1Ù �õª�Ϫ©)
§ 1 �õª
1. O (x2 + ax− b)(x2 − 1) + (x2 − ax+ b)(x2 + 1).
): 2x4 − 2ax+ 2b.
2. Oõª x3 + 2x2 + 3x− 1 3x2 + 2x+ 4�¦È.
): 3x5 + 8x4 + 17x3 + 11x2 + 10x− 4.
3. �
f(x) = 3x2 − 5x+ 3,
g(x) = ax(x− 1) + b(x+ 2)(x− 1) + cx(x+ 2),
Á(½ a, b, c, ¦ f(x) = g(x).
): � x = −2, � a = 25
6
; � x = 0, � b = −3
2
, � x = 1, � c =
1
3
.
4. � f(x), g(x)Ú h(x)Ñ´¢Xêõª, y²: XJ
f 2(x) = xg2(x) + xh2(x),
@o
f(x) = g(x) = h(x) = 0.
y²: X f(x) 6= 0, Kª�gêóê,
mª�gêÛê, gñ, �
f(x) = 0. l
g2(x) + h2(x) = 0.
q, g(x), h(x) �¢Xêõª, l
g2(x), h2(x) �ÄXêÑ´Kê,
ùüêÚ", � g(x), h(x)�ÄXêÑ´", l
g(x) = h(x) = 0.
· 132 · 1Ù �õª�Ϫ©)
§ 2 �Ø�Vg
1. ^ g(x)Ø f(x), ¦û q(x) {ª r(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.
): (1) q(x) = x2 − x+ 4, r(x) = −4x+ 2.
(2) q(x) =
1
9
(3x+ 11), r(x) =
10
9
(x− 2).
2. m, p, q·Üo^, k
(1) x2 +mx+ 1 | x3 + px+ q;
(2) x2 +mx+ 1 | x4 + px2 + q.
): (1) p = 1−m2, q = −m.
(2)
m = 0p = 1 + q ½
p = −m
2 + 2
q = 1
3. ^nÜØ{¦û q(x) 9{ª r(x):
(1) f(x) = x4 − 2x3 + 4x2 − 6x+ 8, g(x) = x− 2;
(2) f(x) = 2x5 − 5x3 − 8x, g(x) = x+ 2.
): (1) q(x) = x3 + 4x+ 2, r(x) = 12.
(2) q(x) = 2x4 − 4x3 + 3x2 − 6x+ 4, r(x) = −8.
4. ^nÜØ{L f(x) x− x0 �:
(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.
): (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. P 〈x〉0 = 1, 〈x〉k = x(x−1)(x−2) · · · (x−k+1), (k > 1). Áò f(x)
L
c0 + c1〈x〉+ c2〈x〉2 + · · ·
�/ª:
(1) f(x) = x4 − 2x3 + x2 − 1;
(2) f(x) = x5.
§ 2 �Ø�Vg · 133 ·
): (1) 1 1 −2 1 0 −1
1 −1 0
2 1 −1 0 0
2 2
3 1 1 2
3
1 4
Ïdf(x) = −1 + 2〈x〉2 + 4〈x〉3 + 〈x〉4.
(2) f(x) = 〈x〉+ 15〈x〉2 + 25〈x〉3 + 10〈x〉4 + 〈x〉5.
6. k´��ê, y²: x | fk(x)�
=� x | f(x);
y²: � f(x)�~ê a,K fk(x)�~ê ak. Ïd x | fk(x) ⇐⇒
ak = 0 ⇐⇒ a = 0 ⇐⇒ x | f(x).
7. �a, büØ��~ê, y²: õª f(x)� (x− a)(x− b)ؤ
�{ª
f(a)− f(b)
a− b x+
af(b)− bf(a)
a− b .
y²: � f(x) = (x− a)(x− b)q(x) +Ax+B, K
f(a) = aA+B, f(b) = bA+B,
dd�
A =
f(a)− f(b)
a− b , B =
af(b)− bf(a)
a− b .
Ïd(ؤá.
8. � f1(x), f2(x), g1(x), g2(x)Ñ´ê K þ�õª, Ù¥ f1(x) 6= 0.
y²: XJ g1(x)g2(x) | f1(x)f2(x), f1(x) | g1(x), K g2(x) | f2(x).
y²: � f1(x)f2(x) = g1(x)g2(x)q1(x), g1(x)=f1(x)q2(x). Kf1(x)f2(x)
= f1(x)q2(x)g2(x)q1(x), du f1(x) 6= 0, � f2(x) = g2(x)q2(x)q1(x), =
g2(x) | f2(x).
