【種花家務(wù)·代數(shù)】1-5-04約分『數(shù)理化自學(xué)叢書(shū)6677版』

2023-09-28 21:38 作者:山嵓 0人讀過(guò) | 我要投稿

【閱前提示】本篇出自『數(shù)理化自學(xué)叢書(shū)6677版』，此版叢書(shū)是“數(shù)理化自學(xué)叢書(shū)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(shū)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(shū)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(lái)自學(xué)。不過(guò)這套叢書(shū)卻很適合像我這樣已接受過(guò)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(xiě)的注解。

【山話嵓語(yǔ)】我在原有“自學(xué)叢書(shū)”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(yú)座，有一定程度的偶雙癥，但“自學(xué)叢書(shū)”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?；二則，我認(rèn)為《微積分初步》這本書(shū)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(kāi)課；而比較生猛的大學(xué)則是直接開(kāi)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(shū)”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

§5-4約分

1、約分的概念

【01】在算術(shù)里，我們已經(jīng)使用過(guò)約分來(lái)化簡(jiǎn)一個(gè)分?jǐn)?shù)。在代數(shù)里，我們也可以應(yīng)用分式的基本性質(zhì)，把分式的分子分母同除以一個(gè)它們的公因式，把分式化簡(jiǎn)（或約簡(jiǎn)），這樣一種變換叫做約分。

【02】例如把分式 $%5Cscriptsize%5Cfrac%7Ba%5E2b%7D%7B2ab%5E2%7D$ 的分子分母同除以 ab，得到 $%5Cscriptsize%5Cfrac%7Ba%5E2b%5Cdiv%20ab%7D%7B2ab%5E2%5Cdiv%20ab%7D%3D%5Cfrac%7Ba%7D%7B2b%7D$ ，這樣，就把原分式 $%5Cscriptsize%5Cfrac%7Ba%5E2b%7D%7B2ab%5E2%7D$ 約簡(jiǎn)了。

2、最簡(jiǎn)分式（既約分式）

【03】一個(gè)分式的分子分母，如果沒(méi)有 1 以外的公因式，就叫做最簡(jiǎn)分式或者既約分式。如 $%5Cscriptsize%5Cfrac%20a%7B3x%7D%2C%5Cfrac%7Bx-y%7D%7Bx%2By%7D$ 等都是最簡(jiǎn)分式，而 $%5Cscriptsize%5Cfrac%7Ba%7D%7Ba%5E%7B2%7D%7D%2C%5Cfrac%7Bx%2By%7D%7Bx%5E%7B2%7D-y%5E%7B2%7D%7D$ 等就不是最簡(jiǎn)分式。

【04】一個(gè)分式如果不是最簡(jiǎn)分式，我們就應(yīng)該進(jìn)行約分，把它化成最簡(jiǎn)分式。

3、約分的方法

(1)分式的分子分母是同底數(shù)的冪的約分.

例1.化簡(jiǎn)： $%5Cscriptsize(1)%5C%3B%5Cfrac%7Ba%5E8%7D%7Ba%5E3%7D%3B%5C%3B(2)%5C%3B%5Cfrac%7Ba%5E5%7D%7Ba%5E5%7D%3B%5C%3B(3)%5C%3B%5Cfrac%7Ba%5E4%7D%7Ba%5E9%7D%3B%5C%3B(4)%5C%3B%5Cfrac%7Ba%5E5%7D%7Ba%5E%7B10%7D%7D.%0A$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3B%5Cfrac%7Ba%5E%7B8%7D%7D%7Ba%5E%7B3%7D%7D%3D%5Cfrac%7Ba%5E%7B8%7D%5Cdiv%20a%5E%7B3%7D%7D%7Ba%5E%7B3%7D%5Cdiv%20a%5E%7B3%7D%7D%3D%5Cfrac%7Ba%5E%7B5%7D%7D%7B1%7D%3Da%5E%7B5%7D%3B%20%5C%5C%0A%26(2)%5C%3B%5Cfrac%7Ba%5E5%7D%7Ba%5E5%7D%3D%5Cfrac%7Ba%5E5%5Cdiv%20a%5E5%7D%7Ba%5E5%5Cdiv%20a%5E5%7D%3D%5Cfrac11%3D1%3B%20%5C%5C%0A%26%20(3)%5C%3B%5Cfrac%7Ba%5E4%7D%7Ba%5E9%7D%3D%5Cfrac%7Ba%5E4%5Cdiv%20a%5E4%7D%7Ba%5E9%5Cdiv%20a%5E4%7D%3D%5Cfrac1%7Ba%5E5%7D%3B%20%20%5C%5C%0A%26(4)%5C%3B%5Cfrac%7Ba%5E%7B5%7D%7D%7Ba%5E%7B10%7D%7D%20%3D%5Cfrac%7Ba%5E5%5Cdiv%20a%5E5%7D%7Ba%5E%7B10%7D%5Cdiv%20a%5E5%7D%3D%5Cfrac1%7Ba%5E5%7D.%20%0A%5Cend%7Baligned%7D$

