
(4) f(y +1) = y4 +4y3 +6y2 +4(k +1)y +2(2k +1),x51p = 2,x1bx58x59x5ax5bx1fx15x7dx71x75,f(y +1)
x60x40x8,x33f(x)x60x40x8.
(5)
f(y?1) = (y?1)p +p(y?1)+1 =
pX
k=0
Ckp(?1)kyp?k +p(y?1)+1
=
p?1X
k=0
Ckp(?1)kyp?k +py?p
= yp?pyp?1 + p(p?1)2 yp?2 +￠￠￠+ p(p?1)2 y2 +2py?p:
x1bx58x59x5ax5bx1fx15x7dx71x75,f(y?1)x60x40x8,x33f(x)x60x40x8.
(6)x1fx2ef(x)xax24x0x78,x33x11f(x)x40x8,xcx44x24
f(x) = (x2 +ax+1)(x2 +bx+1) x6b f(x) = (x2 +ax?1)(x2 +bx?1):
x14x15x1ax7f,x6fx70x2d3x1ax7ex2e1x1ax7ex73x79,x4fa+b = 5,a+b =?5,x60x40x55.
x14x1cx7f,x53x = 1,x4fab =?1,x43a+b = 5,x3bx60x40x55.
x33f(x)x60x40x8.
3,x3cx28x17x2cx13x7fx18x13x5cx24x0x42:
(1) 11+ 3p2+2 3p4 ; (2) 11? 4p2+p2 ;
(3) 11+p2?p3 ;
(4) a2?3a?1a2 +2a+1,x2dx2a,ax2ex3ex6ax3 +x2 +3x+4 = 0x18x78.
x29,(1)x36x37f(x) = 2x2 +x+1x2eg(x) = x3?2,x76x75(f(x);g(x)) = 1,x5dx6fx70x75
(2x2 +x+1) x
2 +7x?3
23?(x
3?2)?2x+13
23 = 1:
x10x16
1
1+ 3p2+2 3p4 =
1
23 (?3+7
3p2? 3p4):
(2) 11? 4p2+p2 = 17 (1+3 4p2+2p2? 4p8).
(3) 11+p2?p3 = 14 (2+p2+p6).
(4) a2?3a?1a2 +2a+1 = 17a2?3a+55.
4,xdf(x)x21x66x76x9x73x79x49x7ex7f,x5ex5f,x34x35f(0)x75f(1)x32x21x1dx79,xcf(x)xax9x79x78.
x27x28,x3ex5e,x34f(x)x24x9x79x78a,xcf(x) = (x?a)g(x),x2dx2ag(x)x2ex9x73x79x49x7ex7f,xc0?ax1c1?a
x2ax5ex5fx24x66x76x21x1bx79,x49x4af(0);f(1)x2ax5ex5fx24x66x76x2ex1bx79,x1ex1f.
5,xdf(x) = x3 +bx2 +cx+dx21x66x76x9x73x79x49x7ex7f,x5ex5f,x34x35bd+cdx2ex1dx79,xcf(x)x71x24x0x79
x1x7x60x40x8.
x27x28,x1bx4fxd,dx1cb+cx32x21x1dx79,x49x4af(0) = dx16x2ef(1) = 1+b+c+dx60x2ex1dx79,x33f(x)xax9
x79x78,x43x1ff(x)x18x21x7ex73x79x2e1,x46degf(x) = 3,x10x16f(x)x60x40x8.
6,x74x75x9x73x79x49x7ex7ff(x) = a0xn +a1xn?1 +￠￠￠+anxax24x0x78,x5ex5f,x34x35x24xbx79p,x4e
￠ 17 ￠
(1) p-a0;
(2) p j ai;i = 2;3;￠￠￠ ;an;
(3) p2 -an
xcf(x)x71Qx7x60x40x8.
x27x28,x34p j a1,xcx1bx58x59x5ax5bx1fx15x7dx71x75f(x)x71Qx7x60x40x8.
x16x17xdp-a1,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],x1bx15f(x)xax24x0x78,x1fx6c2 6 degg(x) 6 n?2,
2 6 degh(x) 6 n?2,xd
g(x) = b0xk +b1xk?1 +￠￠￠+bk; k > 2;m > 2;
h(x) = c0xm +c1xm?1 +￠￠￠+cm; k +m = n:
x1bx15bkcm = an,p j an,p2 -an,x40xdp j bk,p-cm,x43x1fp-b0,xdblx21x49x61x62x63x5ex7bx66x76x60x55x29px9x22
x18x73x79,xc
p-am+l = cmbl +cm?1bl+1 +￠￠￠
x41x1fm+l > 2,p j am+l,x1ex1f,x1fx6cf(x)x71Qx7x60x40x8.
