【種花家務(wù)·代數(shù)】1-5-09分式的除法『數(shù)理化自學(xué)叢書6677版』

2023-09-29 16:20 作者:山嵓 0人讀過 | 我要投稿

【閱前提示】本篇出自『數(shù)理化自學(xué)叢書6677版』，此版叢書是“數(shù)理化自學(xué)叢書編委會(huì)”于1963-1966年陸續(xù)出版，并于1977年正式再版的基礎(chǔ)自學(xué)教材，本系列叢書共包含17本，層次大致相當(dāng)于如今的初高中水平，其最大特點(diǎn)就是可用于“自學(xué)”。當(dāng)然由于本書是大半個(gè)世紀(jì)前的教材，很多概念已經(jīng)與如今迥異，因此不建議零基礎(chǔ)學(xué)生直接拿來自學(xué)。不過這套叢書卻很適合像我這樣已接受過基礎(chǔ)教育但卻很不扎實(shí)的學(xué)酥重新自修以查漏補(bǔ)缺。另外，黑字是教材原文，彩字是我寫的注解。

【山話嵓語】我在原有“自學(xué)叢書”系列17冊(cè)的基礎(chǔ)上又添加了1冊(cè)八五人教中學(xué)甲種本《微積分初步》，原因有二：一則，我是雙魚座，有一定程度的偶雙癥，但“自學(xué)叢書”系列中代數(shù)4冊(cè)、幾何5冊(cè)實(shí)在令我刺撓，因此就需要加入一本代數(shù)，使兩邊能夠?qū)ε计胶猓欢t，我認(rèn)為《微積分初步》這本書對(duì)“準(zhǔn)大學(xué)生”很重要，以我的慘痛教訓(xùn)為例，大一高數(shù)第一堂課，我是直接蒙圈，學(xué)了個(gè)寂寞。另外大學(xué)物理的前置條件是必須有基礎(chǔ)微積分知識(shí)，因此我所讀院校的大學(xué)物理課是推遲開課；而比較生猛的大學(xué)則是直接開課，然后在緒論課中猛灌基礎(chǔ)高數(shù)（例如田光善舒幼生老師的力學(xué)課）。我選擇在“自學(xué)叢書”17本的基礎(chǔ)上添加這本《微積分初步》，就是希望小伙伴升大學(xué)前可以看看，不至于像我當(dāng)年那樣被高數(shù)打了個(gè)措手不及。

第五章分式

§5-9分式的除法

【01】在分?jǐn)?shù)的除法里，我們知道，除以一個(gè)分?jǐn)?shù)，等于乘以這個(gè)分?jǐn)?shù)的倒數(shù)，例如 $%5Cscriptsize%5Cfrac23%5Cdiv%5Cfrac45%3D%5Cfrac23%5Ctimes%5Cfrac54$ 。分式的除法，也依照分?jǐn)?shù)除法的同樣法則來進(jìn)行。

【02】分式的除法法則：一個(gè)代數(shù)式除以一個(gè)分式，等于乘以把這個(gè)分式的分子分母對(duì)調(diào)后所成的分式。即?