一般說(shuō)來(lái)， $%5Cscriptsize%5Cfrac%7Ba%5Em%7D%7Ba%5En%7D$ 可以分三種情況進(jìn)行化簡(jiǎn)：

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5Ctext%7B%20%E5%BD%93%20%7Dm%3En%5Ctext%7B%20%E6%97%B6%2C%7D%5Cfrac%7Ba%5Em%7D%7Ba%5En%7D%3Da%5E%7Bm-n%7D%3B%5C%5C(2)%26%5Ctext%7B%20%E5%BD%93%20%7Dm%3Dn%5Ctext%7B%20%E6%97%B6%2C%7D%5Cfrac%7Ba%5Em%7D%7Ba%5En%7D%3D%5Cfrac%7Ba%5Em%7D%7Ba%5En%7D%3D1%3B%5C%5C(3)%26%5Ctext%7B%20%E5%BD%93%20%7Dm%3Cn%5Ctext%7B%20%E6%97%B6%2C%7D%5Cfrac%7Ba%5Em%7D%7Ba%5En%7D%3D%5Cfrac%7Ba%5Em%5Cdiv%20a%5Em%7D%7Ba%5En%5Cdiv%20a%5Em%7D%3D%5Cfrac1%7Ba%5E%7Bn-m%7D%7D.%5Cend%7Baligned%7D$

例2.化簡(jiǎn)： $%5Cscriptsize%5Cbegin%7Baligned%7D%0A%26(1)%5C%3B%5Cfrac%7Bx%5E%7B8%7D%7D%7Bx%5E%7B2%7D%7D%3B%5C%3B(2)%5C%3B%5Cfrac%7Bx%5E%7B4%7D%7D%7Bx%5E%7B8%7D%7D%3B%5C%3B(3)%5C%3B%5Cfrac%7Bx%5E%7B3%7D%7D%7Bx%5E%7B9%7D%7D%3B%5C%3B(4)%5C%3B%5Cfrac%7Bx%5E%7B4%7D%7D%7Bx%5E%7B12%7D%7D.%0A%5Cend%7Baligned%7D%0A$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3B%5Cfrac%7Bx%5E8%7D%7Bx%5E2%7D%3Dx%5E%7B8-2%7D%3Dx%5E0%3B%5C%5C%0A%26(2)%5C%3B%5Cfrac%7Bx%5E4%7D%7Bx%5E8%7D%3D%5Cfrac1%7Bx%5E%7B8-4%7D%7D%3D%5Cfrac1%7Bx%5E4%7D%3B%5C%5C%0A%26(3)%5C%3B%5Cfrac%7Bx%5E3%7D%7Bx%5E9%7D%3D%5Cfrac1%7Bx%5E%7B9-3%7D%7D%3D%5Cfrac1%7Bx%5E6%7D%3B%5C%5C%0A%26(4)%5C%3B%5Cfrac%7Bx%5E%7B4%7D%7D%7Bx%5E%7B12%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5E%7B12-4%7D%7D%3D%5Cfrac%7B1%7D%7Bx%5E%7B8%7D%7D.%0A%5Cend%7Baligned%7D$