7,x3cx66x58x10x24x18x9x79m,x4ex5 +mx?1x71x24x0x79x1x7x40x8.
x27x28,(a)x34m = 0,xcx5?1x38x13x40x8.
(b)x34f(x)x24x66x1ax1fx7f,xc1+m?1 = 0x6b?1?m?1 = 0,x49x4am = 0x6b?2.
(c)x11f(x)x60x64x66x1ax1fx7f,x41x40x8,xcx40xd
x5 +mx?1 = (x2 +ax§1)(x3 +bx2 +cxcurrency11):
x50x51x40x52x73x79,x4f
a+b = 0; ab+c§1 = 0; ac§bcurrency11 = 0; currency1(a?c) = m:
x33b =?a,8
><
>:
a2 +c = currency11
accurrency1a = §1
m = currency1(a?c)
x71x65x66x66x7dx70x17,c = 0,a =?1,m = 1;x71x65x19x66x7dx70x17,c = 2,a(c+2) =?1,x60x40x55.x10x16mx18x40x55
x51x52x2e0,1,?2,x71x6c3x66x7dx67x17x5 +mx?1x32x40x8.
8,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x9x79,x5ex5f,x49x7ex7f
f(x) = (x?a1)(x?a2)￠￠￠(x?an)?1
x71Qx7x60x40x8.
x27x28,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],degg(x);degh(x) < degf(x),xcf(ai) = g(ai)h(ai) =
1,x33g(ai) =?h(ai) = §1,x49x4ag(ai)+h(ai) = 0,x15x21x49x7ex7f
F(x) = g(x)+h(x)
x24nx76x60x61x18x78,x41degF(x) < n,x6cx55F(x) = 0,g(x) =?h(x),f(x) =?g2(x),x4ax50xx43x13x7cx53,x24
f(x) > 0,?g2(x) 6 0,x1ex1f,x1fx6c
9,xda1;a2;￠￠￠ ;anx2ex1x60x18x61x18x9x79,x5ex5f,x49x7ex7f
f(x) = (x?a1)2(x?a2)2￠￠￠(x?an)2 +1
x71Qx7x60x40x8.
￠ 18 ￠
x27x28,xdf(x) = g(x)h(x),g(x);h(x) 2Z[x],x46
0 < degg(x) < 2n; 0 < degh(x) < 2n:
x43x1fdegg(x) + degh(x) = 2n,x33g(x);h(x)x2ax5ex5fx24x66x76x18x1ax796 n,x60x68xddegh(x) 6 n,x43xd
g(x);h(x)x60x2ex21x66x49x7ex7f.
x1bx15f(x)x71x2x79x7x69x6ax51x57x52,x1fx6cf(x)xax2x78,g(x);h(x)x6bxax2x78,x15x21g(x);h(x)x71x2x79x7
x69x6ax51x57x52.x43x1ff(ai) = 1,x33h(ai) = g(ai) = 1,h(x)?1x24nx76x60x61x18x78a1;￠￠￠ ;an,x10x16
h(x) = (x?a1)￠￠￠(x?an)+1:
x49x4adegg(x) = n,x3ax4a
g(x) = (x?a1)￠￠￠(x?an)+1:
x15x21
g(x)h(x) = [(x?a1)￠￠￠(x?an)+1]2
= (x?a1)2￠￠￠(x?an)2 +2(x?a1)￠￠￠(x?an)+1 6= f(x);
x1ex1f,x1fx6cf(x)x60x40x8.