???????? $%5Cfrac%20ab%5Cdiv%5Cfrac%20cd%3D%5Cfrac%20ab%5Ctimes%5Cfrac%20dc$

例1.計(jì)算： $%5Cscriptsize(1)%5C%3B%5Cfrac%7B3a%5E2b%5E3%7D%7Bc%5E4d%7D%5Cdiv%5Cfrac%7B5a%5E2b%7D%7Bc%5E3d%5E2x%7D%3B%5Cquad(2)%5C%3B%5Cfrac%7Bx%5E2-y%5E2%7D%7Bx%5E2%2By%5E2%7D%5Cdiv%5Cfrac%7Bx%5E3-y%5E3%7D%7Bx%5E6%2By%5E6%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%26%5Cfrac%7B3a%5E2b%5E3%7D%7Bc%5E4d%7D%5Cdiv%5Cfrac%7B5a%5E2b%7D%7Bc%5E3d%5E2x%7D%5C%5C%0A%26%20%3D%5Cfrac%7B3a%5E2b%5E3%7D%7Bc%5E4d%7D%5Ctimes%5Cfrac%7Bc%5E3d%5E2x%7D%7B5a%5E2b%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7B3a%5E%7B2%7Db%5E%7B3%7Dc%5E%7B3%7Dd%5E%7B2%7Dx%7D%7B5a%5E%7B2%7Dbc%5E%7B4%7Dd%7D%3D%5Cfrac%7B3b%5E%7B2%7Ddx%7D%7B5c%7D%3B%0A%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(2)%26%5Cfrac%7Bx%5E%7B2%7D-y%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%5Cdiv%5Cfrac%7Bx%5E%7B3%7D-y%5E%7B3%7D%7D%7Bx%5E%7B6%7D%2By%5E%7B6%7D%7D%3D%5Cfrac%7Bx%5E%7B2%7D-y%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%5Ctimes%5Cfrac%7Bx%5E%7B6%7D%2By%5E%7B6%7D%7D%7Bx%5E%7B3%7D-y%5E%7B3%7D%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(x%2By%5Cright)%5Cleft(x-y%5Cright)%5Cleft(x%5E2%2By%5E2%5Cright)%5Cleft(x%5E4-x%5E2y%5E2%2By%5E4%5Cright)%7D%7B%5Cleft(x%5E2%2By%5E2%5Cright)%5Cleft(x-y%5Cright)%5Cleft(x%5E2%2Bxy%2By%5E2%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(x%2By)(x%5E4-x%5E2y%5E2%2By%5E4)%7D%7Bx%5E2%2Bxy%2By%5E2%7D.%0A%5Cend%7Baligned%7D$

例2.計(jì)算： $%5Cscriptsize(1)%5C%3B%5Cfrac%7Bx%5E2-4x%7D%7Bx%2B3%7D%5Cdiv(4-x)%3B%5Cquad(2)%5C%3B(3x%2B6)%5Cdiv%5Cfrac%7Bx%5E2%2Bx-2%7Dx$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%26%5C%3B%5Cfrac%7Bx%5E2-4x%7D%7Bx%2B3%7D%5Cdiv(4-x)%3D%5Cfrac%7Bx%5E2-4x%7D%7Bx%2B3%7D%5Ctimes%5Cfrac1%7B4-x%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bx(x-4)%7D%7B%5Cleft(x%2B3%5Cright)%5Cleft(4-x%5Cright)%7D%3D-%5Cfrac%20x%7Bx%2B3%7D%3B%20%5C%5C%0A(2)%26%5C%3B(3x%2B6)%5Cdiv%5Cfrac%7Bx%5E2%2Bx-2%7Dx%3D%5Cfrac%7B3%5Cleft(x%2B2%5Cright)%7D1%5Ctimes%5Cfrac%20x%7Bx%5E2%2Bx-2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B3%5Cleft(x%2B2%5Cright)x%7D%7B%5Cleft(x%2B2%5Cright)%5Cleft(x-1%5Cright)%7D%3D%5Cfrac%7B3x%7D%7Bx-1%7D.%0A%5Cend%7Baligned%7D$

例3.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5C%3B%5Cfrac%7Bx%5E2-9%7D%7Bx%5E2-1%7D%5Cdiv%5Cfrac%7Bx%5E2-5x%2B6%7D%7Bx%5E2%2B5x-6%7D%5Cdiv%5Cfrac%7Bx%5E2%2B5x%2B6%7D%7Bx%5E2-5x-6%7D%3B%5C%5C(2)%26%5C%3B%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Cdiv%5Cfrac%7Ba%5E3-b%5E3%7D%7Ba%5E4%2Bb%5E4%7D%5Ctimes%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%2Bb%7D.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%5C%3B%26%5Cfrac%7Bx%5E2-9%7D%7Bx%5E2-1%7D%5Cdiv%5Cfrac%7Bx%5E2-5x%2B6%7D%7Bx%5E2%2B5x-6%7D%5Cdiv%5Cfrac%7Bx%5E2%2B5x%2B6%7D%7Bx%5E2-5x-6%7D%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(x%2B3%5Cright)%5Cleft(x-3%5Cright)%7D%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%5Ctimes%5Cfrac%7Bx%5E2%2B5x-6%7D%7Bx%5E2-5x%2B6%7D%5Ctimes%5Cfrac%7Bx%5E2-5x-6%7D%7Bx%5E2%2B5x%2B6%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(x%2B3)(x-3)(x%2B6)(x-1)(x-6)(x%2B1)%7D%7B(x%2B1)(x-1)(x-2)(x-3)(x%2B2)(x%2B3)%7D%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(x%2B6%5Cright)%5Cleft(x-6%5Cright)%7D%7B%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%7D%3B%0A%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(2)%5C%3B%26%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Cdiv%5Cfrac%7Ba%5E3-b%5E3%7D%7Ba%5E4%2Bb%5E4%7D%5Ctimes%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%2Bb%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Ctimes%5Cfrac%7Ba%5E4%2Bb%5E4%7D%7Ba%5E3-b%5E3%7D%5Ctimes%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%2Bb%7D%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%5Cleft(a%5E4%2Bb%5E4%5Cright)%5Cleft(a%5E2%2Bb%5E2%5Cright)%7D%7B%5Cleft(a%5E2%2Bb%5E2%5Cright)%5Cleft(a-b%5Cright)%5Cleft(a%5E2%2Bab%2Bb%5E2%5Cright)%5Cleft(a%2Bb%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E4%2Bb%5E4%7D%7Ba%5E2%2Bab%2Bb%5E2%7D.%0A%5Cend%7Baligned%7D$