∗9. y²: xd − 1 | xn − 1�
=� d | n.
y²: (⇒) e n = dq, K
xn − 1 = (xd − 1)(xd(q−1) + xd(q−2) + · · ·+ xd + 1).
Ïd xd − 1 | xn − 1.
(⇐) � n = dq + r, 0 ≤ r < d. dþy, xdq − 1 ≡ 0 (mod xd − 1). =
xdq ≡ 1 (mod xd − 1),
· 134 · 1Ù �õª�Ϫ©)
xn ≡ xdq+r ≡ xdq · xr ≡ xr (mod xd − 1),
xn − 1 ≡ xr − 1 (mod xd − 1).
xd − 1 | xr − 1⇔ r = 0, Ïd xd − 1 | xn − 1⇔ r = 0⇔ d | n.
§ 3 úϪ
1. ¦úϪ (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) = x4 + 2x3 + x2 − 2;
(3) f(x) = x4 − x3 − 4x2 + 4x+ 1, g(x) = x2 − x− 1.
): (1) x+ 1.
(2) 1.
(3) 1.
2. ¦ u(x), v(x), ¦ u(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.
): (1) u(x) = −x− 1, v(x) = x+ 2, d(x) = x2 − 2.
(2) u(x) = −1
3
(x− 1), v(x) = 1
3
(2x2 − 2x− 3), d(x) = x− 1.
(3) u(x) = −1
6
(2x2 + 3x), v(x) =
1
6
(2x3 + 5x2 − 6), d(x) = 1.
3. y²: XJ d(x) | f(x), d(x) | g(x),
d(x) f(x) g(x)�|
Ü, @o d(x)´ f(x) g(x)�úϪ.
y²: � d(x) = u(x)f(x) + v(x)g(x), Ké?¿� h(x) ∈ K[x], X
h(x) | f(x), h(x) | g(x), K h(x) | d(x).
q, d(x) f(x) g(x)�úϪ, � d(x)´ f(x) g(x)�
úϪ.
4. y²: XJ h(x)Äõª, K
(f(x)h(x), g(x)h(x)) = (f(x), g(x))h(x).
y²: � d(x) = (f(x), g(x)) 6= 0, K3 u(x), v(x)¦
d(x) = u(x)f(x) + v(x)g(x).
¤±
d(x)h(x) = u(x)f(x)h(x) + v(x)g(x)h(x).
§ 3 úϪ · 135 ·
qÏ d(x)h(x) | f(x)h(x), d(x)h(x) | g(x)h(x), ¤± d(x)h(x)´ f(x)h(x)
g(x)h(x)�úϪ. qÏ d(x), h(x)Ñ´Äõª, � d(x)h(x)
´Äõª, l
(f(x)h(x), g(x)h(x)) = d(x)h(x) = (f(x), g(x))h(x).
qX d(x) = 0, K f(x) = g(x) = 0, ��ªE,¤á.
5. y²: XJ f(x), g(x)Ø�", K(
f(x)
(f(x), g(x))
,
g(x)
(f(x), g(x))
)
= 1.
y²: Ï f(x), g(x)Ø�", � (f(x), g(x)) 6= 0. ¤±
(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))
(dSK 4) ü>� (f(x), g(x)), �(
f(x)
(f(x), g(x))
,
g(x)
(f(x), g(x))
)
= 1.
6. y²: XJ f(x), g(x)Ø�",
u(x)f(x) + v(x)g(x) = (f(x), g(x)),
K (u(x), v(x)) = 1.
y²: Ï f(x), g(x)Ø�", � (f(x), g(x)) 6= 0, Ïd
u(x)
f(x)
(f(x), g(x))
+ v(x)
g(x)
(f(x), g(x))
= 1,
(u(x), v(x)) = 1.
7. y²: XJ (f(x), g(x)) = 1, (f(x), h(x)) = 1, @o
(f(x), g(x)h(x)) = 1.
y²: 3 u(x), v(x), s(x), t(x), ¦
u(x)f(x) + v(x)g(x) = 1,
· 136 · 1Ù �õª�Ϫ©)
s(x)f(x) + t(x)h(x) = 1,
¤±
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. � f1(x), · · · , fm(x), g1(x), · · · , gn(x)Ñ´õª,
(fi(x), gj(x)) =
1 (i = 1, · · · ,m; j = 1, · · · , n), y²:
(f1(x)f2(x) · · · fm(x), g1(x)g2(x) · · · gn(x)) = 1.
y²: d (fi(x), gj(x)) = 1, � (fi(x), g1(x)g2(x)) = 1, . . . ,
(fi(x), g1(x)g2(x) · · · gn(x)) = 1. l
(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. y²: XJ (f(x), g(x)) = 1, @o (f(x) + g(x), f(x)g(x)) = 1.
y²: du (f(x), g(x)) = 1, ¤±
(f(x) + g(x), g(x)) = (f(x), g(x)) = 1,
(f(x) + g(x), f(x)) = (g(x), f(x)) = 1,
Ïd
(f(x) + g(x), f(x)g(x)) = 1.
