首页 高等代数与解析几何 习题解答11

高等代数与解析几何 习题解答11

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高等代数与解析几何 习题解答11 1›˜Ù ˜�õ‘ª�Ϫ©) § 1 ˜�õ‘ª 1. OŽ (x2 + ax− b)(x2 − 1) + (x2 − ax+ b)(x2 + 1). ): 2x4 − 2ax+ 2b. 2. OŽõ‘ª 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, ...

高等代数与解析几何 习题解答11
1›˜Ù ˜�õ‘ª�Ϫ©) § 1 ˜�õ‘ª 1. OŽ (x2 + ax− b)(x2 − 1) + (x2 − ax+ b)(x2 + 1). ): 2x4 − 2ax+ 2b. 2. OŽõ‘ª 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´¢Xêõ‘ª, y²: XJ f 2(x) = xg2(x) + xh2(x), @o f(x) = g(x) = h(x) = 0. y²: X f(x) 6= 0, K†ª�gêóê, mª�gêÛê, gñ, � f(x) = 0. l g2(x) + h2(x) = 0. q, g(x), h(x) �¢Xêõ‘ª, l g2(x), h2(x) �đXêÑ´šKê, ùü‡êƒÚ", � g(x), h(x)�đXêÑ´", l g(x) = h(x) = 0. · 132 · 1›˜Ù ˜�õ‘ª�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·ÜŸo^‡ž, 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)�~ꑏ a,K fk(x)�~ꑏ ak. Ïd x | fk(x) ⇐⇒ ak = 0 ⇐⇒ a = 0 ⇐⇒ x | f(x). 7. �a, bü‡Øƒ��~ê, y²: õ‘ª 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�õ‘ª, 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, Œ� 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 · 1›˜Ù ˜�õ‘ª�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 ŒúϪ 1. ¦ŒúϪ (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)�˜‡ŒúϪ. 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)�˜‡ ŒúϪ. 4. y²: XJ h(x)Ä˜õ‘ª, 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 ŒúϪ · 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)�˜‡ŒúϪ. qÏ d(x), h(x)ѴĘõ‘ª, � d(x)h(x) ´Ä˜õ‘ª, 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>ž� (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 · 1›˜Ù ˜�õ‘ª�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´õ‘ª, … (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, Œ� (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�Œ� (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))�đXêƒÓ, � (f(x), g(x)) = (f1(x), g1(x)). § 3 ŒúϪ · 137 · 11. y²: XJ f(x) † g(x)pƒ, @o f(xm) † g(xm)pƒ. y²: dK�, 3õ‘ª 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 Œ� (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�ŒúϪ. ): - d = (m,n), KŠâSK 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) † xpƒŒ � f(x) | xd − 1. Ïd (xm − 1, xn − 1) = xd − 1, Ù¥ d = (m,n). · 138 · 1›˜Ù ˜�õ‘ª�Ϫ©) ∗14. y²: ‡ f(x) (f(x), g(x)) , g(x) (f(x), g(x)) �gêьu", Ҍ±·� 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: 3õ‘ª 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)) �gêьu", ¤± u(x), v(x)ÑØ´ "õ‘ª. 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�õ‘ª g(x), Ñ3õ‘ª 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) ˜|üüpƒ�õ‘ª, y²: é?