【種花家務(wù)·代數(shù)】1-4-09本章復(fù)習(xí)（因式分解）『數(shù)理化自學(xué)叢書(shū)6677版』

2023-09-27 23:20 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计胶猓欢t，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。

第四章因式分解?

本章提要

1、因式分解的方法及公式

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2、最高公因式與最低公倍式

復(fù)習(xí)題四

1、回答下列問(wèn)題：什么叫做多項(xiàng)式的因式分解？多項(xiàng)式的因式分解有那些方法？有那些公式？因式分解的公式與乘法公式有什么聯(lián)系，又有什么區(qū)別？多項(xiàng)式的因式分解的步驟怎樣？

2、什么叫做幾個(gè)整式的最高公因式？怎樣求法？什么叫做幾個(gè)整式的最低公倍式？怎樣求法？

下列因式分解，是否正確？如有錯(cuò)誤，改正右邊(3~14)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%263%E3%80%81a%5E%7B2%7D-b%5E%7B2%7D%3D(a-b)%5E%7B2%7D.%20%5C%5C%0A%264%E3%80%81a%5E%7B2%7D-4b%5E%7B2%7D%3D(a%2B4b)(a-4b).%20%5C%5C%0A%265%E3%80%8116a%5E%7B16%7D-b%5E%7B2%7D%3D(4a%5E%7B4%7D%2Bb)(4a%5E%7B4%7D-b).%20%5C%5C%0A%266%E3%80%81a%5E%7B2%7D-(b%2Bc)%5E%7B2%7D%3D(a%2Bb%2Bc)(a-b%2Bc).%20%5C%5C%0A%267%E3%80%81-a%5E%7B2%7D%2B2ab-b%5E%7B2%7D%3D(a-b)%5E%7B2%7D.%20%5C%5C%0A%268%E3%80%81a(x-1)%5E%7B2%7D-b(x-1)%3D(x-1)%5E%7B2%7D(a-b).%5C%5C%0A%269%E3%80%81a%5E2%2Ba-b%5E2-b%3Da(a%2B1)-b(b%2B1)%5C%5C%0A%26%3D(a-b)(a%2B1)(b%2B1).%5C%5C%0A%2610%E3%80%81a%5E2-b%5E2%2B2bc-c%5E2%3Da%5E2-(b%2Bc)%5E2%5C%5C%0A%26%3D(a%2Bb%2Bc)(a-b-c).%5C%5C%0A%2611%E3%80%81(a%5E2-b%5E2)(a%5E3-b%5E3).%5C%5C%0A%26%3D(a%2Bb)(a-b)(a-b)(a%5E2%2Bab%2Bb%5E2)%5C%5C%26%3D(a%2Bb)(a-b)(a%5E2%2Bab%2Bb%5E2).%5C%5C%0A%2612%E3%80%81(a%5E3%2Bb%5E3)%5E2%3D(a%2Bb)(a%5E2%2Bab%2Bb%5E2)%5E2.%5C%5C%0A%2613%E3%80%81%20x%5E2-5x-6%3D(x-3)(x-2).%5C%5C%0A%2614%E3%80%81%20x%5E2-x-6%3D(x-2)(x%2B3).%0A%5Cend%7Baligned%7D$