10. � f1(x) = af(x)+ bg(x), g1(x) = cf(x)+ dg(x),
ad− bc 6= 0, y
²:
(f(x), g(x)) = (f1(x), g1(x)).
y²: dK�� (f(x), g(x)) | (f1(x), g1(x)). q
f(x) =
d
ad− bcf1(x)−
b
ad− bcg1(x),
g(x) =
−c
ad− bcf1(x) +
a
ad− bcg1(x),
¤±
(f1(x), g1(x)) | (f(x), g(x)).
qÏ (f1(x), g1(x)) (f(x), g(x))�ÄXêÓ, �
(f(x), g(x)) = (f1(x), g1(x)).
§ 3 úϪ · 137 ·
11. y²: XJ f(x) g(x)p, @o f(xm) g(xm)p.
y²: dK�, 3õª u(x), v(x)¦
u(x)f(x) + v(x)g(x) = 1.
¤±
u(xm)f(xm) + v(xm)g(xm) = 1.
� (f(xm), g(xm)) = 1.
12. y²: é?¿���ê n, Ñk
(f(x), g(x))n = (fn(x), gn(x)).
y²: � (f(x), g(x)) = d(x), f(x) = d(x)f1(x), g(x) = d(x)g1(x), K
(f1(x), g1(x)) = 1.
dSK 8 �
(fn1 (x), g
n
1 (x)) = 1.
u´
(fn(x), gn(x)) = (dn(x)fn1 (x), d
n(x)gn1 (x))
= dn(x)(fn1 (x), g
n
1 (x)) = d
n(x)
= (f(x), g(x))n.
∗13. Á¦ xm − 1 xn − 1�úϪ.
): - d = (m,n), KâSK 10–2.9, xd − 1 | xm − 1, xd − 1 | xn − 1.
� h(x)´ xm − 1 xn − 1�úϪ, Kk
xm − 1 ≡ 0 (mod h(x)), xn − 1 ≡ 0 (mod h(x))
=⇒ xm ≡ 1 (mod h(x)), xn ≡ 1 (mod h(x)).
du d = (m,n), Ïd3 u, v ∈ Z¦� d = um+ vn.
xd = xum+vn ≡ 1 (mod h(x)) =⇒ xd − 1 ≡ 0 (mod h(x)).
q� d = ms− nt, s, t ≥ 0, K d+ nt = ms. u´
xms − 1 = xd+nr − 1 = (xd − 1)xnr + xnr − 1.
e f(x) ∈ K[x] ÷v f(x) | xm − 1, f(x) | xn − 1, K (f(x), x) = 1,
f(x) | xms − 1, f(x) | xnt − 1, u´ f(x) | (xd − 1)xnr. d f(x) xp
� f(x) | xd − 1. Ïd (xm − 1, xn − 1) = xd − 1, Ù¥ d = (m,n).
· 138 · 1Ù �õª�Ϫ©)
∗14. y²:
f(x)
(f(x), g(x))
,
g(x)
(f(x), g(x))
�gêÑu", Ò±·�
ÀJ·Ü�ª
u(x)f(x) + v(x)g(x) = (f(x), g(x))
� u(x) v(x), ¦
deg u(x) < deg
(
g(x)
(f(x), g(x))
)
, deg v(x) < deg
(
f(x)
(f(x), g(x))
)
.
y²: 3õª s(x), t(x) ∈ K[x]¦
s(x)f(x) + t(x)g(x) = (f(x), g(x)).
