【種花家務(wù)·代數(shù)】1-5-07分式的乘法『數(shù)理化自學(xué)叢書(shū)6677版』

2023-09-29 14:55 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(huà)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?

§5-7分式的乘法

【01】分式的乘法，可以依照分?jǐn)?shù)乘法的同樣法則進(jìn)行。

【02】分式的乘法法則：分式相乘，分子分母分別相乘。即 $%5Cfrac%20ab%5Ctimes%5Cfrac%20cd%3D%5Cfrac%7Bac%7D%7Bbd%7D$ ? 。

例1.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%5Cfrac%7B3a%5E2b%7D%7B5xy%5E2%7D%5Ctimes%5Cfrac%7B2ab%5E4c%7D%7B7xy%5E2z%5E3%7D%3B%5C%5C%26(2)%5C%3B%5Cfrac%7B2a%5E3b%5E2%7D%7B5cd%5E4%7D%5Ctimes%5Cfrac%7B3a%5E4c%5E2d%7D%7B4b%5E2%7D%3B%5C%5C%26(3)%5C%3B%5Cfrac%7B-7a%5E2b%7D%7B5cx%5E2%7D%5Ctimes%5Cleft(-%5Cfrac%7B20cx%5E2%7D%7B21a%5E2y%5E2%7D%5Cright)%5Ctimes%5Cleft(-%5Cfrac%7B6ay%5E5%7D%7B5b%5E2x%5E3%7D%5Cright).%5Cend%7Baligned%7D$

【解】

(1)分子分母分別相乘，得

$%5Csmall%5Cfrac%7B3a%5E2b%7D%7B5xy%5E2%7D%5Ctimes%5Cfrac%7B2ab%5E4c%7D%7B7xy%5E2z%5E3%7D%3D%5Cfrac%7B3a%5E2b%5Cboldsymbol%7B%5Ccdot%7D2ab%5E4c%7D%7B5xy%5E3%5Cboldsymbol%7B%5Ccdot%7D7xy%5E2z%5E3%7D%3D%5Cfrac%7B6a%5E3b%5E5c%7D%7B35x%5E2y%5E4z%5E3%7D.$

(2)分子分母分別相乘后，因分子分母有公因式，約成最簡(jiǎn)分式，得

$%5Csmall%5Cfrac%7B2a%5E3b%5E2%7D%7B5cd%5E4%7D%5Ctimes%5Cfrac%7B3a%5E4c%5E2d%7D%7B4b%5E2%7D%3D%5Cfrac%7B6a%5E7b%5E2c%5E2d%7D%7B20b%5E2cd%5E4%7D%3D%5Cfrac%7B3a%5E7c%7D%7B10d%5E3%7D.$

(3)這里有三個(gè)負(fù)號(hào)，乘積有負(fù)號(hào)，放在分式前面，得

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%20-%5Cfrac%7B7a%5E%7B2%7Db%7D%7B5cx%5E%7B2%7D%7D%5Ctimes%5Cleft(-%5Cfrac%7B20cx%5E%7B2%7D%7D%7B21a%5E%7B2%7Dy%5E%7B2%7D%7D%5Cright)%5Ctimes%5Cleft(-%5Cfrac%7B6ay%5E%7B5%7D%7D%7B5b%5E%7B2%7Dx%5E%7B3%7D%7D%5Cright)%20%20%5C%5C%0A%26%3D-%5Cfrac%7B7a%5E2b%E2%80%A220cx%5E2%E2%80%A26ay%5E5%7D%7B5cx%5E2%E2%80%A221a%5E2y%5E2%E2%80%A25b%5E2x%5E3%7D%20%5C%5C%0A%26%3D-%5Cfrac%7B7%C2%B720%C2%B76a%5E3bcx%5E2y%5E5%7D%7B5%C2%B75%C2%B721a%5E%5Cmathbf%7B2%7Db%5E2cx%5E5y%5E%5Cmathbf%7B2%7D%7D%20%5C%5C%0A%26%3D-%5Cfrac%7B8ay%5E3%7D%7B5bx%5E3%7D.%0A%5Cend%7Baligned%7D$