【注意】 $%5Csmall%5Cfrac%7Bx%5E8%7D%7Bx%5E2%7D%5Ctext%7B%20%E4%B8%8D%E7%AD%89%E4%BA%8E%20%7Dx%5E4%5Ctext%7B%2C%7D%5C%3B%5Cfrac%7Bx%5E4%7D%7Bx%5E8%7D%5Ctext%7B%E4%B8%8D%E7%AD%89%E4%BA%8E%7D%5Cfrac1%7Bx%5E2%7D%5Ctext%7B%2C%7D%5C%3B%5Cfrac%7Bx%5E8%7D%7Bx%5E9%7D%5Ctext%7B%E4%B8%8D%E7%AD%89%E4%BA%8E%7D%5Cfrac1%7Bx%5E3%7D.$

習(xí)題5-4(1)

化簡(jiǎn)（m,n 是自然數(shù)）：

$%5Csmall%5Cbegin%7Beqnarray%7D%0A%26%261%E3%80%81%5Cfrac%7Ba%5E%7B12%7D%7D%7Ba%5E4%7D.%26%262%E3%80%81%5Cfrac%7Ba%5E9%7D%7Ba%5E%7B12%7D%7D.%26%263%E3%80%81%5Cfrac%7Bx%5E%7B12%7D%7D%7Bx%5E%7B34%7D%7D.%26%264%E3%80%81%5Cfrac%7B(a%5E4)%5E5%7D%7Ba%5E%7B12%7D%7D.%5C%5C%0A%26%265%E3%80%81%5Cfrac%7B(-a)%5E7%7D%7Ba%5E7%7D.%26%266%E3%80%81%5Cfrac%7B(-x)%5E4%7D%7Bx%5E4%7D.%26%267%E3%80%81%5Cfrac%7Ba%5E8%7D%7B(a%5E4)%5E4%7D.%26%268%E3%80%81%5Cfrac%7B(a%5E6)%5E3%7D%7B(a%5E6)%5E6%7D.%5C%5C%0A%26%269%E3%80%81%5Cfrac%7B(a%5E5)%5E2%7D%7Ba%5E2%7D.%26%2610%E3%80%81%5Cfrac%7B(a%5E5)%5E5%7D%7B(a%5E4)%5E5%7D.%26%2611%E3%80%81%5Cfrac%7Ba%5E%7B2m%7D%7D%7Ba%5Em%7D.%26%2612%E3%80%81%5Cfrac%7Ba%5E%7B2n%7D%7D%7Ba%5E%7B8m%7D%7D.%5C%5C%0A%26%2613%E3%80%81%5Cfrac%7Ba%5E%7B2m%7D%7D%7Ba%5E%7B2m%2Bn%7D%7D.%26%2614%E3%80%81%5Cfrac%7Ba%5E%7B6m%7D%7D%7Ba%5E%7B8m%2Bn%7D%7D.%26%2615%E3%80%81%5Cfrac%7Ba%5E%7Bm-n%7D%7D%7Ba%5E%7Bm%2Bn%7D%7D(m%3En)%2C%26%2616%E3%80%81%5Cfrac%7Ba%5E%7B2m%2B2n%7D%7D%7Ba%5E2%7D.%5Cend%7Beqnarray%7D$

【答案】

$%5Csmall%5Cbegin%7Beqnarray%7D%0A%26%261%E3%80%81%20a%5E8%3B%26%262%E3%80%81%5Cfrac1%7Ba%5E3%7D%3B%26%263%E3%80%81%5Cfrac1%7Bx%5E%7B12%7D%7D%3B%26%264%E3%80%811%3B%5C%5C%0A%26%265%E3%80%81-1%3B%20%26%266%E3%80%811%3B%26%267%E3%80%81%5Cfrac%7B1%7D%7Ba%5E8%7D%3B%26%268%E3%80%81%5Cfrac%7B1%7D%7Ba%5E%7B18%7D%7D%3B%5C%5C%0A%26%269%E3%80%81%20a%5E4%3B%26%2610%E3%80%81%5Cfrac%7B1%7D%7Ba%5E5%7D%3B%26%2611%E3%80%81%20a%5Em%3B%26%2612%E3%80%81%5Cfrac%7B1%7D%7Ba%5En%7D%3B%20%5C%5C%0A%26%2613%E3%80%81%5Cfrac1%7B%5Calpha%5En%7D%3B%26%2614%E3%80%81%5Cfrac1%7B%5Calpha%5E%7B3m%2Bn%7D%7D%3B%26%2615%E3%80%81%5Cfrac1%7B%5Calpha%5E%7B2n%7D%7D%3B%26%2616%E3%80%81a%5E%7B2n%2B2n-2%7D.%20%0A%5Cend%7Beqnarray%7D$