習(xí)題5-9(1)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B14x%5E%7B2%7D%7D%7B27y%5E%7B2%7D%7D%5Cdiv%5Cfrac%7B7x%7D%7B9y%7D.%20%5C%5C%0A%262%E3%80%81%5Cfrac%7Ba%5E%7B2%7D-3a-10%7D%7Ba%5E%7B3%7D-1%7D%5Cdiv%5Cfrac%7Ba%5E%7B2%7D-4%7D%7Ba%5E%7B2%7D%2Ba%2B1%7D.%20%5C%5C%0A%263%E3%80%81(m-1)%5Cdiv%5Cfrac%7Bm-1%7D%7Bm%2B1%7D.%20%5C%5C%0A%264%E3%80%81%5Cfrac%7Bx%5E%7B3%7D%2By%5E%7B3%7D%7D%7Bx%5E%7B2%7D%2By%5E%7B2%7D%7D%5Cdiv(x%2By).%5C%5C%0A%265%E3%80%81%5Cfrac%7B(x%2By)%5E2%7D%7Bxy-y%5E2%7D%5Cdiv%5Cfrac%7Bx%5E2%2B2xy%2By%5E2%7D%7Bx%5E2-2xy%2By%5E2%7D.%20%5C%5C%0A%266%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-3x-4%7D%7Bx%5E%7B2%7D%2B2x-3%7D%5Cdiv%5Cfrac%7Bx%5E%7B2%7D-16%7D%7Bx%5E%7B2%7D%2B6x%2B9%7D.%20%5C%5C%0A%267%E3%80%81%5Cfrac%7Bax%7D%7B(a-x)%5E%7B2%7D%7D%5Cdiv%5Cfrac%7Bab%7D%7Ba%5E%7B2%7D-x%5E%7B2%7D%7D%5Cdiv%5Cfrac%7Bax%7D%7Bb(a-x)%7D.%20%5C%5C%0A%268%E3%80%8116x%5E%7B2%7Dy%5E%7B3%7D%5Cdiv%5CBig(-%5Cfrac%7B20x%5E%7B5%7Dy%5E%7B4%7D%7D%7B3a%5E%7B2%7Db%7D%5CBig)%5Cdiv%5Cfrac%7Bab%7D%7Bx%2By%7D.%5C%5C%0A%269%E3%80%81%5Cfrac%7Ba%5E2%2B5a%2B4%7D%7Ba%5E2%2Ba-2%7D%5Cdiv5%5Cleft(a%2B1%5Cright)%5E2%5Cdiv%5Cfrac1%7Ba%5E2-2a%2B1%7D.%5C%5C%0A%2610%E3%80%81%5Cfrac%7B(a%5E2-x%5E2)%5E3%7D%7Ba%5E2%2Bx%5E2%7D%5Cdiv%5Cfrac%7B(a%5E2%2B2ax%2Bx%5E2)%5E2%7D%7Ba%5E4-x%5E4%7D%5Ctimes%5Cfrac1%7B(a%5E2-2ax%2Bx%5E2)%5E2%7D%0A%5Cend%7Baligned%7D$