例2.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Ccdot%5Cfrac%7Ba%5E3%2Bb%5E3%7D%7Ba%5E3-b%5E3%7D%3B%5C%5C%26(2)%5C%3B%5Cfrac%7Bx%5E2-5x%2B6%7D%7Bx%5E2%2B8x%2B16%7D%5Ccdot%5Cfrac%7Bx%5E2%2B7x%2B12%7D%7Bx%5E2-8x%2B15%7D%5Ccdot%5Cfrac%7Bx%5E2-x-20%7D%7Bx%5E2%2Bx-6%7D%3B%5C%5C%26(3)%5C%3B(x%5E3-4)%C2%B7%5Cfrac%7Bx%5E3%2B8%7D%7Bx%5E3-8%7D%C2%B7%5Cfrac%7Bx%5E2%2B2x%2B4%7D%7Bx%5E2%2B4x%2B4%7D.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%5C%3B%26%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Ccdot%5Cfrac%7Ba%5E3%2Bb%5E3%7D%7Ba%5E3-b%5E3%7D%3D%5Cfrac%7B(a%5E2-b%5E2)(a%5E3%2Bb%5E3)%7D%7B(a%5E2%2Bb%5E2)(a%5E3-b%5E3)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a%2Bb)%7D%7B(a%5E2%2Bb%5E2)%7D%5Cfrac%7B(a-b)(a%2Bb)%7D%7B(a-b)%7D%5Cfrac%7B(a%5E2-ab%2Bb%5E2)%7D%7B(a%5E2%2Bab%2Bb%5E2)%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a%2Bb)%5E2(a%5E2-ab%2Bb%5E2)%7D%7B(a%5E2%2Bb%5E2)(a%5E2%2Bab%2Bb%5E2)%7D.%20%0A%5Cend%7Baligned%7D$

【注意】計(jì)算的結(jié)果，必須約成最簡(jiǎn)分式。

$%5Csmall%5Cbegin%7Baligned%7D(2)%26%5C%3B%5Cfrac%7Bx%5E2-5x%2B6%7D%7Bx%5E2%2B8x%2B16%7D%5Ccdot%5Cfrac%7Bx%5E2%2B7x%2B12%7D%7Bx%5E2-8x%2B15%7D%C2%B7%5Cfrac%7Bx%5E2-x-20%7D%7Bx%5E2%2Bx-6%7D%5C%5C%26%3D%5Cfrac%7B%5Cleft(x-3%5Cright)%5Cleft(x-2%5Cright)%5Cleft(x%2B3%5Cright)%5Cleft(x%2B4%5Cright)%5Cleft(x-5%5Cright)%5Cleft(x%2B4%5Cright)%7D%7B%5Cleft(x%2B4%5Cright)%5E2%5Cleft(x-3%5Cright)%5Cleft(x-5%5Cright)%5Cleft(x%2B3%5Cright)%5Cleft(x-2%5Cright)%7D%5C%5C%26%3D%5Cfrac%7B1%7D%7B1%7D%3D1.%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(3)%5C%3B%26(x%5E2-4)%5Ccdot%5Cfrac%7Bx%5E3%2B8%7D%7Bx%5E3-8%7D%2C%5Cfrac%7Bx%5E2%2B2x%2B4%7D%7Bx%5E2%2B4x%2B4%7D%20%5C%5C%0A%26%3D%5Cfrac%7B%5Cleft(x%2B2%5Cright)%5Cleft(x-2%5Cright)%5Cleft(x%2B2%5Cright)%5Cleft(x%5E%7B2%7D-2x%2B4%5Cright)%5Cleft(x%5E%7B2%7D%2B2x%2B4%5Cright)%7D%7B%5Cleft(x-2%5Cright)%5Cleft(x%5E%7B2%7D%2B2x%2B4%5Cright)%5Cleft(x%2B2%5Cright)%5E%7B2%7D%7D%20%5C%5C%0A%26%3D%5Cfrac%7Bx%5E2-2x%2B4%7D1%3Dx%5E2-2x%2B4.%20%0A%5Cend%7Baligned%7D%0A$