(2)分式的分子分母都是單項(xiàng)式的約分。

例3.化簡(jiǎn)： $%5Cscriptsize%5Cbegin%7Baligned%7D(1)%5Cfrac%7B32x%7D%7B72y%7D%3B%5C%3B(2)%5Cfrac%7B-15ab%5E2x%5E%7B15%7D%7D%7B5abx%5E5%7D%3B%5C%3B(3)%5Cfrac%7B-a%5E3x%5E3%7D%7B-a%5E7x%5E6y%5E3%7D%3B%5C%3B(4)%5Cfrac%7B%2B20a%5E5b%5E2y%5E%7B20%7D%7D%7B-96a%5E3b%5E4x%5E5y%5E%7B10%7D%7D.%5Cend%7Baligned%7D$

【解】把系數(shù)和各相同字母分別約簡(jiǎn)：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5Cfrac%7B32x%7D%7B72y%7D%3D%5Cfrac%7B4x%7D%7B9y%7D%3B%5C%5C%26(2)%5Cfrac%7B-15ab%5E2x%5E%7B15%7D%7D%7B5abx%5E5%7D%3D%5Cfrac%7B-3bx%5E%7B10%7D%7D1%3D-3bx%5E%7B10%7D%3B%5Cend%7Baligned%7D$

【說(shuō)明】分母的因式約去后得 1，分式變成整式。

$%5Csmall(3)%5Cfrac%7B-a%5E3x%5E3%7D%7B-a%5E7x%5E6y%5E3%7D%3D%5Cfrac1%7Ba%5E4x%5E3y%5E3%7D%3B$

【說(shuō)明】分子的因式約去后得 1? 。

$%5Csmall(4)%5Cfrac%7B%2B20a%5E5b%5E%5Cmathbf%7Ba%7Dy%5E%7B20%7D%7D%7B-96a%5E3b%5E4x%5E5y%5E%7B10%7D%7D%3D-%5Cfrac%7B5a%5E2y%5E%7B10%7D%7D%7B24b%5E2x%5E5%7D.$

【說(shuō)明】分母的負(fù)號(hào)一般要移去。

習(xí)題5-4(2)

化簡(jiǎn)：

$%5Csmall%5Cbegin%7Beqnarray%7D%0A%26%261%E3%80%81%5Cfrac%7B16a%5E5b%5E2c%5E3%7D%7B24a%5E2b%5E3c%5E4%7D.%26%262%E3%80%81%5Cfrac%7B15bx%5E2y%5E3z%5E4%7D%7B25ay%5E2z%5E9%7D.%5C%5C%0A%26%263%E3%80%81%5Cfrac%7B-25a%5E%7B25%7D%7D%7B-15a%5E%7B15%7D%7D.%26%264%E3%80%81%5Cfrac%7B144a%5E4b%5E%7B12%7Dc%5E%7B15%7D%7D%7B-128a%5E8b%5E8c%5E%7B18%7D%7D.%5C%5C%0A%26%265%E3%80%81%5Cfrac%7B(a%5E2b%5E3)%5E2%7D%7Ba%5E3b%5E7%7D.%26%266%E3%80%81%5Cfrac%7B(a%5E2b%5E3)%5E3%5Ccdot%20ab%5E2%7D%7B(a%5E5b%5E4)%5E3%7D.%5C%5C%0A%26%267%E3%80%81%5Cfrac%7B-32a%5E3b%5E5x%5E2y%5E3%7D%7B(24a%5E2b%5E8xy%5E4)%5E4%7D.%26%268%E3%80%81%5Cfrac%7B(-a%5E2b%5E5)%5E3%5Cquad(xy)%5E2%7D%7B(-ax%5E3)%5E4%5Ccdot(by%5E2)%5E2%7D.%5C%5C%0A%26%269%E3%80%81%5Cfrac%7B3a%5E%7Bm%2B1%7Db%5E%7Bn%2B2%7D%7D%7B30a%5Emb%5En%7D.%26%2610%E3%80%81%5Cfrac%7B2a%5Emb%5E%7B2n%2B1%7D%7D%7B6a%5E%7B2m%2B1%7Db%5E%7B3n%2B4%7D%7D.%5C%0A%5Cend%7Beqnarray%7D$

