【種花家務(wù)·代數(shù)】1-5-06分式的加減法『數(shù)理化自學(xué)叢書6677版』

2023-09-29 11:01 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冊的基礎(chǔ)上又添加了1冊八五人教中學(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、幾何5冊實(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?；二則，我認(rèn)為《微積分初步》這本書對“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，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?

§5-6分式的加減法

【01】分式的加減法，可以依照分?jǐn)?shù)加減法的同樣法則來進(jìn)行。

1、分母相同的分式的加減法

【02】這和分母相同的分?jǐn)?shù)加減法一樣，我們有下面的分母相同的分式的加減法則：分母相同的分式相加減，分母不變，分子相加減。即

$%5Cfrac%20ba%2B%5Cfrac%20ca%3D%5Cfrac%7Bb%2Bc%7Da%3B%5Cquad%5Cfrac%20ba-%5Cfrac%20ca%3D%5Cfrac%7Bb-c%7Da.$

例1.計(jì)算： $%5Cscriptsize%5Cfrac%7B5a%2B6b%7D%7B3a%5E2bc%7D%2B%5Cfrac%7B3b-4a%7D%7B3a%5E2bc%7D-%5Cfrac%7Ba%2B3b%7D%7B3a%5E2bc%7D$ ? 。

【解】三個(gè)加式的分母相同，只要對分子進(jìn)行加減：

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B5a%2B6b%7D%7B3a%5E2bc%7D%2B%5Cfrac%7B3b-4a%7D%7B3a%5E2bc%7D-%5Cfrac%7Ba%2B3b%7D%7B3a%5E2bc%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(5a%2B6b)%2B(3b-4a)-(a%2B3b)%7D%7B3a%5E2bc%7D%20%5C%5C%0A%26%3D%5Cfrac%7B5a%2B6b%2B3b-4a-a-3b%7D%7B3a%5E2bc%7D%20%5C%5C%0A%26%3D%5Cfrac%7B6b%7D%7B3a%5E%7B%5Cmathbf%7B2%7D%7Dbc%7D%3D%5Cfrac%7B2%7D%7Ba%5E%7B%5Cmathbf%7B2%7D%7Dc%7D.%0A%5Cend%7Baligned%7D$

【注意1】原來各分式的分子如果是多項(xiàng)式，合并成一個(gè)分式后原來的分子要添上括號；如果做的是減法，去括號時(shí)要改變括號內(nèi)各項(xiàng)的符號。這種地方很容易做錯(cuò)，切須注意。

【注意2】加減法合并同類項(xiàng)之后，如果分子分母有公因式，要進(jìn)行約分，化成最簡分式。

例2.計(jì)算： $%5Cscriptsize%5Cfrac%7B12x-7%7D%7Bx%5E2-5x-6%7D-%5Cfrac%7B3x%2B7%7D%7Bx%5E2-5x-6%7D-%5Cfrac%7B8x-15%7D%7Bx%5E2-5x-6%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B12x-7%7D%7Bx%5E2-5x-6%7D-%5Cfrac%7B3x%2B7%7D%7Bx%5E2-5x-6%7D-%5Cfrac%7B8x-15%7D%7Bx%5E2-5x-6%7D%5C%5C%0A%26%3D%5Cfrac%7B(12x-7)-(3x%2B7)-(8x-15)%7D%7Bx%5E2-5x-6%7D%5C%5C%0A%26%3D%5Cfrac%7B12x-7-3x-7-8x%2B15%7D%7Bx%5E2-5x-6%7D%5C%5C%0A%26%3D%5Cfrac%7Bx%2B1%7D%7Bx%5E2-5x-6%7D%5C%5C%0A%26%3D%5Cfrac%7Bx%2B1%7D%7B(x-6)%5Cleft(x%2B1%5Cright)%7D%5C%5C%0A%26%3D%5Cfrac1%7Bx-6%7D.%0A%0A%5Cend%7Baligned%7D$