【答案】 $%5Cscriptsize7%E3%80%81%5Cfrac%7Ba%2Bx%7Da%3B%5C%3B8%E3%80%81-%5Cfrac%7B12a(x%2By)%7D%7B5x%5E3y%7D%3B%5C%3B9%E3%80%81%5Cfrac%7B(a%2B4)(a-1)%7D%7B5(a%2B2)(a%2B1)%7D%3B%5C%3B10%E3%80%811.%0A$

例4.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5Cleft(x%2B%5Cfrac1x%5Cright)%5Cdiv%5Cleft(x-%5Cfrac1x%5Cright)%3B%5C%5C%26(2)%5Cleft(%5Cfrac1a%2B%5Cfrac1b%5Cright)%5Cdiv%5Cleft(%5Cfrac1%7Ba%5E2%7D-%5Cfrac1%7Bb%5E2%7D%5Cright).%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5Cleft(x%2B%5Cfrac1x%5Cright)%5Cdiv%5Cleft(x-%5Cfrac1x%5Cright)%3D%5Cfrac%7Bx%5E2%2B1%7Dx%5Cdiv%5Cfrac%7Bx%5E2-1%7Dx%5C%5C%26%3D%5Cfrac%7Bx%5E2%2B1%7Dx%5Ctimes%5Cfrac%20x%7Bx%5E2-1%7D%3D%5Cfrac%7Bx%5E2%2B1%7D%7Bx%5E2-1%7D%3B%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D(2)%26%5Cleft(%5Cfrac1a%2B%5Cfrac1b%5Cright)%5Cdiv%5Cleft(%5Cfrac1%7Ba%5E2%7D-%5Cfrac1%7Bb%5E2%7D%5Cright)%3D%5Cfrac%7Bb%2Ba%7D%7Bab%7D%5Cdiv%5Cfrac%7Bb%5E2-a%5E2%7D%7Ba%5E2b%5E2%7D%5C%5C%26%3D%5Cfrac%7Bb%2Ba%7D%7Bab%7D%5Ctimes%5Cfrac%7Ba%5E2b%5E2%7D%7Bb%5E2-a%5E2%7D%3D%5Cfrac%7Ba%5E2b%5E2(b%2Ba)%7D%7Bab%5Cleft(b%2Ba%5Cright)%5Cleft(b-a%5Cright)%7D%5C%5C%26%3D%5Cfrac%7Bab%7D%7Bb-a%7D.%5Cend%7Baligned%7D$

例5.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%5Cfrac%7Ba-b%7D%7Ba%2Bb%7D%5Cdiv%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%5E2-b%5E2%7D-%5Cfrac%7Ba%2Bb%7D%7Bab%7D%5Cdiv%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%5E2b%2Bab%5E2%7D%3B%5C%5C%26(2)%5C%3B%5Cfrac%20x%7Bx-1%7D%5Cdiv%5Cfrac%20x%7Bx%2B1%7D-%5Cfrac%7Bx%2B1%7D%7Bx%5E3-1%7D%5Cdiv%5Cfrac1%7Bx%5E2%2Bx%2B1%7D.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%26%5C%3B%5Cfrac%7Ba-b%7D%7Ba%2Bb%7D%5Cdiv%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%5E2-b%5E2%7D-%5Cfrac%7Ba%2Bb%7D%7Bab%7D%5Cdiv%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7Ba%5E2b%2Bab%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba-b%7D%7Ba%2Bb%7D%5Ctimes%5Cfrac%7B(a%2Bb)%5Cleft.(a-b)%5Cright.%7D%7Ba%5E2%2Bb%5E2%7D-%5Cfrac%7Ba%2Bb%7D%7Bab%7D%5Ctimes%5Cfrac%7Bab%5Cleft(a%2Bb%5Cright)%7D%7Ba%5E2%2Bb%5E2%7D%20%5C%5C%0A%20%5C%5C%0A%26%3D%5Cfrac%7B(a-b)%5E2%7D%7Ba%5E2%2Bb%5E2%7D-%5Cfrac%7B(a%2Bb)%5E2%7D%7Ba%5E2%2Bb%5E2%7D.(%5Cfrac16%2B%5Cfrac26)%5Cdiv%5Cleft(%5Cfrac16%2B%5Cfrac16%5Cright).x%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2-2ab%2Bb%5E2-(a%5E2%2B2ab%2Bb%5E2)%7D%7Ba%5E2%2Bb%5E2%7D%3D%5Cfrac%7B-4ab%7D%7Ba%5E2%2Bb%5E2%7D%3B%20%0A%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(2)%26%5C%3B%5Cfrac%20x%7Bx-1%7D%5Cdiv%5Cfrac%20x%7Bx%2B1%7D-%5Cfrac%7Bx%2B1%7D%7Bx%5E3-1%7D%5Cdiv%5Cfrac1%7Bx%5E2%2Bx%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%20x%7Bx-1%7D%5Ctimes%5Cfrac%7Bx%2B1%7Dx-%5Cfrac%7Bx%2B1%7D%7B(x-1)(x%5E2%2Bx%2B1)%7D%20%0A%5Ctimes%5Cfrac%7Bx%5E%7B2%7D%2Bx%2B1%7D%7B1%7D%5C%5C%0A%26%3D%5Cfrac%7Bx%2B1%7D%7Bx-1%7D-%5Cfrac%7Bx%2B1%7D%7Bx-1%7D%3D0.%20%20%0A%5Cend%7Baligned%7D$