K
s(x)
f(x)
(f(x), g(x))
+ t(x)
g(x)
(f(x), g(x))
= 1. (∗)
-
s(x) =
g(x)
(f(x), g(x))
q(x) + u(x),
Ù¥ u(x) = 0½ deg u(x) < deg
g(x)
(f(x), g(x))
. P v(x) =
f(x)
(f(x), g(x))
q(x) +
t(x), Kd (*),
u(x)
f(x)
(f(x), g(x))
+ v(x)
g(x)
(f(x), g(x))
= 1. (∗∗)
db�,
f(x)
(f(x), g(x))
g(x)
(f(x), g(x))
�gêÑu", ¤± u(x), v(x)ÑØ´
"õª. u´
deg u(x) < deg
g(x)
(f(x), g(x))
.
d (**)
deg
(
u(x)
f(x)
(f(x), g(x))
)
= deg
(
v(x)
g(x)
(f(x), g(x))
)
,
l
deg v(x) < deg
f(x)
(f(x), g(x))
.
§ 4 ؽ§Ó{ª · 139 ·
§ 4 ؽ§Ó{ª
1. � (f(x),m(x)) = 1, y²: é?Û�õª g(x), Ñ3õª h(x),
¦
h(x)f(x) ≡ g(x) (mod m(x)).
y²: db�, 3 u(x), v(x) ∈ K[x], ¦
u(x)f(x) + v(x)m(x) = 1.
¤±
g(x)u(x)f(x) + g(x)v(x)m(x) = g(x).
u´
g(x)u(x)f(x) ≡ g(x) (mod m(x)).
- h(x) = g(x)u(x), K
h(x)f(x) ≡ g(x) (mod m(x)).
∗2. � m1(x), · · · ,ms(x) |üüp�õª, y²: é?Û�õ
ª f1(x), · · · , fs(x), Ñ3õª F (x), ¦
F (x) ≡ fi(x) (mod mi(x)), i = 1, · · · , s.
y²: -M(x) = m1(x)m2(x) · · ·ms(x), Ri(x) = M(x)
mi(x)
.
K (Ri(x),mi(x)) = 1, mj(x) | Ri(x), i 6= j. 3 hi(x)¦ (SK 1)
hi(x)Ri(x) ≡ fi(x) (mod mi(x))
-
F (x) =
s∑
i=1
hi(x)Ri(x),
K
F (x) ≡
s∑
i=1
hi(x)Ri(x) (mod mk(x))
≡ hk(x)Rk(x) (mod mk(x))
≡ fk(x) (mod mk(x)).
· 140 · 1Ù �õª�Ϫ©)
∗3. � m(x) EXêõª,
m(0) 6= 0. y²: 3EXêõª
f(x), ¦
f 2(x) ≡ x (mod m(x)).
y²: (a) Äky²é?¿� a 6= 0, Ó{ª
f 2(x) ≡ x (mod (x− a)m)
k). �
√
a´ a�?¿², K
(x− a)m = ((√x−√a)(√x+√a))m = (√x−√a)m(√x+√a)m
= (h(x)
√
x− g(x))(h(x)√x+ g(x)) = h2(x)x− g2(x).
u´
g2(x) ≡ h2(x)x (mod (x− a)m)
h(a)
√
a + g(a) = (
√
a +
√
a)m 6= 0,
h(a)√a − g(a) = (√a − √a)m =
0, Ïd g(a)h(a) 6= 0, l
(h(x), (x − a)m) = 1, 3 h1(x) ∈ K[x] ¦
h1(x)h(x) ≡ 1 (mod (x− a)m). u´
(h1(x)g(x))
2 ≡ x (mod (x− a)m)
� f(x) = h1(x)g(x), Kk
f 2(x) ≡ x (mod (x− a)m).
(b) �m(x) = (x− a1)m1(x− a2)m2 · · · (x− as)ms , ai 6= aj é i 6= j. K
(x− a1)m1 , · · · , (x− as)ms üüp. d (a), 3 fi(x) ∈ K[x], ¦
f 2i (x) ≡ x (mod (x− ai)mi).
dSK 2, 3 f(x)¦
f(x) ≡ fi(x) (mod (x− ai)mi)
u´
f 2(x) ≡ x (mod (x− ai)mi)
d (x− a1)m1 , · · · , (x− as)ms üüp�
f 2(x) ≡ x (mod m(x)).
§ 5 Ϫ©)½n · 141 ·
§ 5 Ϫ©)½n
1. y²: gm(x) | fm(x) ⇐⇒ g(x) | f(x).
y²: �
f(x) = apl11 (x)p
l2
2 (x) · · · plss (x),
g(x) = bpk11 (x)p
k2
2 (x) · · · pkss (x),
Ù¥ a, b ∈ K, p1(x), · · · , ps(x) ´üüp�Ø�õª,
li, ki ≥ 0,
i = 1, · · · , s. K
g(x) | f(x) ⇐⇒ ki ≤ li, i = 1, · · · , s
⇐⇒ mki ≤ mli, i = 1, · · · , s
⇐⇒ gm(x) | fm(x).