例3.計(jì)算： $%5Cscriptsize%5Cfrac%7Ba%5E2-2a%2B1%7D%7Ba%5E2%2Ba%2B1%7D-%5Cfrac%7Ba%5E2%2B3a-3%7D%7Ba%5E2%2Ba%2B1%7D%2B%5Cfrac%7B5a-4%7D%7Ba%5E2%2Ba%2B1%7D$ 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7Ba%5E2-2a%2B1%7D%7Ba%5E2%2Ba%2B1%7D-%5Cfrac%7Ba%5E2%2B3a-3%7D%7Ba%5E2%2Ba%2B1%7D%2B%5Cfrac%7B5a-4%7D%7Ba%5E2%2Ba%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a%5E%7B2%7D-2a%2B1)-(a%5E%7B2%7D%2B3a-3)%2B(5a-4)%7D%7Ba%5E%7B2%7D%2Ba%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2-2a%2B1-a%5E2-3a%2B3%2B5a-4%7D%7Ba%5E2%2Ba%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac0%7Ba%5E2%2Ba%2B1%7D%3D0%0A%5Cend%7Baligned%7D$

【說明】分子合并同類項(xiàng)，各項(xiàng)恰好正負(fù)相消，得 0，因?yàn)?0 除以任何不等于 0 的數(shù)是 0，所以結(jié)果等于 0? 。

例4.計(jì)算： $%5Cscriptsize%5Cfrac%7B3a%5E2-5a%7D%7Ba%5E2%2B1%7D-%5Cfrac%7B2a%5E2-5a%2B1%7D%7Ba%5E2%2B1%7D%2B%5Cfrac%7B2(a%5E2%2B2)%7D%7Ba%5E2%2B1%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B3a%5E2-5a%7D%7Ba%5E2%2B1%7D-%5Cfrac%7B2a%5E2-5a%2B1%7D%7Ba%5E2%2B1%7D%2B%5Cfrac%7B2a%5E2%2B4%7D%7Ba%5E2%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(3a%5E2-5a)-(2a%5E2-5a%2B1)%2B(2a%5E2%2B4)%7D%7Ba%5E2%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B3a%5E2-5a-2a%5E2%2B5a-1%2B2a%5E2%2B4%7D%7Ba%5E2%2B1%7D%20%5C%5C%0A%26%3D%5Cfrac%7B3a%5E2%2B3%7D%7Ba%5E2%2B1%7D%3D%5Cfrac%7B3%5Cleft(a%5E2%2B1%5Cright)%7D%7Ba%5E2%2B1%7D%3D3.%0A%5Cend%7Baligned%7D$