10,xdx7ex72x49x7ex7ff(x)x71x24x0x79x1x7x60x40x8,x5ex5f,f(x2)x71x24x0x79x1x7x40x8x18x43x13x44x45x46x47x21
x44x71x9x79c 6= 0x2ex9x73x79x49x7ex7fg(x);h(x),x4e
cf(x) = g2(x)?xh2(x):
x27x28,x43x13x28x38x13,x16x17x5ex44x45x28.
xdg(x)x2ef(x2)x18x34x66x60x40x8x1fx7f,xcx1bg(x) j f(x2)x40x4fg(?x) j f(x2),x38x13g(?x)x3bx60x40x8.
g(x)x1cg(?x)x18x29x73x61x24x16x173x66x40x55:
(a) g(x) = g(?x); (b) g(x) =?g(?x); (3) (g(x);g(?x)) = 1.
(a) x34 g(x) = g(?x),xc g(x) = h(x2),x1b h(x2) j f(x2) x4f h(x) j f(x),x4a f(x) x60x40x8,x10x16
h(x) = cf(x),g(x) = cf(x2),x1cf(x2)x40x8x1ex1f,x1fx6cg(x) 6= g(?x).
(b)x34g(x) =?g(?x),xcg(x) =?xh(x2),xh(x2) j f(x2),x33x j f(x),x15x21§f(x) =?x = 02?x￠12,
x38x39x3ax3b.
(c)x34(g(x);g(?x)) = 1,xcg(x)g(?x) j f(x2),xdg(x) = u(x2)+xv(x2),xc
g(x)g(?x) = u2(x2)?x2v2(x2):
x4au2(x2)?x2v2(x2) j f(x2),x1fx6c
u2(x)?xv2(x) j f(x):
x33x44x71c 6= 0x4ecf(x) = u2(x)?xv2(x),x5ex6c.
11,x5ex5f,x14x10x24x18x57x9x79n,f(x) = x2n?x2n?1 +1x71x24x0x79x1x7x60x40x8,(x6dx6e,x6fnx70x71x72x73
x74x75x70x76x210)
x27x28,x21x7bx45x77x4ex4f10x18x38x39x3cx78x2e,x50f(x)x21x7ex72x49x7ex7fx53,x40x51c = 1,x2ex5ex6bx66x38x39,x36x37
f(x2) = c?1(g2(x2)?x2h2(x2)) = c?1(g(x2)+xh(x2))(g(x2)?xh(x2));
x79x35x3ex11g(x2)+xh(x2) = r(g1(x2)+xh1(x2)),x2dx2ag1(x2)+xh1(x2)x21x7ex72x49x7ex7f,xcg1(x2)?xh1(x2)
x3bx21x7ex72x49x7ex7f,x15x21
f(x2) = c?1r2(g1(x2)+xh1(x2))(g1(x2)?xh1(x2)) = c?1r2(g21(x2)?x2h21(x2));
￠ 19 ￠
x78x2dx7ax5ax7bx0,c?1r2 = 1,x15x21f(x) = g21(x)?xh21(x).
x14nx58x36x37x71,x30x35x58x3cx78x7cx18x4ex4f10.
x50n = 1x53,x76x75x2?x+1x71x24x0x79x1x7x60x40x8.
x39xdx2n?x2n?1 +1x71x24x0x79x1x7x60x40x8,x4ax2n+1?x2n +1x71x24x0x79x1x7x40x8,xcx78x2dx3cx78x18x4e
x4f10,x44x71g(x);h(x) 2Z[x],x4e
x2n?x2n?1 +1 = g2(x)?xh2(x);
x40x52x4cx3dx4f
2nx2n?1?2n?1x2n?1?1 = 2g(x)g0(x)?h2(x)?2xh(x)h0(x):
xc2 j h2(x),2 j h(x),x10x16
x2n?x2n?1 +1 = g2(x)+4p(x):
x53
g(x) = x2n?1?x2n?2 +1+k(x)+2l(x);
x2dx2ak(x)x18x16x7ex73x79x32x210x6b1,xc
x2n?x2n?1 +1 = x2n?x2n?1 +1+4x2n?1?2x2n?2?2x3￠2n?2 +k2(x)+4p2(x):
x1fx6c2 j k(x),4 j k2(x),x3ax4a
x2n?x2n?1 +1 = x2n?x2n?1 +1?2x2n?2?2x3￠2n?2 +4p3(x);
4p3(x) = 2(x2n?2 +x3￠2n?2;
x6bx60x40x55,x49x4ax75x2n+1?x2n +1x71x24x0x79x1x7x60x40x8.
￠ 20 ￠