例6.計(jì)算： $%5Cscriptsize%5Cleft(%5Cfrac%7B2a%7D%7B2a%2Bb%7D-%5Cfrac%7B4a%5E2%7D%7B4a%5E2%2B4ab%2Bb%5E2%7D%5Cright)%5Cdiv%5Cleft(%5Cfrac%7B2a%7D%7B4a%5E2-b%5E2%7D%2B%5Cfrac1%7Bb-2a%7D%5Cright)$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cleft(%5Cfrac%7B2a%7D%7B2a%2Bb%7D-%5Cfrac%7B4a%5E2%7D%7B4a%5E2%2B4ab%2Bb%5E2%7D%5Cright)%5Cdiv%5Cleft(%5Cfrac%7B2a%7D%7B4a%5E2-b%5E2%7D%2B%5Cfrac1%7Bb-2a%7D%5Cright)%5C%5C%0A%26%3D%5Cleft%5B%5Cfrac%7B2a%7D%7B2a%2Bb%7D-%5Cfrac%7B4a%5E2%7D%7B(2a%2Bb)%5E2%7D%5Cright%5D%5Cdiv%5Cleft%5B%5Cfrac%7B2a%7D%7B%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D-%5Cfrac1%7B2a-b%7D%5Cright%5D%5C%5C%0A%26%3D%5Cfrac%7B2a%5Cleft(2a%2Bb%5Cright)-4a%5E%7B2%7D%7D%7B%5Cleft(2a%2Bb%5Cright)%5E%7B2%7D%7D%5Cdiv%5Cfrac%7B2a-%5Cleft(2a%2Bb%5Cright)%7D%7B%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B4a%5E2%2B2ab-4a%5E2%7D%7B%5Cleft(2a%2Bb%5Cright)%5E2%7D%5Cfrac%7B2a-2a-b%7D%7B%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D%5C%5C%0A%26%3D%5Cfrac%7B2ab%7D%7B%5Cleft(2a%2Bb%5Cright)%5E%7B2%7D%7D%5Cdiv%5Cfrac%7B-b%7D%7B%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2ab%7D%7B%5Cleft(2a%2Bb%5Cright)%5E2%7D%5Ctimes%5Cfrac%7B%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D%7B-b%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2ab%5Cleft(2a%2Bb%5Cright)%5Cleft(2a-b%5Cright)%7D%7B-b%5Cleft(2a%2Bb%5Cright)%5E%7B2%7D%7D%3D-%5Cfrac%7B2a%5Cleft(2a-b%5Cright)%7D%7B2a%2Bb%7D.%20%20%0A%5Cend%7Baligned%7D$