【說明】約分后分母變?yōu)?1，分子是 3，結(jié)果等于3? 。

習(xí)題5-6(1)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B2a-3b%7D%7B3a%5E%7B3%7Db%5E%7B5%7Dc%5E%7B2%7D%7D-%5Cfrac%7B3b-5c%7D%7B3a%5E%7B3%7Db%5E%7B5%7Dc%5E%7B2%7D%7D%2B%5Cfrac%7B4a%5E%7B2%7D-5c%7D%7B3a%5E%7B3%7Db%5E%7B5%7Dc%5E%7B2%7D%7D.%20%20%5C%5C%0A%262%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-ax%2Ba%5E%7B2%7D%7D%7B12a%5E%7B3%7Dx%5E%7B4%7D%7D-%5Cfrac%7B2x%5E%7B2%7D-3ax%2B3a%5E%7B2%7D%7D%7B12a%5E%7B3%7Dx%5E%7B4%7D%7D-%5Cfrac%7B5x%5E%7B2%7D%2B2ax-2a%5E%7B3%7D%7D%7B12a%5E%7B3%7Dx%5E%7B4%7D%7D.%20%20%5C%5C%0A%263%E3%80%81%5Cfrac%7Bx%5E2%2B4%7D%7Bx%2B3%7D%2B%5Cfrac%7B2x-7%7D%7Bx%2B3%7D-%5Cfrac%7Bx%5E2%2Bx-6%7D%7Bx%2B3%7D.%20%5C%5C%0A%264%E3%80%81%5Cfrac%7B3x-5%7D%7Bx%5E%7B2%7D-3x-4%7D%2B%5Cfrac%7Bx%5E%7B2%7D-7x%2B2%7D%7Bx%5E%7B2%7D-3x-4%7D-%5Cfrac%7B2(x%2B2)%7D%7Bx%5E%7B2%7D-3x-4%7D.%5C%5C%0A%265%E3%80%81%5Cfrac%7B5x-7%7D%7Bx%5E%7B2%7D%2B3x%7D-%5Cfrac%7B2x%2B5%7D%7Bx%5E%7B2%7D%2B3x%7D-%5Cfrac%7B2x-15%7D%7Bx%5E%7B2%7D%2B3x%7D.%20%5C%5C%0A%266%E3%80%81%5Cfrac%7Bx%5E3-4x%7D%7Bx%5E2-3x%2B2%7D-%5Cfrac%7B3x%5E2-6x%7D%7Bx%5E2-3x%2B2%7D.%20%5C%5C%0A%267%E3%80%81%5Cfrac%7B3a%5E2%2B3a-1%7D%7Ba%5E2-5a-3%7D-%5Cfrac%7Ba%5E2%2B2a%2B7%7D%7Ba%5E2-5a-3%7D-%5Cfrac%7Ba%5E2%2B6a-5%7D%7Ba%5E2-5a-3%7D.%20%5C%5C%0A%268%E3%80%81%5Cfrac%7B3x%5E%7B2%7D-5ax%2Ba%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2B3ax%2B2a%5E%7B2%7D%7D-%5Cfrac%7Bx%5E%7B2%7D%2Bax%2B3a%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2B3ax%2B2a%5E%7B2%7D%7D-%5Cfrac%7B2x%5E%7B2%7D-6ax-2a%5E%7B2%7D%7D%7Bx%5E%7B2%7D%2B3ax%2B2a%5E%7B2%7D%7D.%20%5C%5C%0A%269%E3%80%81%5Cfrac%7Bx%5E2%2B3x-5%7D%7Bx%5E2-3x-18%7D-%5Cfrac%7B2x%5E2-4x%2B7%7D%7Bx%5E2-3x-18%7D-%5Cfrac%7B7x-12-x%5E2%7D%7Bx%5E2-3x-18%7D.%20%5C%5C%0A%2610%E3%80%81%5Cfrac%7Bx%5E2-a%5E2%7D%7Bx%5E2-9a%5E2%7D-%5Cfrac%7B(x-2a)%5E2%7D%7Bx%5E2-9a%5E2%7D%2B%5Cfrac%7B5a%5E2-4ax%7D%7Bx%5E2-9a%5E2%7D.%0A%0A%5Cend%7Baligned%7D$

【答案】6、x；7、1；8、0；9、0；10、0? 。

2、分母不相同的分式的加減法

和異分母分?jǐn)?shù)的加減法一樣，做分母不相同的分式的加減法可以用下面的分母不相同的分式的加減法則：分母不相同的分式相加減，先通分，再照分母相同的分式加減法進(jìn)行。

例5.計(jì)算： $%5Cscriptsize%5Cfrac%7B4a%5E2-5ab%7D%7B2b%5E2%7D-%5Cfrac%7B2b%5E2-5ab%7D%7B3a%5E2%7D%2B%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7B5ab%7D$ 。

【解】各分式的最簡公分母是 30a2b2，

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B4a%5E2-5ab%7D%7B2b%5E2%7D-%5Cfrac%7B2b%5E2-5ab%7D%7B3a%5E2%7D%2B%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7B5ab%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(4a%5E2-5ab)15a%5E2-(2b%5E2-5ab)10b%5E2%2B(a%5E2%2Bb%5E2)6ab%7D%7B30a%5E2b%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B60a%5E4-75a%5E3b-20b%5E4%2B50ab%5E3%2B6a%5E3b%2B6ab%5E3%7D%7B30a%5E2b%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B60a%5E4-69a%5E3b%2B56ab%5E3-20b%5E4%7D%7B30a%5E2b%5E2%7D.%20%0A%0A%5Cend%7Baligned%7D$

