【種花家務·代數(shù)】1-5-10繁分式『數(shù)理化自學叢書6677版』

2023-09-29 17:34 作者:山嵓 0人讀過 | 我要投稿

【閱前提示】本篇出自『數(shù)理化自學叢書6677版』，此版叢書是“數(shù)理化自學叢書編委會”于1963-1966年陸續(xù)出版，并于1977年正式再版的基礎(chǔ)自學教材，本系列叢書共包含17本，層次大致相當于如今的初高中水平，其最大特點就是可用于“自學”。當然由于本書是大半個世紀前的教材，很多概念已經(jīng)與如今迥異，因此不建議零基礎(chǔ)學生直接拿來自學。不過這套叢書卻很適合像我這樣已接受過基礎(chǔ)教育但卻很不扎實的學酥重新自修以查漏補缺。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“自學叢書”系列17冊的基礎(chǔ)上又添加了1冊八五人教中學甲種本《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“自學叢書”系列中代數(shù)4冊、幾何5冊實在令我刺撓，因此就需要加入一本代數(shù)，使兩邊能夠?qū)ε计胶?；二則，我認為《微積分初步》這本書對“準大學生”很重要，以我的慘痛教訓為例，大一高數(shù)第一堂課，我是直接蒙圈，學了個寂寞。另外大學物理的前置條件是必須有基礎(chǔ)微積分知識，因此我所讀院校的大學物理課是推遲開課；而比較生猛的大學則是直接開課，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課）。我選擇在“自學叢書”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前可以看看，不至于像我當年那樣被高數(shù)打了個措手不及。

第五章分式?

§5-10繁分式

【01】我們前面所學到的分式，分子和分母都是整式。有時，我們也會遇到另外一種形式的分式。例如 $%5Cscriptsize%5Cfrac%7B%5Cfrac%20ab%7Dc%2C%5C%3B%5Cfrac%20a%7B%5Cfrac%20bc%7D%2C%5C%3B%5Cfrac%7B%5Cfrac%20ab%7D%7B%5Cfrac%20cd%7D%2C%5C%3B%5Cfrac%7Ba%2B%5Cfrac1a%7D%7Ba-%5Cfrac1a%7D$ 等等。在這種分式里，分子或分母本身是一個分式，我們把這樣的分式，叫做繁分式。

【02】繁分式實際上是分式除法的另一種寫法，因此可利用分式除法的法則，把它化成普通分式（分子分母都是整式的分式）。這種變換的過程，叫做把繁分式化簡。

例1.化簡繁分式：

$%5Csmall(1)%5Cfrac%7B%5Cfrac%7Ba%2Bb%7Dc%7D%7B%5Cfrac%7Ba-b%7Dc%7D%3B%5C%3B(2)%5Cfrac%7Bx%2B%5Cfrac1a%7D%7Bx-%5Cfrac1a%7D%3B%5C%3B(3)%5Cbegin%7Barray%7D%7Bc%7D%5Cfrac%20a%7B1%2B%5Cfrac1a%7D%3B%5C%3B(4)%5C%5C%5Cend%7Barray%7D%5Cfrac%7B1%2B%5Cfrac1a%7Da.$

【解】

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5Cfrac%7B%5Cfrac%7Ba%2Bb%7Dc%7D%7B%5Cfrac%7Ba-b%7Dc%7D%3D%5Cfrac%7Ba%2Bb%7Dc%5Cdiv%5Cfrac%7Ba-b%7Dc%5C%5C%26%3D%5Cfrac%7Ba%2Bb%7Dc%5Ctimes%5Cfrac%20c%7Ba-b%7D%3D%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%3B%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D(2)%26%5Cfrac%7Bx%2B%5Cfrac1a%7D%7Bx-%5Cfrac1a%7D%3D%5Cleft(x%2B%5Cfrac1a%5Cright)%5Cdiv%5Cleft(x-%5Cfrac1a%5Cright)%5C%5C%26%3D%5Cfrac%7Bax%2B1%7Da%5Cdiv%5Cfrac%7Bax-1%7Da%5C%5C%26%3D%5Cfrac%7Bax%2B1%7Da%5Ctimes%5Cfrac%20a%7Bax-1%7D%3D%5Cfrac%7Bax%2B1%7D%7Bax-1%7D%3B%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A%5Cbegin%7Baligned%7D(3)%26%5Cfrac%20a%7B1%2B%5Cfrac1a%7D%5Cend%7Baligned%7D%26%20%3Da%5Cdiv%5Cleft(1%2B%5Cfrac1a%5Cright)%3Da%5Cdiv%5Cfrac%7Ba%2B1%7Da%20%20%5C%5C%0A%26%3Da%5Ctimes%5Cfrac%20a%7Ba%2B1%7D%3D%5Cfrac%7Ba%5E2%7D%7Ba%2B1%7D%3B%0A%5Cend%7Baligned%7D$