【注意】x2－4 就是 $%5Cscriptsize%5Cfrac%7Bx%5E2-4%7D%7B1%7D%20$ ? 。

例3.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5Cfrac%7B(b%5E2-a%5E2)%5E2%7D%7B(a%2Bb)%5E2%7D%5Ccdot%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7B(a-b)%5E2%7D%3B%5C%5C(2)%26%5Cfrac%7B(b%2Ba)%5E3%7D%7B(a%5E3-b%5E3)%5E2%7D%5Ccdot%5Cfrac%7Ba%5E2%2Bab%2Bb%5E2%7D%7Ba%5E2%2B2ab%2Bb%5E2%7D%C2%B7(b-a)%5E3.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%E2%88%B5%26(b%5E2-a%5E2)%5E2%3D%5Cbegin%7Bbmatrix%7D-(a%5E2-b%5E2)%5Cend%7Bbmatrix%7D%5E2%3D(a%5E2-b%5E2)%5E2%2C%20%5C%5C%0A%E2%88%B4%26%5Cfrac%7B(b%5E2-a%5E2)%5E2%7D%7B(a%2Bb)%5E2%7D%5Ccdot%5Cfrac%7Ba%5E2%2Bb%5E2%7D%7B(a-b)%5E2%7D%3D%5Cfrac%7B(a%5E2-b%5E2)%5E2(a%5E2%2Bb%5E2)%7D%7B(a%2Bb)%5E2(a-b)%5E2%7D%20%5C%5C%0A%26%5Cbegin%7Baligned%7D%3D%5Cfrac%7B%5Cleft%5B%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%5Cright%5D%5E2%5Cleft(a%5E2%2Bb%5E2%5Cright)%7D%7B%5Cleft(a%2Bb%5Cright)%5E2%5Cleft(a-b%5Cright)%5E2%7D%5Cend%7Baligned%7D%20%5C%5C%0A%26%3D%5Cfrac%7B(a%2Bb)%5E2(a-b)%5E2(a%5E2%2Bb%5E2)%7D%7B(a%2Bb)%5E2(a-b)%5E2%7D%3Da%5E2%2Bb%5E2%3B%0A%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(2)%E2%88%B5%26(b-a)%5E3%3D%5B-(a-b)%5D%5E3%3D-(a-b)%5E3%2C%20%5C%5C%0A%E2%88%B4%26%5Cfrac%7B(b%2Ba)%5E3%7D%7B(a%5E3-b%5E3)%5E2%7D%C2%B7%5Cfrac%7Ba%5E2%2Bab%2Bb%5E2%7D%7Ba%5E2%2B2ab%2Bb%5E2%7D%C2%B7(b-a)%5E3%20%5C%5C%0A%26%3D%5Cfrac%7B(a%2Bb)%5E3%7D%7B%5Cleft%5B(a-b)%5Cleft(a%5E2%2Bab%2Bb%5E2%5Cright)%5Cright%5D%5E2%7D%20%C2%B7%5Cfrac%7Ba%5E2%2Bab%2Bb%5E2%7D%7B(a%2Bb)%5E2%7D%C2%B7%5Cfrac%7B%3D(a-b)%5E3%7D%7B1%7D%5C%5C%26%3D-%5Cfrac%7B(a%2Bb)%5E3(a%5E2%2Bab%2Bb%5E2)(a-b)%5E3%7D%7B(a-b)%5E2(a%5E2%2Bab%2Bb%5E2)%5E2(a%2Bb)%5E2%7D%5C%5C%26%3D-%5Cfrac%7B(a%2Bb)(a-b)%7D%7Ba%5E2%2Bab%2Bb%5E2%7D.%20%0A%5Cend%7Baligned%7D$