例6.計(jì)算： $%5Cscriptsize%5Cfrac%7Ba%5E2-2ac%2Bc%5E2%7D%7Ba%5E2c%5E2%7D-%5Cfrac%7Bb%5E2-2bc%2Bc%5E2%7D%7Bb%5E2c%5E2%7D$ 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7Ba%5E2-2ac%2Bc%5E2%7D%7Ba%5E2c%5E2%7D-%5Cfrac%7Bb%5E2-2bc%2Bc%5E2%7D%7Bb%5E2c%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(a%5E2-2ac%2Bc%5E2%5Cright)b%5E2-%5Cleft(b%5E2-2bc%2Bc%5E2%5Cright)a%5E2%7D%7Ba%5E2b%5E2c%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2b%5E2-2ab%5E2c%2Bb%5E2c%5E2-a%5E2b%5E2%2B2a%5E2bc-a%5E2c%5E2%7D%7Ba%5E2b%5E2c%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2a%5E2bc-a%5E2c%5E2-2ab%5E2c%2Bb%5E2c%5E2%7D%7Ba%5E2b%5E2c%5E2%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bc%5Cleft(2a%5E2b-a%5E2c-2ab%5E2%2Bb%5E2c%5Cright)%7D%7Ba%5E2b%5E2c%5E2%7D%5C%5C%0A%26%3D%5Cfrac%7B2a%5E2b-a%5E2c-2ab%5E2%2Bb%5E2c%7D%7Ba%5E2b%5E2c%7D.%0A%5Cend%7Baligned%7D$

例7.計(jì)算： $%5Cscriptsize%5Cfrac1%7Bx-y%7D-%5Cfrac%7B3xy%7D%7Bx%5E3-y%5E3%7D$ 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7B1%7D%7Bx-y%7D-%5Cfrac%7B3xy%7D%7Bx%5E%7B3%7D-y%5E%7B3%7D%7D%3D%5Cfrac%7B1%7D%7Bx-y%7D-%5Cfrac%7B3xy%7D%7B(x-y)(x%5E%7B2%7D%2Bxy%2By%5E%7B2%7D)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bx%5E2%2Bxy%2By%5E2-3xy%7D%7B(x-y)(x%5E2%2Bxy%2By%5E2)%7D%3D%5Cfrac%7Bx%5E2-2xy%2By%5E2%7D%7B(x-y)(x%5E2%2Bxy%2By%5E2)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(x-y)%5E2%7D%7B(x-y)%5Cleft(x%5E2%2Bxy%2By%5E2%5Cright)%7D%3D%5Cfrac%7Bx-y%7D%7Bx%5E2%2Bxy%2By%5E2%7D.%20%20%0A%0A%5Cend%7Baligned%7D$

例8.計(jì)算： $%5Cscriptsize%5Cfrac%7Bx%2B2%7D%7Bx%5E2-9x%2B20%7D-%5Cfrac%7Bx-1%7D%7Bx%5E2-11x%2B30%7D$ 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7Bx%2B2%7D%7Bx%5E2-9x%2B20%7D-%5Cfrac%7Bx-1%7D%7Bx%5E2-11x%2B30%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bx%2B2%7D%7B%5Cleft(x-5%5Cright)%5Cleft(x-4%5Cright)%7D-%5Cfrac%7Bx-1%7D%7B%5Cleft(x-5%5Cright)%5Cleft(x-6%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(x%2B2%5Cright)%5Cleft(x-6%5Cright)-%5Cleft(x-1%5Cright)%5Cleft(x-4%5Cright)%7D%7B%5Cleft(x-5%5Cright)%5Cleft(x-4%5Cright)%5Cleft(x-6%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(x%5E2-4x-12)-(x%5E2-5x%2B4)%7D%7B(x-5)%5Cleft(x-4%5Cright)%5Cleft(x-6%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bx%5E2-4x-12-x%5E2%2B5x-4%7D%7B%5Cleft(x-5%5Cright)%5Cleft(x-4%5Cright)%5Cleft(x-6%5Cright)%7D%5C%5C%0A%26%3D%5Cfrac%7Bx-16%7D%7B%5Cleft(x-5%5Cright)%5Cleft(x-4%5Cright)%5Cleft(x-6%5Cright)%7D.%0A%5Cend%7Baligned%7D$