【答案】

$%5Csmall%5Cbegin%7Beqnarray%7D%0A%26%266%E3%80%81%5Cfrac1%7Ba%5E8b%7D%3B%26%267%E3%80%81%5Cfrac%7B-1%7D%7B18aby%5E5%7D%3B%26%268%E3%80%81%5Cfrac%7B-a%5E2b%5E%7B13%7D%7D%7Bx%5E%7B10%7Dy%5E2%7D%3B%5C%5C%0A%26%269%E3%80%81%5Cfrac%7Bab%5E2%7D%7B10%7D%3B%26%2610%E3%80%81%5Cfrac1%7B3a%5E%7Bn%2B1%7Db%5E%7Bn%2B3%7D%7D.%0A%5Cend%7Beqnarray%7D$

(3)分式的分子分母是多項(xiàng)式的約分。

例4.化簡(jiǎn)： $%5Cscriptsize%5Cbegin%7Baligned%7D%0A(1)%5Cfrac%7Bx-y%7D%7Bx%5E2-y%5E2%7D%3B%5C%3B(2)%5Cfrac%7B(a-b)%5E3%7D%7Ba%5E3-b%5E3%7D%3B%5C%3B(3)%5Cfrac%7Bx%5E2%2B4x-5%7D%7Bx%5E2-3x%2B2%7D%3B%5C%3B(4)%5Cfrac%7Bx%5E3-x%5E2%2Bx-1%7D%7Bx-1%7D.%0A%5Cend%7Baligned%7D$

【解】先分解因式，再約簡(jiǎn)：

$%5Csmall(1)%5Cfrac%7Bx-y%7D%7Bx%5E2-y%5E2%7D%3D%5Cfrac%7Bx-y%7D%7B%5Cleft(x%2By%5Cright)%5Cleft(x-y%5Cright)%7D%3D%5Cfrac1%7Bx%2By%7D.$

【注意1】不要把 $%5Cscriptsize%5Cfrac%7Bx-y%7D%7Bx%5E2-y%5E2%7D$ 約做 $%5Cscriptsize%5Cfrac%7B%5Ccancel%7Bx%7D-%5Ccancel%7By%7D%7D%7Bx%5E%7B%5Ccancel%7B2%7D%7D-y%5E%7B%5Ccancel%7B2%7D%7D%7D%3D%5Cfrac1%7Bx-y%7D$ ，這是錯(cuò)誤的。

【注意2】最后結(jié)果不要寫(xiě)作 $%5Cscriptsize%5Cfrac1%7B(x-y)%7D$ ，因?yàn)榉质降臋M線就表示括號(hào)，再用括號(hào)就多余了。

$%5Csmall%5Cbegin%7Baligned%7D%26(2)%5Cfrac%7B(a-b)%5E3%7D%7Ba%5E3-b%5E3%7D%3D%5Cfrac%7B(a-b)%5E3%7D%7B(a-b)%5Cleft(a%5E2%2Bab%2Bb%5E2%5Cright)%7D%3D%5Cfrac%7B(a-b)%5E2%7D%7Ba%5E2%2Bab%2Bb%5E2%7D%3B%5C%5C%26(3)%5Cfrac%7Bx%5E2%2B4x-5%7D%7Bx%5E2-3x%2B2%7D%3D%5Cfrac%7B(x%2B5)(x-1)%7D%7B(x-2)(x-1)%7D%3D%5Cfrac%7Bx%2B5%7D%7Bx-2%7D.%5Cend%7Baligned%7D$