2. � f(x), g(x) ∈ K[x],
k©)ª
f(x) = apr11 (x)p
r2
2 (x) · · · prss (x), ri > 0, i = 1, · · · , s;
g(x) = bpt11 (x)p
t2
2 (x) · · · ptss (x), ti > 0, i = 1, · · · , s,
Ù¥ p1(x), · · · , ps(x)´ØÓ�ÄØ�õª. y²:
[f(x), g(x)] = pmax(r1,t1)1 (x)p
max(r2,t2)
2 (x) · · · pmax(rs,ts)s (x).
y²: -mi = max(ri, ti), i = 1, · · · , s.
m(x) = pm11 (x)p
m2
2 (x) · · · pmss (x),
KÏ ri ≤ mi, ti ≤ mi, Ïd
f(x) | m(x), g(x) | m(x) =⇒ [f(x), g(x)] | m(x).
� s(x) ∈ K[x]´ f(x), g(x)�ú�ª, Kk
s(x) = pl11 (x)p
l2
2 (x) · · · plss (x)h(x),
li ≤ ri, li ≤ ti, (h(x), pi(x)) = 1, i = 1, · · · , s.
u´
li ≥ max(ri, ti), i = 1, · · · , s, =⇒ m(x) | s(x).
· 142 · 1Ù �õª�Ϫ©)
Ïd
[f(x), g(x)] = pm11 (x)p
m2
2 (x) · · · pmss (x).
3. � f(x), g(x) ∈ K[x]Ñ´Äõª, y²:
[f(x), g(x)] =
f(x)g(x)
(f(x), g(x))
.
y²: �
f(x) = pr11 (x)p
r2
2 (x) · · · prss (x), ri > 0, i = 1, · · · , s;
g(x) = pt11 (x)p
t2
2 (x) · · · ptss (x), ti > 0, i = 1, · · · , s,
Ù¥ p1(x), · · · , ps(x)´ØÓ�ÄØ�õª. -
mi = max(ri, ti), li = min(ri, ti), i = 1, · · · , s.
K
f(x)g(x) = pr1+t11 (x)p
r2+t2
2 (x) · · · prs+tss (x),
(f(x), g(x)) = pl11 (x)p
l2
2 (x) · · · plss (x),
du ri + ti − li = mi, i = 1, · · · , s. Ïd
f(x)g(x)
(f(x), g(x))
= pm11 (x)p
m2
2 (x) · · · pmss (x) = [f(x), g(x)].
4. ¦e�õª��ú�ª:
(1) f(x) = x4 − 4x3 + 1, g(x) = x3 − 3x2 + 1;
(2) f(x) = x4 − x− 1 + i. g(x) = x2 + 1.
): (1) du (f(x), g(x)) = 1, [f(x), g(x)] = f(x)g(x) = x7 − 7x6 +
12x5 + x4 − 3x3 − 3x2 + 1.
(2) du (f(x), g(x)) = x− i, [f(x), g(x)] = f(x)(x+i) = x5+ix4−x2−
x− (1 + i).
5. � p(x)´gêu"�õª. y²: XJéu?Ûõª f(x), g(x),
d p(x) | f(x)g(x) ±íÑ p(x) | f(x)½ö p(x) | g(x), K p(x)´Ø�õ
ª.
y²: e p(x) �, K3gê�u p(x) �~êõª f(x), g(x) ¦
p(x) = f(x)g(x). l
p(x) | f(x)g(x). �Ï
deg f(x) < deg p(x), deg g(x) < deg p(x),
§ 6 Ϫ · 143 ·
p(x) ∤ f(x), p(x) ∤ g(x), b�gñ, Ïd p(x)Ø�.
∗6. y²: gêu 0�Äõª f(x)´,Ø�õª��¿
©7^´, é?¿�õª g(x)½ök (f(x), g(x)) = 1, ½öé,��
êm, f(x) | gm(x).
y²: (⇒) � f(x) = pm(x), Ù¥ p(x) Ø�, Ke g(x) ∈ K[x] ÷v
p(x) | g(x), k
f(x) = pm(x) | gm(x).
X p(x) ∤ g(x), K (p(x), g(x)) = 1, l
(pm(x), g(x)) = 1, = (f(x), g(x)) =
1.