習(xí)題5-6(2)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac5%7B3x%7D-%5Cfrac%7B2x-7%7D%7B12x%5E2%7D.%5C%5C%0A%262%E3%80%81%5Cfrac%7Ba%2Bb%7D%7B3a%5E2b%7D-%5Cfrac%7Ba-b%7D%7B5ab%5E2%7D.%20%5C%5C%0A%263%E3%80%81%5Cfrac%7B3xy-4%7D%7Bx%5E2y%5E2%7D-%5Cfrac%7B5y%5E2%2B7%7D%7Bxy%5E3%7D-%5Cfrac%7B6x%5E2-11%7D%7Bx%5E3y%7D.%5C%5C%0A%264%E3%80%81%5Cfrac1%7Bx-7%7D-%5Cfrac1%7Bx-3%7D.%20%5C%5C%0A%26%205%E3%80%81%5Cfrac%20x%7B1%2Bx%7D%2B%5Cfrac%20x%7B1-x%7D.%5C%5C%0A%266%E3%80%81%5Cfrac%7B1%2Bx%7D%7B1%2Bx%2Bx%5E2%7D-%5Cfrac%7B1-x%7D%7B1-x%2Bx%5E2%7D.%20%20%5C%5C%0A%267%E3%80%81%5Cfrac%20a%7Ba-b%7D-%5Cfrac%7Ba%5E3%2B2ab%5E3%7D%7Ba%5E3-b%5E3%7D.%5C%5C%0A%268%E3%80%81%5Cfrac1%7Ba-b%7D-%5Cfrac%7Ba%2Bb%7D%7B(a-b)%5E2%7D-%5Cfrac%7Ba%2B2b%7D%7Ba%5E2-b%5E2%7D.%20%20%5C%5C%0A%269%E3%80%81%5Cfrac%7Bx%2B1%7D%7Bx%5E2-5x%2B6%7D-%5Cfrac%7Bx-1%7D%7Bx%5E2-7x%2B12%7D.%5C%5C%0A%2610%E3%80%81%5Cfrac%7B3%7D%7Ba%5E2%2B7a%2B12%7D-%5Cfrac%7B2%7D%7Ba%5E2%2B2a-8%7D.%20%5C%5C%0A%2611%E3%80%81%5Cfrac%7B1%7D%7Bx-1%7D-%5Cfrac%7B1%7D%7Bx%2B2%7D-%5Cfrac%7Bx%2B3%7D%7B(x%2B1)(x%2B2)%7D.%5C%5C%0A%2612%E3%80%81%5Cfrac%7Bx%2By%7D%7By%7D-%5Cfrac%7B2x%7D%7Bx%2By%7D%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7By(x%5E%7B2%7D-y%5E%7B2%7D)%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%261%E3%80%81%5Cfrac%7B18x%2B7%7D%7B12x%5E2%7D%3B%26%262%E3%80%81%5Cfrac%7B-3a%5E2%2B8ab%2B5b%5E2%7D%7B15a%5E2b%5E2%7D%3B%5C%5C%0A%26%263%E3%80%81%5Cfrac%7B-8x%5E2y%5E2-4xy-7x%5E2%2B11y%5E2%7D%7Bx%5E3y%5E3%7D%3B%26%264%E3%80%81%5Cfrac4%7B(x-7)(x-3)%7D%3B%5C%5C%0A%26%265%E3%80%81%5Cfrac%7B2x%7D%7B(1%2Bx)(1-x)%7D%3B%26%266%E3%80%81%5Cfrac%7B2x%5E3%7D%7B(1%2Bx%2Bx%5E2)(1-x%2Bx%5E2)%7D%3B%5C%5C%0A%26%267%E3%80%81%5Cfrac%7Bab%7D%7Ba%5Ez%2Bab%2Bb%5Ez%7D%3B%26%268%E3%80%81%5Cfrac%7B-(a%5E%7B2%7D%2B3ab)%7D%7B(a%2Bb)(a-b)%5E%7B2%7D%7D%3B%5C%5C%0A%26%269%E3%80%81%5Cfrac%7B-6%7D%7B(x-3)(x-2)(x-4)%7D%3B%20%26%26%0A10%E3%80%81%5Cfrac%7Ba-12%7D%7B(a%2B3)(a%2B4)(a-2)%7D%3B%5C%5C%0A%26%2611%E3%80%81%5Cfrac%7B-(x-3)%7D%7B(x-1)(x%2B1)%7D%3B%20%26%2612%E3%80%81%5Cfrac%7B2x%5E3-x%5E2y%2Bxy%5E2-y%5E3%7D%7By(x%2By)(x-y)%7D.%0A%0A%5Cend%7Beqnarray%7D$