【注意】不要把 $%5Cscriptsize%5Cfrac%7Bx%5E2%2B4x-5%7D%7Bx%5E2-3x%2B2%7D$ 約做 $%5Cscriptsize%5Cfrac%7B4x-5%7D%7B-3x%2B2%7D$ ，因?yàn)檫@樣分子分母都減去x2，不是同除以相同的代數(shù)式。

$%5Csmall%5Cbegin%7Baligned%7D(4)%5Cfrac%7Bx%5E3-x%5E2%2Bx-1%7D%7Bx-1%7D%3D%5Cfrac%7Bx%5E3(x-1)%2B(x-1)%7D%7Bx-1%7D%3D%5Cfrac%7B(x-1)(x%5E3%2B1)%7D%7Bx-1%7D%3Dx%5E2%2B1.%5Cend%7Baligned%7D$

【注意】不要把 $%5Cscriptsize%5Cfrac%7Bx%5E3-x%5E2%2B%5Ccancel%7Bx-1%7D%7D%7B%5Ccancel%7Bx-1%7D%7D$ 約做 x3－x2，這是錯(cuò)誤的。

例5.化簡(jiǎn)： $%5Cscriptsize(1)%5Cfrac%7B(1-x)%5E2(1%2Bx)%5E2%7D%7B(x%5E2-1)%5E2%7D%3B%5C%3B(2)%5Cfrac%7B(x%5E3-x-2)%5E3%7D%7B(x%5E2-1)%5E3(2-x)%5E3%7D.$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5Cfrac%7B(1-x)%5E%7B2%7D(1%2Bx)%5E%7B2%7D%7D%7B(x%5E%7B2%7D-1)%5E%7B2%7D%7D%5C%5C%26%20%3D%5Cfrac%7B(x-1)%5E2(x%2B1)%5E2%7D%7B%5B(x%2B1)(x-1)%5D%5E2%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7B(x-1)%5E2(x%2B1)%5E2%7D%7B(x%2B1)%5E2(x-1)%5E2%7D%3D1%3B%0A%5Cend%7Baligned%7D$

【說(shuō)明】(1－x)2＝(x－1)2? 。

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(2)%5Cfrac%7B(x%5E2-x-2)%5E3%7D%7B(x%5E2-1)%5E3(2-x)%5E3%7D%5C%5C%0A%26%20%3D%5Cfrac%7B%5Cleft%5B%5Cleft(x-2%5Cright)%5Cleft(x%2B1%5Cright)%5Cright%5D%5E%7B3%7D%7D%7B-%5Cleft%5B%5Cleft(x%2B1%5Cright)%5Cleft(x-1%5Cright)%5Cright%5D%5E%7B3%7D%5Cleft(x-2%5Cright)%5E%7B3%7D%7D%20%20%5C%5C%0A%26%3D%5Cfrac%7B(x-2)%5E3(x%2B1)%5E3%7D%7B-(x%2B1)%5E3(x-1)%5E3(x-2)%5E3%7D%20%5C%5C%0A%26%3D%5Cfrac%7B-1%7D%7B(x-1)%5E3%7D.%5C%0A%5Cend%7Baligned%7D$

【說(shuō)明】(2－x)3＝－(x－2)3? 。

習(xí)題5-4(3)