習(xí)題5-9(2)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cleft(%5Cfrac%7B1%7D%7Ba%7D%2B%5Cfrac%7B1%7D%7Bb%7D%5Cright)%5Cdiv%5Cleft(%5Cfrac%7B1%7D%7Ba%7D-%5Cfrac%7B1%7D%7Bb%7D%5Cright).%20%5C%5C%0A%262%E3%80%81%5Cleft(%5Cfrac%20xy%2B%5Cfrac%20yz%2B%5Cfrac%20zx%5Cright)%5Cdiv%5Cleft(%5Cfrac%20yx%2B%5Cfrac%20zy%2B%5Cfrac%20xz%5Cright).%20%5C%5C%0A%263%E3%80%81%5Cleft(5-%5Cfrac%7Ba%5E%7B2%7D-19b%5E%7B2%7D%7D%7Ba%5E%7B2%7D-4b%5E%7B2%7D%7D%5Cright)%5Cdiv%5Cleft(3-%5Cfrac%7Ba-5b%7D%7Ba-2b%7D%5Cright).%20%5C%5C%0A%264%E3%80%81%5Cleft(%5Cfrac%7Ba%7D%7Ba%2B1%7D%2B1%5Cright)%5Cdiv%5Cleft(1-%5Cfrac%7B3a%5E%7B2%7D%7D%7B1-a%5E%7B2%7D%7D%5Cright).%5C%5C%0A%265%E3%80%81%5Cleft%5B1-%5Cfrac%7B(a-b)%5E%7B2%7D-c%5E%7B2%7D%7D%7B(a%2Bb)%5E%7B2%7D-c%5E%7B2%7D%7D%5Cright%5D%5Cdiv%5Cleft%5B1-%5Cfrac%7Ba%5E%7B2%7D-(b-c)%5E%7B2%7D%7D%7Ba%5E%7B2%7D-(b%2Bc)%5E%7B2%7D%7D%5Cright%5D.%20%20%5C%5C%0A%266%E3%80%81%5Cfrac%7Bx%5E2-(y-z)%5E2%7D%7B(x-y)%5E2-z%5E2%7D%5Cdiv%5Cfrac%7B(x%2By)%5E2-z%5E2%7D%7Bx%5E2-(y%2Bz)%5E2%7D-1.%20%5C%5C%0A%267%E3%80%81%5Cleft%5B%5Cfrac%7Bm%5E%7B2%7D-n%5E%7B2%7D%7D%7Bm%5E%7B2%7D%2B2mn%2Bn%5E%7B2%7D%7D%2B%5Cfrac%7B2%7D%7Bmn%7D%5Cdiv%5Cleft(%5Cfrac%7B1%7D%7Bm%7D%2B%5Cfrac%7B1%7D%7Bn%7D%5Cright)%5E%7B2%7D%5Cright%5D%5Cdiv%5Cfrac%7B1%7D%7Bm%2Bn%7D.%20%20%5C%5C%0A%268%E3%80%81%5Cleft%5B%5Cfrac%7B2%7D%7B3x%7D-%5Cfrac%7B2%7D%7Bx%2By%7D%5CBig(%5Cfrac%7Bx%2By%7D%7B3x%7D-x-y%5CBig)%5Cright%5D%5Cdiv%5Cfrac%7Bx-y%7D%7Bx%7D.%5C%5C%0A%269%E3%80%81%5Cleft(a-%5Cfrac%7B4ab%7D%7Ba%2Bb%7D%2Bb%5Cright)%5Cdiv%5Cleft(%5Cfrac%7Ba%7D%7Ba%2Bb%7D-%5Cfrac%7Bb%7D%7Bb-a%7D-%5Cfrac%7B2ab%7D%7Ba%5E2-b%5E2%7D%5Cright).%5C%5C%0A%2610%E3%80%81(x-a)%5Cdiv%5Cleft%5Bx-%5Cfrac%7B(x-b)%5Cleft(x-c%5Cright)%7D%7Bx%2Ba%7D%5Cright%5D.%0A%5Cend%7Baligned%7D$

【答案】 $%5Cscriptsize6%E3%80%810%3B%5C%3B7%E3%80%81%5Cfrac%7Bm%5E2-n%5E2%2B2mn%7D%7Bm%2Bn%7D%3B%5C%3B8%E3%80%81%5Cfrac%7B2x%7D%7Bx-y%7D%3B%5C%3B9%E3%80%81%20a-b%3B%5C%3B10%E3%80%81%5Cfrac%7Bx%5E2-a%5E2%7D%7Bax%2Bbx%2Bcx-bc%7D.$

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