例9.計(jì)算： $%5Cscriptsize%5Cfrac1%7B%5Cleft(a-b%5Cright)%5Cleft(b-c%5Cright)%7D%2B%5Cfrac1%7B%5Cleft(b-a%5Cright)%5Cleft(a-c%5Cright)%7D$ 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%20%5Cfrac1%7B%5Cleft(a-b%5Cright)%5Cleft(b-c%5Cright)%7D%2B%5Cfrac1%7B%5Cleft(b-a%5Cright)%5Cleft(a-c%5Cright)%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7B1%7D%7B%5Cleft(a-b%5Cright)%5Cleft(b-c%5Cright)%7D-%5Cfrac%7B1%7D%7B%5Cleft(a-b%5Cright)%5Cleft(a-c%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a-c)-(b-c)%7D%7B(a-b)%5Cleft(b-c%5Cright)%5Cleft(a-c%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba-c-b%2Bc%7D%7B(a-b)%5Cleft(b-c%5Cright)%5Cleft(a-c%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba-b%7D%7B%5Cleft(a-b%5Cright)%5Cleft(b-c%5Cright)%7D%5Cfrac%7B%7D%7B%5Cleft(a-c%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac1%7B%5Cleft(b-c%5Cright)%5Cleft(a-c%5Cright)%7D.%0A%5Cend%7Baligned%7D$

例10.計(jì)算： $%5Cscriptsize%5Cfrac%7Ba%5E2-bc%7D%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a%2Bc%5Cright)%7D%2B%5Cfrac%7Bb%5E2-ac%7D%7B%5Cleft(b%2Bc%5Cright)%5Cleft(a%2Bb%5Cright)%7D%2B%5Cfrac%7Bc%5E2-ab%7D%7B%5Cleft(a%2Bc%5Cright)%5Cleft(b%2Bc%5Cright)%7D$ ? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cfrac%7Ba%5E2-bc%7D%7B(a%2Bb)%7D%2B%5Cfrac%7Bb%5E2-ac%7D%7B(b%2Bc)%5Cleft(a%2Bb%5Cright)%7D%2B%5Cfrac%7Bc%5E2-ab%7D%7B(a%2Bc)%5Cleft(b%2Bc%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a%5E%7B2%7D-bc)(b%2Bc)%2B(b%5E%7B2%7D-ac)(a%2Bc)%2B(c%5E%7B2%7D-ab)(a%2Bb)%7D%7B(a%2Bb)(a%2Bc)(b%2Bc)%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2b%2Ba%5E2c-b%5E2c-bc%5E2%2Bab%5E2%2Bb%5E2c-a%5E2c-ac%5E2%2Bac%5E2%2Bbc%5E2-a%5E2b-ab%5E2%7D%7B(a%2Bb)%5Cleft(a%2Bc%5Cright)%5Cleft(b%2Bc%5Cright)%7D%20%5C%5C%0A%26%3D%5Cfrac0%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a%2Bc%5Cright)%5Cleft(b%2Bc%5Cright)%7D%3D0.%0A%5Cend%7Baligned%7D$