【注】遇到這類(lèi)題目，先把分子分母中的字母，按照同樣的順序排列，相乘后就容易發(fā)現(xiàn)應(yīng)該怎樣把它約簡(jiǎn)。

習(xí)題5-7(1)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B3x%5E2y%7D%7B4z%5E2%7D%5Ccdot%5Cfrac%7B5z%7D%7B6xy%7D%5Ccdot%5Cleft(-%5Cfrac%7B12x%5E2%7D%7By%5E2%7D%5Cright).%20%5C%5C%0A%262%E3%80%81%5Cfrac%7B3ab%5E2%7D%7B2cx%5E2%7D%5Ccdot%5Cleft(-%5Cfrac%7B5by%5E2%7D%7B4ax%5E2%7D%5Cright)%5Cleft(-%5Cfrac%7B3cx%5E2%7D%7B2by%5E2%7D%5Cright).%5C%5C%0A%263%E3%80%81%5Cfrac%7Ba-b%7D%7Ba%5E%7B2%7D%2Bab%7D%5Ccdot%5Cfrac%7Ba%5E%7B4%7D%2Ba%5E%7B2%7Db%5E%7B2%7D%7D%7Ba%5E%7B2%7D-ab%7D.%20%5C%5C%0A%264%E3%80%81%5Cfrac%7Bx-5%7D%7Bx%2B3%7D%5Ccdot%5Cfrac%7Bx%5E2-9%7D%7Bx%5E2-25%7D.%5C%5C%0A%265%E3%80%81%5Cfrac%7Bx%5E%7B3%7D%2By%5E%7B3%7D%7D%7B(x-y)%5E%7B2%7D%7D%5Ccdot%5Cfrac%7Bx%5E%7B3%7D-y%5E%7B3%7D%7D%7B(x%2By)%5E%7B2%7D%7D.%20%5C%5C%0A%266%E3%80%81%5Cfrac%7Ba%5E%7B2%7Db%5E%7B2%7D%2B3ab%7D%7B4a%5E%7B2%7D-1%7D%5Ccdot%5Cfrac%7B2a%2B1%7D%7Bab%2B3%7D.%20%5C%5C%0A%267%E3%80%81%5Cfrac%7B16x%5E%7B2%7D-9a%5E%7B2%7D%7D%7Bx%5E%7B2%7D-4%7D%5Ccdot%5Cfrac%7Bx-2%7D%7B4x-3a%7D.%5C%5C%0A%268%E3%80%81%5Cfrac%7Ba(x-b)%7D%7Bbx%2Bab%7D%5Ccdot%5Cfrac%7Bb%5E%7B2%7Dx%2Bab%5E%7B2%7D%7D%7Ba%5E%7B2%7Dx-a%5E%7B3%7D%7D%5Ccdot%5Cfrac%7Bx%5E%7B2%7D-a%5E%7B2%7D%7D%7Bx%5E%7B2%7D-b%5E%7B2%7D%7D.%5C%5C%0A%269%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-x-20%7D%7Bx%5E%7B2%7D%2B2x-8%7D%5Ccdot%5Cfrac%7Bx%5E%7B2%7D-x-2%7D%7Bx%5E%7B2%7D-25%7D%5Ccdot%5Cfrac%7Bx%5E%7B2%7D%2B5x%7D%7Bx%2B1%7D.%20%5C%5C%0A%2610%E3%80%81%5Cfrac%7Bx%5E2-8x-9%7D%7Bx%5E2%2B4x-5%7D%5Ccdot%5Cfrac%7Bx%5E2-1%7D%7Bx%5E2-17x%2B72%7D%5Ccdot%5Cfrac%7Bx%5E2-9x%2B8%7D%7Bx%5E2-25%7D.%20%5C%5C%0A%2611%E3%80%81%5Cfrac%7Bx%5E4-8x%7D%7Bx%5E2-4x-5%7D%5Ccdot%5Cfrac%7Bx%5E2%2B2x%2B1%7D%7Bx%5E2%2B2x%2B4%7D%5Ccdot%5Cfrac%7Bx-5%7D%7Bx%5E3-x%5E2-2x%7D.%20%5C%5C%0A%2612%E3%80%81%5Cfrac%7B(a-b)%5E2-c%5E2%7D%7B(a%2Bb)%5E2-c%5E2%7D%5Ccdot%5Cfrac%7Ba%5E2-(b%2Bc)%5E2%7D%7Ba%5E2-(b-c)%5E2%7D.%5C%5C%0A%2613%E3%80%81%5Cfrac%7Bx%5E2-(y-z)%5E2%7D%7By%5E2-(z-x)%5E2%7D%5Ccdot%5Cfrac%7Bx%5E2-(x-y)%5E2%7D%7B(y%2Bz)%5E2-x%5E2%7D.