$%5Csmall(4)%5Cfrac%7B1%2B%5Cfrac1a%7Da%3D%5Cleft(1%2B%5Cfrac1a%5Cright)%5Cdiv%20a%3D%5Cfrac%7Ba%2B1%7Da%5Ctimes%5Cfrac1a%3D%5Cfrac%7Ba%2B1%7D%7Ba%5E2%7D.$

????????繁分式也可以應用分式的基本性質(zhì)來化簡，如

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3B%5Cfrac%7B%5Cfrac%7Ba%2Bb%7Dc%7D%7B%5Cfrac%7Ba-b%7Dc%7D%3D%5Cfrac%7B%5Cfrac%7Ba%2Bb%7D%7Bc%7D%5Ctimes%20c%7D%7B%5Cfrac%7Ba-b%7Dc%5Ctimes%20c%7D%3D%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%3B%5C%5C%0A%26(2)%5C%3B%5Cfrac%7Bx%2B%5Cfrac1a%7D%7Bx-%5Cfrac1a%7D%3D%5Cfrac%7B%5Cleft(x%2B%5Cfrac1a%5Cright)%5Ctimes%20a%7D%7B%5Cleft(x-%5Cfrac1a%5Cright)%5Ctimes%20a%7D%3D%5Cfrac%7Bax%2B1%7D%7Bax-1%7D%3B%5C%5C%0A%26(3)%5C%3B%5Cfrac%20a%7B1%2B%5Cfrac1a%7D%3D%5Cfrac%7Ba%5Ctimes%20a%7D%7B%5Cleft(1%2B%5Cfrac1a%5Cright)%5Ctimes%20a%7D%3D%5Cfrac%7Ba%5E2%7D%7Ba%2B1%7D%3B%5C%5C%0A%26(4)%5C%3B%5Cfrac%7B1%2B%5Cfrac1a%7Da%3D%5Cfrac%7B%5Cleft(1%2B%5Cfrac1a%5Cright)%5Ctimes%20a%7D%7Ba%5Ctimes%20a%7D%3D%5Cfrac%7Ba%2B1%7D%7Ba%5E2%7D.%0A%5Cend%7Baligned%7D$

例2.化簡： $%5Cscriptsize%5Cfrac%7B2%2B%5Cfrac1%7Bx-1%7D-%5Cfrac1%7Bx%2B1%7D%7D%7Bx%2B%5Cfrac%20x%7Bx%5E2-1%7D%7D$ ? 。

【解1】應用分式除法化簡：

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B2%2B-%5Cfrac%7B1%7D%7Bx-1%7D-%5Cfrac%7B1%7D%7Bx%2B1%7D%7D%7Bx%2B%5Cfrac%7Bx%7D%7Bx%5E2-1%7D%7D%3D%5Cleft(2%2B%5Cfrac%7B1%7D%7Bx-1%7D-%5Cfrac%7B1%7D%7Bx%2B1%7D%5Cright)%5Cdiv%5Cleft(x%2B%5Cfrac%7Bx%7D%7Bx%5E2-1%7D%5Cright)%20%5C%5C%0A%26%3D%5Cfrac%7B2%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%2B%5Cleft(x%2B1%5Cright)-%5Cleft(x-1%5Cright)%7D%7B%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%7D%5Cdiv%5Cfrac%7Bx%5Cleft(x%5E%7B2%7D-1%5Cright)%2Bx%7D%7Bx%5E%7B2%7D-1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2x%5E2-2%2Bx%2B1-x%2B1%7D%7B%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%7D%5Cdiv%5Cfrac%7Bx%5E3-x%2Bx%7D%7Bx%5E2-1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2x%5E2%7D%7B%5Cleft(x-1%5Cright)%5Cleft(x%2B1%5Cright)%7D%5Ctimes%5Cfrac%7B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%7D%7Bx%5E3%7D%3D%5Cfrac2x.%20%0A%5Cend%7Baligned%7D$