Û�õ‘ ª f1(x), · · · , fs(x), Ñ3õ‘ª 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 · 1›˜Ù ˜�õ‘ª�Ϫ©) ∗3. � m(x) EXêõ‘ª, … m(0) 6= 0. y²: 3EXêõ‘ª 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Š, 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 üüpƒ. 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 üüpƒŒ� 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) ´üüpƒ�،�õ‘ª, … 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�Ę،�õ‘ª. 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 · 1›˜Ù ˜�õ‘ª�Ϫ©) Ïd [f(x), g(x)] = pm11 (x)p m2 2 (x) · · · pmss (x). 3. � f(x), g(x) ∈ K[x]ѴĘõ‘ª, 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�Ę،�õ‘ª. - 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�õ‘ª��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)´gêŒu"�õ‘ª. y²: XJéu?Ûõ‘ª f(x), g(x), d p(x) | f(x)g(x) Œ±íÑ p(x) | f(x)½ö p(x) | g(x), K p(x)´ØŒ�õ ‘ª. y²: e p(x) Œ�, K3gê�u p(x) �š~êõ‘ª 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 ­Ïª · 143 · p(x) ∤ f(x), p(x) ∤ g(x), †b�gñ, Ïd p(x)،�. ∗6. y²: gêŒu 0�Ęõ‘ª f(x)´,˜ØŒ�õ‘ª�˜�¿ ©7‡^‡´, é?¿�õ‘ª g(x)½ök (f(x), g(x)) = 1, ½öé,˜�� êm, f(x) | gm(x). y²: (⇒) � f(x) = pm(x), Ù¥ p(x) ،�, 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)�˜‡Ä˜ØŒ�Ïf, K (p(x), f(x)) = p(x), l 3,‡��ê m, ¦ f(x) | pm(x), ù`² p(x)´ f(x)�˜ØŒ�Ï f. ¤± f(x) = cpr(x). qÏ f(x), p(x)�đXêÑ´ 1, � c = 1. l f(x) = pr(x). ∗7. y²: gêŒu 0 �Ęõ‘ª f(x) ´,˜ØŒ�õ‘ª�˜� ¿©7‡^‡´, é?¿�õ‘ª g(x), h(x), d f(x) | g(x)h(x) Œ±íÑ f(x) | g(x), ½öé,˜��ê m, f(x) | hm(x). y²: (⇒) � f(x) = pm(x), Ù¥ p(x)´Ä˜ØŒ�õ‘ª, Kd f(x) | g(x)h(x), Œ� 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)�˜‡Ä˜ØŒ�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)�˜ØŒ�Ïf. ¤± f(x) = cpr(x). qÏ f(x), p(x) �đXêÑ´ 1, � c = 1. l f(x) = pr(x). § 6 ­Ïª 1. �Oe�knXêõ‘ªkíϪ, ek, K¦Ñ­Ïª: (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 · 1›˜Ù ˜�õ‘ª�Ϫ©) (4) x+ 3, 2­, x− 1, 3­. 2. a, b A÷vŸo^‡, e�õ‘ªk­Ïª? (1) f(x) = x3 + 3ax+ b; (2) f(x) = x4 + 4ax+ b. ): (1) � a = b = 0 k 3­Ïª x, � 4a3 = −b2 … a 6= 0, k 2­Ïª 2ax+ b. (2) � a = b = 0 k 4 ­Ïª x, � 27a4 = b3 … a 6= 0, k 2 ­Ïª 3ax+ b. 3. � p(x)´ f ′(x)� k ­Ïª, UÄ` p(x)´ f(x)� k + 1­Ïª,  Ÿo? ): ØU. ϏqŒU f ′(x)?˜­ÏªÑØ´ f(x)�Ϫ. ~X f(x) = x4 − 1, f ′(x) = 4x3. 4. y²: XJ (f ′(x), f ′′(x)) = 1, @o, f(x)�­ÏªÑ´ f(x)��­ Ϫ. y²: du (f ′(x), f ′′(x)) = 1, f ′(x)�?