習(xí)題5-6(3)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7Bz%7D%7Bx%5E%7B2%7D-xz%7D%2B%5Cfrac%7Bx%7D%7Bz%5E%7B2%7D-xz%7D.%20%5C%5C%0A%262%E3%80%81%5Cfrac%7Bx%5E3%2Bx%5E2y%7D%7Bx%5E2y-y%5E3%7D-%5Cfrac%7Bx(x-y)%7D%7Bxy%2Bx%5E2%7D-%5Cfrac%7B2xy%7D%7Bx%5E2-y%5E2%7D.%20%5C%5C%0A%263%E3%80%81%5Cfrac%20a%7Ba-b%7D-%5Cfrac%20b%7Ba%2B2b%7D%2B%5Cfrac%7Ba%5E2%2Bb%5E2%2Bab%7D%7B(b-a)(2b%2Ba)%7D.%5C%5C%0A%264%E3%80%81%5Cfrac1%7B(a-b)(b-c)%7D%2B%5Cfrac1%7B(b-a)(a-c)%7D-%5Cfrac1%7B(c-b)(c-a)%7D.%20%5C%5C%0A%265%E3%80%81%5Cfrac%7B1%7D%7Ba%5E2-(b-c)%5E2%7D-%5Cfrac%7B1%7D%7Bb%5E2-(c-a)%5E2%7D-%5Cfrac%7B1%7D%7Bc%5E2-(a-b)%5E2%7D.%20%5C%5C%0A%266%E3%80%81%5Cfrac%7Ba-c%7D%7B(a%2Bb)%5E2-c%5E2%7D-%5Cfrac%7Ba-b%7D%7B(a%2Bc)%5E2-b%5E2%7D.%5C%5C%0A%267%E3%80%81%5Cfrac%7B3%2B2x%7D%7B2-x%7D-%5Cfrac%7B2-3x%7D%7B2%2Bx%7D%2B%5Cfrac%7B16x-x%5E2%7D%7Bx%5E2-4%7D.%5C%5C%0A%268%E3%80%81%5Cfrac%7B1%7D%7B(2-a)(3-a)%7D-%5Cfrac%7B2%7D%7B(a-1)(a-3)%7D%2B%5Cfrac%7B1%7D%7B(a-1)(a-2)%7D.%5C%5C%0A%269%E3%80%81%5Cfrac%7Ba%2Bb%7D%7B(b-c)(c-a)%7D%2B%5Cfrac%7Bb%2Bc%7D%7B(b-a)(a-c)%7D%2B%5Cfrac%7Ba%2Bc%7D%7B(a-b)(b-c)%7D.%5C%5C%0A%2610%E3%80%81%5Cfrac%7Ba%5E2-bc%7D%7B(a-b)(a-c)%7D%2B%5Cfrac%7Bb%5E2%2Bac%7D%7B(b%2Bc)(b-a)%7D%2B%5Cfrac%7Bc%5E2%2Bab%7D%7B(c-a)(c%2Bb)%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%261%E3%80%81-%5Cfrac%7Bx%2Bz%7D%7Bxz%7D%3B%5C%3B2%E3%80%81%5Cfrac%7Bx%5E2%2Bxy%2By%5E2%7D%7By(x%2By)%7D%3B%5C%3B3%E3%80%810%3B%5C%3B4%E3%80%810%3B%20%5C%5C%0A%265%E3%80%81%5Cfrac%7Bb%2Bc-3a%7D%7B(a-b%2Bc)(a%2Bb-c)(b%2Bc-a)%7D%3B%5C%5C%0A%266%E3%80%81%5Cfrac%7Bb%5E2-c%5E2%2Bac-ab%7D%7B(a%2Bb%2Bc)(a%2Bb-c)(a%2Bc-b)%7D%3B%5C%5C%0A%267%E3%80%81%5Cfrac%7B1%7D%7Bx%2B2%7D%3B%5Cquad8%E3%80%810%3B%5Cquad9%E3%80%810%3B%5Cquad10.0.%0A%5Cend%7Baligned%7D$

例11.計(jì)算： $%5Cscriptsize%20x%2B8-%5Cfrac%7Bx%5E2%2B4%7D%7Bx-2%7D$ 。

【分析】這是一個(gè)整式和一個(gè)分式的代數(shù)和，把整式 x＋8 當(dāng)做 $%5Cscriptsize%5Cfrac%7Bx%2B8%7D%7B1%7D%20$ 再通分。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26x%2B8-%5Cfrac%7Bx%5E%7B2%7D%2B4%7D%7Bx-2%7D%5C%5C%0A%26%20%3D%5Cfrac%7B(x%2B8)(x-2)-(x%5E2%2B4)%7D%7Bx-2%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7Bx%5E2%2B6x-16-x%5E2-4%7D%7Bx-2%7D%20%5C%5C%0A%26%3D%5Cfrac%7B6x-20%7D%7Bx-2%7D.%0A%5Cend%7Baligned%7D$

例12.計(jì)算： $%5Cscriptsize%20a%2B%5Cfrac1%7Ba-1%7D$ ? 。

【解】 $%5Csmall%20a%2B%5Cfrac1%7Ba-1%7D%3D%5Cfrac%7Ba%5Cleft(a-1%5Cright)%2B1%7D%7Ba-1%7D%3D%5Cfrac%7Ba%5E2-a%2B1%7D%7Ba-1%7D$ ? 。