(⇐) � p(x) ´ f(x)�ÄØ�Ïf, K (p(x), f(x)) = p(x), l
3,��ê m, ¦ f(x) | pm(x), ù`² p(x)´ f(x)�Ø�Ï
f. ¤± f(x) = cpr(x). qÏ f(x), p(x)�ÄXêÑ´ 1, � c = 1. l
f(x) = pr(x).
∗7. y²: gêu 0 �Äõª f(x) ´,Ø�õª��
¿©7^´, é?¿�õª g(x), h(x), d f(x) | g(x)h(x) ±íÑ
f(x) | g(x), ½öé,��ê m, f(x) | hm(x).
y²: (⇒) � f(x) = pm(x), Ù¥ p(x)´ÄØ�õª, Kd f(x) |
g(x)h(x), � p(x) | g(x)h(x), l
p(x) | g(x)½ p(x) | h(x). u´ f(x) =
pm(x) | gm(x)½ f(x) = pm(x) | hm(x).
(⇐) � p(x)´ f(x)�ÄØ�Ïf, K f(x) = p(x)f1(x). l
f(x) | p(x)f1(x).
f(x) ∤ f1(x), l
3,��ê m, ¦ f(x) | pm(x),
ù`² p(x)´ f(x)�Ø�Ïf. ¤± f(x) = cpr(x). qÏ f(x), p(x)
�ÄXêÑ´ 1, � c = 1. l
f(x) = pr(x).
§ 6 Ϫ
1. �Oe�knXêõªkÃϪ, ek, K¦ÑϪ:
(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.
): (1) x+ 1, 4.
(2) x− 2, 3.
(3) x2 − 2x+ 2, 2.
· 144 · 1Ù �õª�Ϫ©)
(4) x+ 3, 2, x− 1, 3.
2. a, b A÷vo^, e�õªkϪ?
(1) f(x) = x3 + 3ax+ b; (2) f(x) = x4 + 4ax+ b.
): (1) � a = b = 0 k 3Ϫ x, � 4a3 = −b2
a 6= 0, k 2Ϫ
2ax+ b.
(2) � a = b = 0 k 4 Ϫ x, � 27a4 = b3
a 6= 0, k 2 Ϫ
3ax+ b.
3. � p(x)´ f ′(x)� k Ϫ, UÄ` p(x)´ f(x)� k + 1Ϫ,
o?
): ØU. ÏqU f ′(x)?ϪÑØ´ f(x)�Ϫ. ~X f(x) =
x4 − 1, f ′(x) = 4x3.
4. y²: XJ (f ′(x), f ′′(x)) = 1, @o, f(x)�ϪѴ f(x)��
Ϫ.
y²: du (f ′(x), f ′′(x)) = 1, f ′(x)�?ϪÑØ´ f ′′(x)�Ϫ. �
p(x)´ f(x)�Ϫ, K p(x) | f ′(x), u´ p(x) ∤ f ′′(x), `² p(x)´ f ′(x)
�üϪ, � p(x)´ f(x)��Ϫ.
5. y²: K[x]¥Ø�õª p(x)´ f(x) ∈ K[x]� k (k > 1) Ϫ
�¿©7^´ p(x) ´ f(x), f ′(x), · · · , f (k−1)(x) �Ϫ, �Ø´ f (k)(x)
�Ϫ.
y²: (⇒) é k ^8B{. � k = 1(Øw,¤á. y�(Øé k−1¤
á. � p(x)´ f(x)� k Ϫ, K f(x) = pk(x)g(x), Ù¥ (p(x), g(x)) = 1.
K
f ′(x) = kpk−1(x)g(x) + pk(x)g′(x) = pk−1(x)(kg(x) + p(x)g′(x)).
d (p(x), g(x)) = 1 � (p(x), kg(x) + p(x)g′(x)) = 1, Ïd p(x)´ f ′(x)�
k − 1 Ϫ. â8Bb�, p(x) ´ f ′(x), · · · , f (k−1)(x) �Ϫ, �Ø´
f (k)(x)�Ϫ.
p(x)´ f(x)�Ϫ´®�.
(⇐) X p(x)´ f(x), f ′(x), · · · , f (k−1)(x)�Ϫ, �Ø´ f (k)(x)�Ϫ,
K p(x)´ f (k−1)(x)�Ϫ, ?
, p(x)´ f (k−2)(x)��Ϫ, ga
í, p(x)´ f(x)� kϪ.
6. Á¦õª x1999 + 1ر (x− 1)2 ¤�{ª.