%5C%5C%0A%2614%E3%80%81%5Cfrac%7Bx%5E2-2xy%2By%5E2-z%5E2%7D%7Bx%5E2%2B2xy%2By%5E2-z%5E2%7D%5Ccdot%5Cfrac%7Bx%2By-z%7D%7Bx-y%2Bz%7D.%5C%5C%0A%2615%E3%80%81%5Cfrac%7Bx%5E3%2By%5E3%2B3xy(x%2By)%7D%7Bx%5E3-y%5E3-3xy(x-y)%7D%5Ccdot%5Cfrac%7Bx(x-2y)%2By%5E2%7D%7Bx(x%2B2y)%2By%5E2%7D.%5C%5C%0A%2616%E3%80%81(x%5E2%2B2xy%2By%5E2)%5Ccdot%5Cfrac%7Bx-y%7D%7Bx%5E3%2By%5E3%7D%5Ccdot%5Cfrac%7Bx%5E2-xy%2By%5E2%7D%7Bx%5E2-y%5E2%7D.%5C%5C%0A%2617%E3%80%81%5Cfrac%7Bx%5E2%2B2x-3%7D%7Bx%5E2-3x%2B2%7D%5Ccdot%5Cfrac%7Bx%5E2%2B5x-6%7D%7Bx%5E2%2Bx-6%7D%5Ccdot%5Cfrac%7Bx%5E2-4x%2B4%7D%7Bx%5E2%2B6x%7D%5Ccdot%5Cfrac%20x%7Bx-1%7D.%20%5C%5C%0A%2618%E3%80%81%5Cfrac%7Ba%2B2b%7D%7Ba%5E3%2Ba%5E2b%2Bab%5E2%2Bb%5E3%7D%C2%B7%5Cfrac%7Ba%5E3%2Ba%5E2b-ab%5E2-b%5E3%7D%7Ba%5E2-4b%5E2%7D%C2%B7%5Cfrac%7Ba-2b%7D%7Ba%5E4-b%5E4%7D.%5C%5C%0A%2619%E3%80%81%5Cfrac%7B(x%2By)%5E2%7D%7By%5E2-xy%7D%5Ccdotp%5Cleft%5B-%5Cfrac%7B(x-y)%5E2%7D%7By%5E2%2Bxy%7D%5Cright%5D.%5C%5C%0A%2620%E3%80%81%5Cfrac%7Bx%5E2-1%7D%7Bx%5E2%7D%5Ccdotp%5Cfrac%20x%7B1-x%7D.%5C%5C%0A%2621%E3%80%81%5Cfrac%7Ba%5E2-4a%2B3%7D%7B16-a%5E2%7D%5Ccdotp%5Cfrac%7Ba%5E2-3a-4%7D%7Ba%5E2-1%7D.%5C%5C%0A%2622%E3%80%81%5Cfrac%7Bx%5E2-2x-3%7D%7Bx%5E2%2B3x-10%7D%5Ccdotp%5Cfrac%7B9x-x%5E2-14%7D%7Bx%5E2-7x%2B12%7D.%5C%5C%0A%2623%E3%80%81%5Cfrac%7B(a-b)%5E3%7D%7B(a%2Bb)%5E3%7D%5Ccdotp%5Cfrac%7B(b%2Ba)%5E3%7D%7B(b-a)%5E3%7D.%5C%5C%0A%2624%E3%80%81%5Cfrac%7B(a%5E2-b%5E2)%5E2%7D%7Bb%5E6-a%5E6%7D%5Ccdotp%5Cfrac%7B(b%5E3%2Ba%5E6)%5E2%7D%7B(b%5E2-a%5E2)%5E3%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%2617%E3%80%811%3B%5C%3B18%E3%80%81%5Cfrac1%7B(a%5E2%2Bb%5E2)%5E2%7D%3B%5C%3B19%E3%80%81%5Cfrac%7Bx%5E2-y%5E2%7D%7By%5E2%7D%3B%5C%3B20%E3%80%81-%5Cfrac%7Bx%2B1%7Dx%3B%5C%5C%0A%2621%E3%80%81-%5Cfrac%7Ba-3%7D%7Ba%2B4%7D%3B%5C%3B22%E3%80%81-%5Cfrac%7B(x%2B1)(x-7)%7D%7B(x%2B5)(x-4)%7D%3B%5C%3B23%E3%80%81-1%3B%5C%3B24%E3%80%81%5Cfrac%7Ba%5E2-ab%2Bb%5E2%7D%7B(a%5E3-b%5E3)(a-b)%7D.%0A%5Cend%7Baligned%7D$