【注意】

????????我們在整式一章中，學(xué)習(xí)過有余式的除法，如 (a2－a＋1)÷(a－1) 時(shí)得部分的商 a，剩下余式 1? 。如果要表達(dá)它們之間的關(guān)系，我們可以寫做 $%5Cscriptsize%5Cfrac%7Ba%5E%7B2%7D-a%2B1%7D%7Ba-1%7D%3Da%2B%5Cfrac%7B1%7D%7Ba-1%7D$ ，和算術(shù)里的 $%5Cscriptsize%5Cfrac%7B36%7D5%3D7%5Cfrac15$ 相類似。

????????這里應(yīng)該注意，在算術(shù)里 $%5Cscriptsize7%5Cfrac15$ 就是 $%5Cscriptsize7%2B%5Cfrac15$ ，普通寫做帶分?jǐn)?shù) $%5Cscriptsize7%5Cfrac15$ ，加號可以省去，但在代數(shù)里，不能把 $%5Cscriptsize%20a%2B%5Cfrac%7B1%7D%7Ba-1%7D$ 寫做 $%5Cscriptsize%20a%5Cfrac%7B1%7D%7Ba-1%7D$ ，如果把加號省去了，就和 $%5Cscriptsize%20a%C3%97%5Cfrac%7B1%7D%7Ba-1%7D$ 要混淆了。

習(xí)題5-6(4)

計(jì)算(1~8)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B1%7D%7Ba-1%7D%2B1.%20%5C%5C%0A%262%E3%80%81a%2Bb-%5Cfrac%7Ba%5E%7B2%7D%7D%7Ba-b%7D.%20%5C%5C%0A%263%E3%80%81x%2B3%2B%5Cfrac%7B6-x-2x%5E2%7D%7Bx-2%7D.%20%5C%5C%0A%264%E3%80%81%20a%2Bb%2B%5Cfrac%7Ba%5E%7B3%7D-b%5E%7B3%7D%7D%7Ba%5E%7B2%7D-ab%2Bb%5E%7B2%7D%7D.%20%5C%5C%0A%265%E3%80%81(a-b)%5E%7B2%7D-%5Cfrac%7Ba%5E%7B3%7D%2Bb%5E%7B3%7D%7D%7Ba-b%7D.%5C%5C%0A%266%E3%80%81a%5E%7B2%7D-b%5E%7B2%7D%2B%5Cfrac%7Ba%5E%7B3%7D-b%5E%7B3%7D%7D%7Ba%2Bb%7D.%5C%5C%0A%267%E3%80%81%5Cfrac%7By%5E%7B3%7D%7D%7Bx-y%7D%2Bx%5E%7B2%7D%2Bxy%2By%5E%7B2%7D.%5C%5C%0A%268%E3%80%81%5Cfrac%7Bx%2B5%7D%7Bx%5E%7B2%7D%2B3x%2B2%7D%2B3.%0A%5Cend%7Baligned%7D$

把下面的分式化成為一個(gè)整式與一個(gè)分式的代數(shù)和（分式的分子的次數(shù)要低于分母的次數(shù)）：

$%5Csmall%5Cbegin%7Baligned%7D%0A%269%E3%80%81%5Cfrac%7Bx%5E2%2Bx%2B1%7D%7Bx-1%7D.%5C%5C%0A%2610%E3%80%81%5Cfrac%7Ba%5E2%7D%7Ba-1%7D.%5C%5C%0A%2611%E3%80%81(2x%5E2%2B3x-4)%5Cdiv(x%2B1).%5C%5C%0A%2612%E3%80%81(x%5E3-1)%5Cdiv(x%5E2%2Bx-2).%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%269.%5Cquad%20x%2B2%2B%5Cfrac3%7Bx-1%7D%3B%5Cquad10.%5Cquad%20a%2B1%2B%5Cfrac1%7Ba-1%7D%3B%5C%5C%2611.%5Cquad2x%2B1-%5Cfrac5%7Bx%2B1%7D%3B%5Cquad12.%5Cquad%20x-1%2B%5Cfrac%7B3x-3%7D%7Bx%5E2%2Bx-2%7D.%5Cend%7Baligned%7D$

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【種花家務(wù)·代數(shù)】1-5-06分式的加減法『數(shù)理化自學(xué)叢書6677版』

習(xí)題5-6(1)

習(xí)題5-6(2)

習(xí)題5-6(3)

習(xí)題5-6(4)