): � x1999 + 1 = (x− 1)2q(x) + ax+ b, Kü>¦���
1999x1998 = 2(x− 1)q(x) + (x− 1)2q(x) + a.
§ 7 õª� · 145 ·
± x = 1 \þüª, �
a = 1999, b = −1997.
�¤¦{ª 1999x − 1997.
§ 7 õª�
1. ¦e�õª�ú�:
(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.
): (1) 1 +
√
2i, 1−√2i.
(2)
−1 +√3i
2
,
−1−√3i
2
.
2. XJ (x− 1)2 | Ax4 +Bx2 + 1, ¦ A,B.
): A = 1, B = −2.
3. ® x4 − 3x3 + 6x2 + ax+ bU� x2 − 1�Ø, ¦ a, b.
): a = 3, b = −7.
4. y²: XJ f(x) | f(xn), @o f(x)�U´"½ü .
y²: � a ´ f(x) �, K f(a) = 0, u´ f(an) = 0, q��
f((an)n) = f(an
2
) = 0, . . . , f(an
n
) = 0. Ï
a, an, an
2
, · · · , ann Ñ´ f(x)
�. � f(x)�ØÓ=kkõ, �7k k < l¦ an
k
= an
l
, =
an
k
(an
l−nk − 1) = 0.
u´ a = 0½ an
l−nk = 1, � a 0½ü .
5. y²: sinxØ´õª.
y²: sinx kÃõØÓ� kpi, k ∈ Z,
õªkkõ. Ï
d sinxØ´õª.
6. ®õª f(x) = x5 − 10x2 + 15x− 6 k, Á¦§�¤k¿(
½�ê.
):
−3 +√15i
2
,
−3−√15i
2
, 1,1,1.
7. ¦ t�, ¦ f(x) = x3 − 3x2 + tx− 1 k.
): t = 3, 1 3; t = −15
4
, −1
2
2.
8. ¦õª f(x) = x3 + px+ q k�^.
): 4p3 + 27q2 = 0.
· 146 · 1Ù �õª�Ϫ©)
9. y²: e�õªvk:
(1) f(x) = 1 + x+
x2
2!
+ · · ·+ x
n
n!
;
∗(2) f(x) = 1 + 2x+ 3x2 + · · ·+ (n+ 1)xn.
y²: (1)
(f(x), f ′(x)) =
(
1 + x+
x2
2!
+ · · ·+ x
n
n!
, 1 + x+
x2
2!
+ · · · + x
n−1
(n − 1)!
)
=
(
xn
n!
, 1 + x+
x2
2!
+ · · · + x
n−1
(n − 1)!
)
= 1.
¤± f(x)Ã.
(2) �
g(x) = (1−x)2(1+2x+3x2+ · · ·+(n+1)xn) = 1−(n+2)xn+1+(n+1)xn+2,
g′(x) = (n+ 2)(n + 1)xn+1 − (n+ 2)(n + 1)xn,
(g(x), g′(x)) = x− 1.
¤± g(x)=k�´ x = 1. q f(x)�w,Ñ´ g(x)�,
x = 1
Ø´ f(x)�, � f(x)Ã.
10. y²: f(x) = xn + axn−m + b (n > 2, n > m > 0)ØUk"�
êu 2�.
y²: f ′(x) = xn−m−1[nxm + (n−m)a].
(a) � a 6= 0, nxm + (n −m)a�Ñ´ü, ¤± f(x)�êu 2
�U´ x = 0.
(b) � a = 0, f ′(x)�=k� x = 0, � f(x)�êu 2�
U´ x = 0.
11. XJ a´ f ′′′(x)� k, y²: a´
g(x) =
x− a
2
[f ′(x) + f ′(a)]− f(x) + f(a)
� k + 3.
y²:
g(x) =
x− a
2
[f ′(x) + f ′(a)]− f(x) + f(a),
g′(x) =
1
2
[f ′(a)− f ′(x)] + x− a
2
f ′′(x),
g′′(x) =
x− a
2
f ′′′(x),
§ 7 õª� · 147 ·
w, a´ g(x), g′(x), g′′(x)�, q a´ f ′′′(x)� k , Ïd a´ g′′(x)�
k + 1, ´ g(x)� k + 3.
12. y²: x0 ´ f(x)� k�¿©7^´f(x0) = f
′(x0) = · · · =
f (k−1)(x0) = 0
f
(k)(x0) 6= 0.
y²: x0 ´ f(x)� k ⇐⇒ x− x0 ´ f(x)� kϪ
⇐⇒ x− x0 ´ f(x), f ′(x), · · · , f (k−1)(x)�Ϫ, �Ø´ f (k)(x)�Ϫ
⇐⇒ f(x0) = f ′(x0) = · · · = f (k−1)(x) = 0, f (k)(x0) 6= 0.