約簡(jiǎn)(1~16)：

$%5Csmall%5Cbegin%7Beqnarray%7D%0A%26%261%E3%80%81%5Cfrac%7Ba%2Bb%7D%7Ba%5E3%2Bb%5E3%7D.%26%262%E3%80%81%5Cfrac%7Ba%2Bb%7D%7Ba%5E4-b%5E4%7D.%5C%5C%0A%26%263%E3%80%81%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2B2ab%2Bb%5E2%7D.%26%264%E3%80%81%5Cfrac%7Ba%5E3-b%5E3%7D%7B(a-b)%5E3%7D.%5C%5C%0A%26%265%E3%80%81%5Cfrac%7Bab%7D%7Ba%5E2b-ab%5E2%7D.%26%266%E3%80%81%5Cfrac%7Bx%5E3-x%5E2y%7D%7B2xy%7D.%5C%5C%0A%26%267%E3%80%81%5Cfrac%7Bx%5E2-1%7D%7B(x-1)%5E2%7D.%26%268%E3%80%81%5Cfrac%7Bx%5E3-1%7D%7Bx%5E2-1%7D.%5C%5C%0A%26%269%E3%80%81%5Cfrac%7Bx%5E2%2B3x-28%7D%7Bx%5E3-64%7D.%26%2610%E3%80%81%5Cfrac%7Ba%5E2%2Ba-6%7D%7Ba%5E2%2B5a%2B6%7D.%5C%5C%0A%26%2611%E3%80%81%5Cfrac%7Ba%5E3-a%5E2%2Ba-1%7D%7Ba(a-1)%7D.%26%2612%E3%80%81%5Cfrac%7B(x%5E3%2By%5E3)(x%5E2%2Bxy%2By%5E2)%7D%7B(x%5E3-y%5E3)(x%5E2-xy%2By%5E2)%7D.%20%5C%5C%0A%26%2613%E3%80%81%5Cfrac%7B(a-b)(a%5E2-b%5E2)%7D%7B(a%5E2%2Bab)(a%5E2-ab)%7D.%26%26%2014%E3%80%81%5Cfrac%7B(a%5E2-b%5E2)(a%5E2%2Bb%5E2)%7D%7B(a%5E3-b%5E3)(a%5E3%2Bb%5E3)%7D.%20%20%5C%5C%0A%26%2615%E3%80%81%5Cfrac%7Ba%5E2-(b%2Bc%2Bd)%5E2%7D%7B(a-b)%5E2-(c%2Bd)%5E2%7D.%26%26%2016%E3%80%81%5Cfrac%7B%5Cleft%5Bx%5E2-(y-z)%5E2%5Cright%5D%5Cleft%5Bz%5E2-(x-y)%5E2%5Cright%5D%7D%7B%5Cleft%5By%5E2-(z-x)%5E2%5Cright%5D%5Cleft%5B(y%2Bz)%5E2-x%5E2%5Cright%5D%7D%20.%0A%5Cend%7Beqnarray%7D$

化成最簡(jiǎn)分式后，再求值(17~18)：

$%5Csmall%5Cbegin%7Baligned%7D%2617.%5Cquad%5Cfrac%7Bx%5E2-2x%2B1%7D%7Bx%5E3-3x%5E2%2B3x-1%7D%2C%5Cquad(1)x%3D31%3B%5Cquad(2)x%3D-49.%5C%5C%2618.%5Cquad%5Cfrac%7Bx%5E2-7x%2B6%7D%7Bx%5E3-9x%2B18%7D%2C%5Cquad(1)x%3D-7%3B%5Cquad(2)x%3D3.1.%5Cend%7Baligned%7D$