例4.計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%5Cleft(a%2B%5Cfrac%7Bb%5E2%7Da%5Cright)%5Cleft(1-%5Cfrac%7Bb%5E2-a%5E2%7D%7Bb%5E2%2Ba%5E2%7D%5Cright)%3B%5C%5C%26(2)%5C%3B%5Cfrac2%7Ba%2B2b%7D%5Ccdot%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Ccdot%5Cfrac%7Ba%5E4-b%5E4%7D%7B(b-a)%5E2%7D-%5Cfrac%7Ba%5E2-b%5E2%7D%7B(a%2B2b)%5E2%7D%5Ccdot%5Cfrac%7B4b%5E2-a%5E2%7D%7Bb-a%7D.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D(1)%26%5C%3B%5Cleft(a%2B%5Cfrac%7Bb%5E2%7Da%5Cright)%5Cleft(1-%5Cfrac%7Bb%5E2-a%5E2%7D%7Bb%5E2%2Ba%5E2%7D%5Cright)%5C%5C%26%3D%5Cfrac%7Ba%5E2%2Bb%5E2%7Da%5Ccdot%5Cfrac%7Bb%5E2%2Ba%5E2-(b%5E2-a%5E2)%7D%7Bb%5E2%2Ba%5E2%7D%5C%5C%26%3D%5Cfrac%7Ba%5E2%2Bb%5E2%7Da%5Ccdot%5Cfrac%7B2a%5E2%7D%7Ba%5E2%2Bb%5E2%7D%3D2a%3B%5Cend%7Baligned%7D$