13. y²: XJ f ′(x) | f(x), K f(x) k n, Ù¥ n = deg f(x).
y²: db�,
f(x)
(f(x), f ′(x))
= c(x − a). l
x − a f(x)=k�Ø
�Ϫ (íØ 6.4), ¤± f(x) = c(x− a)n, f(x) k n.
14. ÁUeL¤�ê, ¦gê$�õª:
x 1 2 3 4
y 2 1 4 3
): f(x) = −4
3
x3 + 10x2 − 65
3
x+ 15.
15. e n gõª f(x)� x1, x2, · · · , xn,
ê cØ´ f(x)�, y
²:
n∑
i=1
1
xi − c = −
f ′(c)
f(c)
.
y²: õª f(x) = (x− x1)(x− x2) · · · (x− xn), K
f ′(x) =
n∑
i=1
f(x)
x− xi ,
f ′(x)
f(x)
=
n∑
i=1
1
x− xi ,
l
n∑
i=1
1
xi − c = −
f ′(c)
f(c)
.
∗16. A^.%{K�Ñ.KF�úª.
y²: �¤¦õª
f(x) = c0 + c1x+ · · ·+ cn−1xn−1,
· 148 · 1Ù �õª�Ϫ©)
Ù¥ ci ½. ò ai, bi \þªü>, � c0, c1, · · · , cn−1 �5§|:
c0 + c1a1 + · · ·+ cn−1an−11 = b1
c0 + c1a2 + · · ·+ cn−1an−12 = b2
. . . . . . . . . . . . . . . . . . . . . . .
c0 + c1an + · · ·+ cn−1an−1n = bn
d5§|�XêÝ
A´��Ý
:
A =
1 a1 a
2
1 · · · an−11
1 a2 a
2
2 · · · an−12
...
...
...
. . .
...
1 an a
2
n · · · an−1n
,
|A| =
∏
1≤iÓ¦± x− ai, 2- x = ai, �
Ai =
1
F ′(ai)
.
Ïd�ð�ª
1
F (x)
=
1
(x− a1)F ′(a1) +
1
(x− a2)F ′(a2) + · · ·+
1
(x− an)F ′(an) .
l
1 =
n∑
i=1
F (x)
(x− ai)F ′(ai) .
-
f(x) = (x− ai)fi(x) + f(ai),
· 150 · 1Ù �õª�Ϫ©)
K
f(x) =
n∑
i=1
[(x− ai)fi(x) + f(ai)] F (x)
(x− ai)F ′(ai)
=
n∑
i=1
fi(x)F (x)
F ′(ai)
+
n∑
i=1
f(ai)F (x)
(x− ai)F ′(ai)
= F (x)
(
n∑
i=1
fi(x)
F ′(ai)
)
+
n∑
i=1
f(ai)F (x)
(x− ai)F ′(ai) .
du
n∑
i=1
f(ai)F (x)
(x− ai)F ′(ai) ∈ K[x],
deg
n∑
i=1
f(ai)F (x)
(x− ai)F ′(ai) ≤ n − 1, ¤±^
F (x)Ø f(x) ¤��{ª
n∑
i=1
f(ai)F (x)
(x− ai)F ′(ai) .
∗18. ® a1, · · · , an; b1, · · · , bn pØÓ�ê, ¦)e�§|:
1
b1 − a1x1 +
1
b1 − a2x2 + · · ·+
1
b1 − anxn = −1,
1
b2 − a1x1 +
1
b2 − a2x2 + · · ·+
1
b2 − anxn = −1,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
bn − a1x1 +
1
bn − a2x2 + · · ·+
1
bn − anxn = −1.
): � x1, · · · , xn ´d§|�?), kn©ª
F (x) = 1 +
x1
x− a1 +
x2
x− a2 + · · · +
xn
x− an , (∗)
K F (bi) = 0, i = 1, · · · , n.
- F (x) =
g(x)
(x− a1)(x− a2) · · · (x− an) , K deg g(x) = n,
g(x) �Ä
xn. du F (bi) = 0, � g(bi) = 0, i = 1, · · · , n, ¤±
g(x) = (x− b1)(x− b2) · · · (x− bn).
F (x) =
(x− b1)(x− b2) · · · (x− bn)
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