【解2】應用分式的基本性質(zhì)，把分子分母都乘以 x2－1，得

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B2%2B%5Cfrac1%7Bx-1%7D-%5Cfrac1%7Bx%2B1%7D%7D%7Bx%2B%5Cfrac%20x%7Bx%5E2-1%7D%7D%3D%5Cfrac%7B2(x%5E2-1)%2B(x%2B1)-(x-1)%7D%7Bx(x%5E2-1)%2Bx%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2x%5E2-2%2Bx%2B1-x%2B1%7D%7Bx%5E3-x%2Bx%7D%3D%5Cfrac%7B2x%5E2%7D%7Bx%5E3%7D%3D%5Cfrac2x.%20%0A%5Cend%7Baligned%7D$

例3.化簡： $%5Cscriptsize(1)%5C%3B%5Cfrac%7B%5Cfrac%20cb%7Da%3B%5C%3B(2)%5C%3B%5Cfrac%20c%7B%5Cfrac%20ba%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5Cfrac%7B%5Cfrac%20cb%7Da%3D%5Cfrac%7B%5Cfrac%20cb%5Ctimes%20b%7D%7Ba%5Ctimes%20b%7D%3D%5Cfrac%20c%7Bab%7D%2C%20%5C%5C%0A%26%E6%88%96%5C%3B%5Cfrac%7B%5Cfrac%7Bc%7D%7Bb%7D%7D%7Ba%7D%3D%5Cfrac%7Bc%7D%7Bb%7D%5Cdiv%20a%3D%5Cfrac%7Bc%7Db%5Ctimes%5Cfrac%7B1%7Da%3D%5Cfrac%7Bc%7D%7Bab%7D%3B%0A%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(2)%5Cfrac%20c%7B%5Cfrac%20ba%7D%3D%5Cfrac%7Bc%5Ctimes%20a%7D%7B%5Cfrac%20ba%5Ctimes%20a%7D%3D%5Cfrac%7Bac%7Db%2C%20%5C%5C%0A%26%E6%88%96%5C%3B%5Cfrac%20c%7B%5Cfrac%20ba%7D%3Dc%5Cdiv%5Cfrac%20ba%3Dc%5Ctimes%5Cfrac%20ab%3D%5Cfrac%7Bac%7Db.%0A%5Cend%7Baligned%7D$

【注意】必須區(qū)別 $%5Cscriptsize%5Cfrac%7B%5Cfrac%20cb%7Da%E4%B8%8E%5Cfrac%20c%7B%5Cfrac%20ba%7D$ ，這兩個式子不相等，不同之處是兩條分數(shù)線的長短，較短的線是小括號，較長的線是大括號。

例4.化簡： $%5Cscriptsize%5Cfrac1%7B1-%5Cfrac%7B1%2Bx%7D%7Bx-%5Cfrac1x%7D%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac1%7B1-%5Cfrac%7B1%2Bx%7D%7Bx-%5Cfrac1x%7D%7D%3D%5Cfrac1%7B1-%5Cfrac%7B1%2Bx%7D%7B%5Cfrac%7Bx%5E2-1%7Dx%7D%7D%3D%5Cfrac1%7B1-(1%2Bx)%5Ccdot%5Cfrac%20x%7Bx%5E2-1%7D%7D%5C%5C%0A%26%3D%5Cfrac1%7B1-%5Cfrac%7B(1%2Bx)%5Ccdot%20x%7D%7B(x%2B1)%5Ccdot(x-1)%7D%7D%3D%5Cfrac1%7B1-%5Cfrac%20x%7Bx-1%7D%7D%20%5C%5C%0A%26%3D%5Cfrac1%7B%5Cfrac%7Bx-1-x%7D%7Bx-1%7D%7D%3D1%5Ccdot%5Cfrac%7Bx-1%7D%7B-1%7D%3D1-x.%20%0A%5Cend%7Baligned%7D$