$%5Csmall%5Cbegin%7Baligned%7D%0A(2)%5C%3B%26%5Cfrac2%7Ba%2B2b%7D%5Ccdot%5Cfrac%7Ba%5E2-b%5E2%7D%7Ba%5E2%2Bb%5E2%7D%5Ccdot%5Cfrac%7Ba%5E4-b%5E4%7D%7B(b-a)%5E2%7D-%5Cfrac%7Ba%5E2-b%5E2%7D%7B(a%2B2b)%5E2%7D%5Ccdot%5Cfrac%7B4b%5E2-a%5E2%7D%7Bb-a%7D%5C%5C%0A%26%3D%5Cfrac%7B2%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%5Cleft(a%5E2%2Bb%5E2%5Cright)%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%7D%7B%5Cleft(a%2B2b%5Cright)%5Cleft(a%5E2%2Bb%5E2%5Cright)%5Cleft(a-b%5Cright)%5E2%7D-%5Cfrac%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%5Cleft(2b%2Ba%5Cright)%5Cleft(2b-a%5Cright)%7D%7B%5Cleft(a%2B2b%5Cright)%5E2%5Cleft(b-a%5Cright)%7D%5C%5C%0A%26%5Cbegin%7Baligned%7D%3D%5Cfrac%7B2%5Cleft(a%2Bb%5Cright)%5E%7B2%7D%7D%7Ba%2B2b%7D-%5Cfrac%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright)%5Cleft(a%2B2b%5Cright)%5Cleft(a-2b%5Cright)%7D%7B%5Cleft(a%2B2b%5Cright)%5E%7B2%7D%5Cleft(a-b%5Cright)%7D%5Cend%7Baligned%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2%5Cleft(a%2Bb%5Cright)%5E%7B%5Cmathbf%7B2%7D%7D%7D%7Ba%2B2b%7D-%5Cfrac%7B%5Cleft(a%2Bb%5Cright)%5Cleft(a-2b%5Cright)%7D%7Ba%2B2b%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2%5Cleft(a%2Bb%5Cright)%5E%7B2%7D-%5Cleft(a%2Bb%5Cright)%5Cleft(a-2b%5Cright)%7D%7Ba%2B2b%7D%5C%5C%0A%26%3D%5Cfrac%7B2a%5E2%2B4ab%2B2b%5E2-(a%5E2-ab-2b%5E2)%7D%7Ba%2B2b%7D%20%5C%5C%0A%26%3D%5Cfrac%7B2a%5E2%2B4ab%2B2b%5E2-a%5E2%2Bab%2B2b%5E2%7D%7Ba%2B2b%7D%20%5C%5C%0A%26%3D%5Cfrac%7Ba%5E2%2B5ab%2B4b%5E2%7D%7Ba%2B2b%7D.%0A%5Cend%7Baligned%7D%0A$

【說(shuō)明】這是加減法與乘法的混合運(yùn)算，仍按照以前講的運(yùn)算次序。

習(xí)題5-7(2)