˜ÏªÑØ´ f ′′(x)�Ϫ. � p(x)´ f(x)�­Ïª, K p(x) | f ′(x), u´ p(x) ∤ f ′′(x), `² p(x)´ f ′(x) �üϪ, � p(x)´ f(x)��­Ïª. 5. y²: K[x]¥ØŒ�õ‘ª p(x)´ f(x) ∈ K[x]� k (k > 1) ­Ïª �¿©7‡^‡´ p(x) ´ f(x), f ′(x), · · · , f (k−1)(x) �Ϫ, �Ø´ f (k)(x) �Ϫ. y²: (⇒) é k ^8B{. � k = 1ž(Øw,¤á. y�(Øé k−1¤ á. � p(x)´ f(x)� k ­Ïª, 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 Œ� (p(x), kg(x) + p(x)g′(x)) = 1, Ïd p(x)´ f ′(x)� k − 1 ­Ïª. Šâ8Bb�, 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)�˜­Ïª, ? , p(x)´ f (k−2)(x)��­Ïª, ga í, Œ p(x)´ f(x)� k­Ïª. 6. Á¦õ‘ª 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 õ‘ª�Š · 145 · ± x = 1 “\þüª, � a = 1999, b = −1997. �¤¦{ª 1999x − 1997. § 7 õ‘ª�Š 1. ¦e�õ‘ª�ú�Š: (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)�ŠU´"½ü Š. y²: � a ´ f(x) �˜‡Š, K f(a) = 0, u´ f(an) = 0, qŒ�� f((an)n) = f(an 2 ) = 0, . . . , f(an n ) = 0. Ï a, an, an 2 , · · · , ann Ñ´ f(x) �Š. � f(x)�Øӊ=kkõ‡, �7k k < l¦ an k = an l , = an k (an l−nk − 1) = 0. u´ a = 0½ an l−nk = 1, � a 0½ü Š. 5. y²: sinxØ´õ‘ª. y²: sinx kÁõ‡ØÓ�Š kpi, k ∈ Z, õ‘ªkkõ‡Š. Ï d sinxØ´õ‘ª. 6. ®õ‘ª f(x) = x5 − 10x2 + 15x− 6 k­Š, Á¦§�¤kŠ¿( ½Š�­ê. ): −3 +√15i 2 , −3−√15i 2 , 1,1,1. 7. ¦ t�Š, ¦ f(x) = x3 − 3x2 + tx− 1 k­Š. ): t = 3ž, 1 3­Š; t = −15 4 ž, −1 2  2­Š. 8. ¦õ‘ª f(x) = x3 + px+ q k­Š�^‡. ): 4p3 + 27q2 = 0. · 146 · 1›˜Ù ˜�õ‘ª�Ϫ©) 9. y²: e�õ‘ªvk­Š: (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)íŠ. (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�­Š´ x = 1. q f(x)�­Šw,Ñ´ g(x)�­Š, x = 1 Ø´ f(x)�Š, � f(x)íŠ. 10. y²: f(x) = xn + axn−m + b (n > 2, n > m > 0)ØUkš"�­ êŒu 2�Š. y²: f ′(x) = xn−m−1[nxm + (n−m)a]. (a) � a 6= 0ž, nxm + (n −m)a�ŠÑ´üŠ, ¤± f(x)�­êŒu 2 �ŠŒU´ x = 0. (b) � a = 0ž, f ′(x)�=k�­Š x = 0, � f(x)�­êŒu 2�Š ŒU´ x = 0. 11. XJ a´ f ′′′(x)�˜‡ k­Š, y²: a´ g(x) = x− a 2 [f ′(x) + f ′(a)]− f(x) + f(a) �˜‡ k + 3­Š. 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 õ‘ª�Š · 147 · w, a´ g(x), g′(x), g′′(x)�Š, q a´ f ′′′(x)� k ­Š, Ïd a´ g′′(x)� k + 1­Š, ´ g(x)� k + 3­Š. 12. y²: x0 ´ f(x)� k­Š�¿©7‡^‡´f(x0) = f ′(x0) = · · · = f (k−1)(x0) = 0 f (k)(x0) 6= 0. y²: x0 ´ f(x)� k­Š ⇐⇒ x− x0 ´ f(x)� k­Ïª ⇐⇒ 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 n­Š, Ù¥ n = deg f(x). y²: db�, f(x) (f(x), f ′(x)) = c(x − a). l x − a f(x)=k�، �Ϫ (íØ 6.4), ¤± f(x) = c(x− a)n, f(x) k n­Š. 14. ÁUeL¤‰�êŠ, ¦gê$�õ‘ª: x 1 2 3 4 y 2 1 4 3 ): f(x) = −4 3 x3 + 10x2 − 65 3 x+ 15. 15. e n gõ‘ª f(x)�Š x1, x2, · · · , xn, ê cØ´ f(x)�Š, y ²: n∑ i=1 1 xi − c = − f ′(c) f(c) . y²:  õ‘ª 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^Ž.%{K�Ñ.‚KF�Šúª. y²: �¤¦õ‘ª f(x) = c0 + c1x+ · · ·+ cn−1xn−1, · 148 · 1›˜Ù ˜�õ‘ª�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 d‚5§|�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, Œ� Ai = 1 F ′(ai) . ÏdŒ�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 · 1›˜Ù ˜�õ‘ª�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 p؃Ó�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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