例5.化簡： $%5Cscriptsize%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac1x%7D%7D%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac1x%7D%7D%7D%3D%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac1%7B%5Cfrac%7Bx%2B1%7Dx%7D%7D%7D%5C%5C%0A%26%3D%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B1%5Ccdot%5Cfrac%20x%7Bx%2B1%7D%7D%7D%3D%5Cfrac1%7B1%2B%5Cfrac1%7B1%2B%5Cfrac%20x%7Bx%2B1%7D%7D%7D%5C%5C%0A%26%3D%5Cfrac1%7B1%2B%5Cfrac1%7B%5Cfrac%7Bx%2B1%2Bx%7D%7Bx%2B1%7D%7D%7D%3D%5Cfrac1%7B1%2B1%5Ccdot%5Cfrac%7Bx%2B1%7D%7B2x%2B1%7D%7D%5C%5C%0A%26%3D%5Cfrac1%7B1%2B%5Cfrac%7Bx%2B1%7D%7B2x%2B1%7D%7D%3D%5Cfrac1%7B%5Cfrac%7B(2x%2B1)%2B(x%2B1)%7D%7B2x%2B1%7D%7D%20%5C%5C%0A%26%3D%5Cfrac%7B1%7D%7B%5Cfrac%7B%203x%2B2%7D%7B2x%2B1%7D%7D%3D1%5Ccdot%5Cfrac%7B2x%2B1%7D%7B3x%2B2%7D%3D%5Cfrac%7B2x%2B1%7D%7B3x%2B2%7D.%20%0A%5Cend%7Baligned%7D$

習題5-10

化簡：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B1%2B%5Cfrac1a%7D%7B1-%5Cfrac1a%7D.%5C%3B2%E3%80%81%5Cfrac%7B1-c%5E2%7D%7B1%2B%5Cfrac1c%7D.%5C%3B3%E3%80%81%5Cfrac%7B%5Cfrac1a%2B%5Cfrac1b%7D%7B%5Cfrac1%7Bb%5E2%7D-%5Cfrac1%7Ba%5E2%7D%7D.%5C%5C%0A%264%E3%80%81%5Cfrac%7B3a-%5Cfrac%7B3x%5E2%7Da%7D%7B1%2B%5Cfrac%20xa%7D.%5C%3B5%E3%80%81%5Cfrac%7B%5Cfrac1x-%5Cfrac2%7Bx%5E2%7D-%5Cfrac3%7Bx%5E3%7D%7D%7B%5Cfrac9x-x%7D.%5C%3B6%E3%80%81%5Cfrac%7B2x%5E2-x%7D%7B%5Cfrac1%7Bx%5E2%7D-4%7D.%5C%5C%0A%267%E3%80%81%5Cfrac%7Bx-1%2B%5Cfrac6%7Bx-6%7D%7D%7Bx-2%2B%5Cfrac3%7Bx-6%7D%7D.%5Cquad%208%E3%80%81%5Cfrac1%7Ba-%5Cfrac%7Ba%5E2-1%7D%7Ba%2B%5Cfrac1%7Ba-1%7D%7D%7D.%5C%5C%0A%269%E3%80%81%5Cfrac%7Bx%2B%5Cfrac%7By-x%7D%7B1%2Bxy%7D%7D%7B1-x%5Cleft(%5Cfrac%7By-x%7D%7B1%2Bxy%7D%5Cright)%7D.%5Cquad10%E3%80%81%5Cfrac1%7B1%2B%5Cfrac%20x%7B1%2Bx%2B%5Cfrac%7B2x%5E2%7D%7B1-x%7D%7D%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7Ba%2B1%7D%7Ba-1%7D%3B%5C%3B2%E3%80%81%20c(1-c)%3B%5C%3B3%E3%80%81%5Cfrac%7Bab%7D%7Ba-b%7D%3B%5C%3B4%E3%80%813(a-x)%3B%20%5C%5C%0A%265%E3%80%81-%5Cfrac%7Bx%2B1%7D%7Bx%5E%7B2%7D(x%2B3)%7D%3B%5C%3B6%E3%80%81-%5Cfrac%7Bx%5E%7B3%7D%7D%7B2x%2B1%7D%3B%5C%3B7%E3%80%81%5Cfrac%7Bx-4%7D%7Bx-5%7D%3B%5C%5C%268%E3%80%81%5Cfrac%7Ba%5E%7B2%7D-a%2B1%7D%7B2a-1%7D%3B%20%5C%3B9%E3%80%81y%3B%5C%3B10%E3%80%81%5Cfrac%7B1%2Bx%5E%7B2%7D%7D%7B1%2Bx%7D.%0A%5Cend%7Baligned%7D$

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【種花家務·代數(shù)】1-5-10繁分式『數(shù)理化自學叢書6677版』

習題5-10