計(jì)算：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cleft(1-%5Cfrac%7B2b%7Da%2B%5Cfrac%7Bb%5E2%7D%7Ba%5E2%7D%5Cright)%5CBig(1%2B%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%5CBig).%5C%5C%0A%262%E3%80%81%5Cleft(x%2B%5Cfrac1x%5Cright)%5CBig(1-%5Cfrac1%7Bx%5E2%2B1%7D%5CBig).%5C%5C%0A%263%E3%80%81(x%5E2-1)%5CBig(%5Cfrac1%7Bx-1%7D-%5Cfrac1%7Bx%2B1%7D-1%5CBig).%5C%5C%0A%264%E3%80%81%5CBig(1%2B%5Cfrac%7B2ab%7D%7Ba%5E2%2Bb%5E2%7D%5CBig)%5CBig%5B1-%5Cfrac%7B2ab%7D%7B(a%2Bb)%5E2%7D%5CBig%5D.%5C%5C%0A%265%E3%80%81%5Cfrac%7Bx%5E%7B2%7D-4%7D%7Bx%5E%7B2%7D-5x%2B6%7D%5Ccdot%5Cfrac%7Bx%5E%7B2%7D-9%7D%7Bx%5E%7B2%7D%2B5x%2B6%7D-%5Cfrac%7Bx%5E%7B2%7D%2B9x%2B20%7D%7Bx%5E%7B2%7D-16%7D%5Ccdot%5Cfrac%7Bx%5E%7B2%7D-9x%2B20%7D%7Bx%5E%7B2%7D-25%7D.%20%5C%5C%0A%266%E3%80%81%5Cleft(1%2B%5Cfrac%20ax%2B%5Cfrac%7Ba%5E2%7D%7Bx%5E2%7D%5Cright)%5Cbiggl(1-%5Cfrac%20ax%5Cbiggr)%5Cbiggl(1-%5Cfrac%7Ba%5E3-2x%5E3%7D%7Ba%5E3-x%5E3%7D%5Cbiggr).%20%5C%5C%0A%26%207%E3%80%81%5Cleft(%5Cfrac%7B5a%7D%7Ba%2Bx%7D%2B%5Cfrac%7B5x%7D%7Ba-x%7D%2B%5Cfrac%7B10ax%7D%7Ba%5E2-x%5E2%7D%5Cright)%5CBig(%5Cfrac%7Ba%7D%7Ba%2Bx%7D%2B%5Cfrac%7Bx%7D%7Ba-x%7D%2B%5Cfrac%7B2ax%7D%7Bx%5E2-a%5E2%7D%5CBig).%20%20%5C%5C%0A%26%208%E3%80%81%5Cleft(%5Cfrac%7B2a%7D%7Ba%2B1%7D-%5Cfrac%7B2%7D%7B1-a%7D%2B%5Cfrac%7B4a%7D%7Ba%5E2-1%7D%5Cright)%5Cbiggl(%5Cfrac%7B2a%7D%7B1%2Ba%7D-%5Cfrac%7B2%7D%7B1-a%7D-%5Cfrac%7B4a%7D%7Ba%5E2-1%7D%5Cbiggr).%20%20%5C%5C%0A%269%E3%80%81%5Cleft(1%2B%5Cfrac%20y%7Bx-y%7D%5Cright)%5Cleft(y-%5Cfrac%7By%5E2%7D%7Bx%2By%7D%5Cright)-%5Cfrac%7Bx%5E4y%7D%7Bx%5E4-y%5E4%7D%5Ccdot%5Cfrac%7Bx%5E2%2By%5E2%7D%7Bx%5E2%7D.%20%5C%5C%0A%2610%E3%80%81%5Cleft%5B%5Cfrac%7Ba%5E%7B2%7D%7D%7Ba%5E%7B2%7D-b%5E%7B2%7D%7D-%5Cfrac%7Ba%5E%7B2%7Db%7D%7Ba%5E%7B2%7D%2Bb%5E%7B2%7D%7D%5Cbigg(%5Cfrac%7Ba%7D%7Bab%2Bb%5E%7B2%7D%7D%2B%5Cfrac%7Bb%7D%7Ba%5E%7B2%7D%2Bab%7D%5Cbigg)%5Cright%5D%5Ctimes%5Cfrac%7Ba-b%7D%7Bb%7D.%0A%5Cend%7Baligned%7D$

【答案】

$%5Cscriptsize%5Cbegin%7Baligned%7D%0A%261%E3%80%81%5Cfrac%7B2a-2b%7Da%3B%5C%3B2%E3%80%81x%3B%5C%3B3%E3%80%813-x%5E2%3B%5C%3B4%E3%80%811%3B%5C%3B5%E3%80%810%3B%5C%5C%0A%266%E3%80%81-1%3B%5C%3B7%E3%80%815%3B%5C%3B8%E3%80%814%3B%5C%3B9%E3%80%810%3B%5C%3B10%E3%80%81%5Cfrac%7Ba%7D%7Ba%2Bb%7D.%0A%5Cend%7Baligned%7D$

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【種花家務(wù)·代數(shù)】1-5-07分式的乘法『數(shù)理化自學(xué)叢書(shū)6677版』

習(xí)題5-7(1)

習(xí)題5-7(2)