化簡(jiǎn)(19~30)：

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%2619%E3%80%81%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba(b%5E2-a%5E2)%7D.%5C%5C%0A%2620%E3%80%81%5Cfrac%7B(a%2Bb)%5E3(b-a)%5E3%7D%7B(a%5E2-b%5E2)%5E3%7D.%5C%5C%0A%2621%E3%80%81%5Cfrac%7B(a%5E2-b%5E2)%5E3%7D%7B(a%5E2-2ab%2Bb%5E2)%5E3%7D.%5C%5C%0A%2622%E3%80%81%5Cfrac%7B(a-b)%5E5%7D%7B(b%5E2-2ab%2Ba%5E2)%5E3%7D.%5C%5C%0A%2623%E3%80%81%5Cfrac%7B(a%5E2-b%5E2)(ab-a%5E2-b%5E2)%7D%7B(b%5E3%2Ba%5E9)(b%5E3-a%5E9)%7D.%5C%5C%0A%2624%E3%80%81%5Cfrac%7B(a%5E2-b%5E2)%5E7%7D%7B(b-a)%5E7(b%2Ba)%5E5%7D.%5C%5C%0A%2625%E3%80%81%5Cfrac%7B(a%5E3-b%5E3)%5E3%7D%7B(b-a)%5E3%7D.%5C%5C%0A%2626%E3%80%81%5Cfrac%7B(6y-x)(y%2Bx)%5E3%7D%7B(x%5E2-5xy-6y%5E2)%5E2%7D.%5C%5C%0A%2627%E3%80%81%5Cfrac%7B(1-a)(2%2Ba)(a%2B3)%7D%7B(a-1)(a-2)(a-3)%7D.%5C%5C%0A%2628%E3%80%81%5Cfrac%7B(b-a)(c-b)(a%2Bc)%7D%7B(a-b)(b-c)(c-a)%7D.%5C%5C%0A%2629%E3%80%81%5Cfrac%7B(b-a)%5E2(b%2Bc)%5E2(a-c)%5E2%7D%7B(a-b)%5E2(b-c)%5E2(c-a)%5E2%7D.%5C%5C%0A%2630%E3%80%81%5Cfrac%7B(b-a)%5E5(a%2Bb)%5E4%7D%7B(a-b)%5E5(b%2Ba)%5E5%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%2611%E3%80%81%5Cfrac%7Ba%5E2%2B1%7Da%3B%5Cquad12%E3%80%81%5Cfrac%7Bx%2By%7D%7Bx-y%7D%3B%5Cquad13%E3%80%81%5Cfrac%7Ba-b%7D%7Ba%5E2%7D%3B%20%5C%5C%0A%2614%E3%80%81%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7B(a%5E2%2Bab%2Bb%5E2)(a%5E2-ab%2Bb%5E2)%7D%3B15%E3%80%81%5Cfrac%7Ba%2Bb%2Bc%2Bd%7D%7Ba-b%2Bc%2Bd%7D%3B%5C%5C%0A%2616%E3%80%81%5Cfrac%7B(x-y%2Bz)%5E2%7D%7B(y%2Bz%2Bx)(y%2Bz-x)%7D%3B17%E3%80%81%5Cfrac%7B1%7D%7Bx-1%7D%2C(1)%5Cfrac%7B1%7D%7B30%7D%2C(2)-%5Cfrac%7B1%7D%7B50%7D%3B%5C%5C%0A%2618%E3%80%81%5Cfrac%7Bx-1%7D%7Bx-3%7D%2C(1)%5Cfrac%7B4%7D%7B5%7D%2C(2)21%3B19%E3%80%81-%5Cfrac%7B1%7D%7Ba%7D%3B20%E3%80%81-1%3B%5C%5C%0A%2621%E3%80%81%5Cfrac%7B(a%2Bb)%5E3%7D%7B(a-b)%5E3%7D%3B22%E3%80%81%5Cfrac%7B1%7D%7Ba-b%7D%3B23%E3%80%81%5Cfrac%7B1%7D%7Ba%5E2%2Bab%2Bb%5E2%7D%3B24%E3%80%81-(a%2Bb)%5E2%3B%5C%5C%0A%2625%E3%80%81-(a%5E2%2Bab%2Bb%5E2)%5E3%3B26%E3%80%81-%5Cfrac%7Bx%2By%7D%7Bx-6y%7D%3B27%E3%80%81-%5Cfrac%7B(a%2B2)(a%2B3)%7D%7B(a-2)(a-3)%7D%3B%5C%5C%0A%2628%E3%80%81-%5Cfrac%7Ba%2Bc%7D%7Ba-c%7D%3B%5Cquad29%E3%80%81%5Cfrac%7B(b%2Bc)%5E%7B2%7D%7D%7B(b-c)%5E%7B2%7D%7D%3B%5Cquad30%E3%80%81%5Cfrac%7B-1%7D%7Ba%2Bb%7D.%0A%5Cend%7Baligned%7D$

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【種花家務(wù)·代數(shù)】1-5-04約分『數(shù)理化自學(xué)叢書(shū)6677版』

習(xí)題5-4(1)

習(xí)題5-4(2)

習(xí)題5-4(3)