分解下列各式的因式(15~44)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2615%E3%80%81a%5E%7B4%7D-a%5E%7B2%7Db%5E%7B2%7D.%5C%5C%0A%2616%E3%80%81ax%5E2(m-n)%2B3x(n-m)_%5Cbullet%20%20%20%5C%5C%0A%2617%E3%80%813ab(c-d)-9b%5E%7B2%7D(d-c).%5C%5C%0A%20%2618%E3%80%81(x%2By)%5E3-4xy(x%2By)_%5Cbullet%20%20%20%5C%5C%0A%2619%E3%80%81x%5E%7B5%7D%2Bx%5E%7B4%7D%2Bx%5E%7B2%7D%2Bx.%5C%5C%0A%2620%E3%80%81ax-a%2Bx-1.%5C%5C%0A%2621%E3%80%81bx%2B1-b-x.%5C%5C%0A%2622%E3%80%81ax-bx-b%2Ba.%20%20%20%5C%5C%0A%26%2023%E3%80%81ax-ay%2Bbx%2Bcy-cx-by.%5C%5C%0A%2624%E3%80%813x%5E%7B2%7Dy%5E%7B2%7D-9xy-12.%20%20%5C%5C%0A%26%2025%E3%80%81a%5E2-b%5E2-a%2Bb.%20%5C%5C%0A%20%2626%E3%80%81a%5E%7B2%7D%2B4ab%2B4b%5E%7B2%7D-9c%5E%7B2%7D%E3%80%82%20%20%5C%5C%0A%2627%E3%80%81a%5E%7B2%7D%2B2ab%2Bb%5E%7B2%7D-4a-4b.%5C%5C%0A%2628%E3%80%81a%5E%7B4%7Dc-c%5E%7B5%7D.%20%20%5C%5C%0A%2629%E3%80%81a%5E2-x%5E2-ab-bx%5C%5C%0A%20%2630%E3%80%81x%5E4%2B6x%5E2-135.%20%5C%5C%0A%2631%E3%80%81x%5E3-5x%5E2-24x.%5C%5C%0A%2632%E3%80%81a%5E4x%5E4-a%5E3x%5E3-a%5E2x%5E2%2B1.%5C%5C%0A%2633%E3%80%815a%5E2-5b%5E2-3a%2B3b.%5C%5C%0A%2634%E3%80%814-x%5E2%2B2x%5E3%2Bx%5E4.%5C%5C%0A%2635%E3%80%81a%5E%7B12%7Db%2Bb%5E%7B13%7D.%5C%5C%0A%2636%E3%80%81a%5E2%2Bb%5E2%2B2(ab%2Bac%2Bbc).%5C%5C%0A%2637%E3%80%81a%5E3-b%5E3%2Ba(a%5E2-b%5E2)%2Bb(a-b)%5E2.%5C%5C%0A%2638%E3%80%81(x%5E2-y%5E2-z%5E2)%5E2-4y%5E2z%5E2.%5C%5C%0A%2639%E3%80%81x%5E2%2B2xy%2By%5E2-x-y-6.%5C%5C%0A%2640%E3%80%814a%5E2b%5E2-(a%5E2%2Bb%5E2-c%5E2)%5E2.%5C%5C%0A%2641%E3%80%81a%5E2-b%5E2%2Bx%5E2-y%5E2%2B2(ax-by).%5C%5C%0A%2642%E3%80%81x%5E6-27y%5E6-x%5E4%2B9y%5E4.%5C%5C%0A%2643%E3%80%81x%5E%7B24%7D-y%5E%7B24%7D.%5C%5C%0A%2644.x%5E%7B4%7D-x%5E%7B3%7D%2B8x-8.%0A%5Cend%7Baligned%7D$

利用因式分解，計(jì)算下列各式(45~48)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2645%E3%80%81354%5E2-353%5E2.%5C%5C%0A%2646%E3%80%8182%5E2-80%5E2.%5C%5C%0A%2647%E3%80%811.36%5E2-0.36%5E2.%5C%5C%0A%2648%E3%80%8154.1%5E2-51.1%5E2.%0A%5Cend%7Baligned%7D$

求下列各代數(shù)式的值(49~50)：

49、a2－b2，其中 a＝－5.6，b＝4.6? 。

50、a2－4b2，其中 a＝－3.8，b＝－3.1? 。

用兩種不同方法，求出下列的積(51~52)：

51、(a2＋2ab＋b2)(a2－2ab＋b2)? 。[提示：(1) (a＋b)2(a－b)2＝[(a＋b)(a－b)]2＝(a2－b2)＝…，(2) [(a2＋b2)＋2ab][(a2＋b2)－2ab]＝…? .]

52、(a3＋3a2b＋3ab2＋b3)(a3－3a2b＋3ab2－b3) .

化簡(jiǎn)(53~55)：

53、(a＋2b)3－(a－2b)3 .

54、(8a?＋27b?)÷(2a2＋3b2)－(8a?－27b?)÷(2a2－3b2) .

55、(a2＋b2)(a?－a2b2＋b?)－(a2－b2)(a?＋a2b2＋b?) .

用簡(jiǎn)便的方法求下列各代數(shù)式的值(56~60)：

56、(a2－b2)÷(a－b)＋(a－b)2÷(a－b)，當(dāng) a＝ $%5Cscriptsize3%5Cfrac%7B3%7D%7B5%7D%20$ ? 。

57、(a3＋b3)÷(a＋b)－(a＋b)3÷(a＋b)，當(dāng) a＝ $%5Cscriptsize-3%5Cfrac%7B1%7D%7B3%7D%20$ ，b＝ $%5Cscriptsize%7C-2%5Cfrac%7B1%7D%7B2%7D%20%7C$ ? 。

58、(a－b)(a2＋ab＋b2)＋3ab(b－a)，當(dāng) a＝－3.7，b＝6.3? 。

59、xy＋1－x－y，當(dāng) x＝0.99，y＝1.03? 。

60、(a＋b)(a2－ab＋b2)－9b3，當(dāng) a＝－4,368，b＝－2.184? 。

*分解因式(61~65)：

61、x?＋x2y2＋y?.[提示：變成 x?＋2x2y2＋y?－x2y2]

62、4x?＋1.[提示：變成 4x?＋4x2＋1－4x2]

63、x?＋5x2y2＋9y?.

64、4a?－24a2b2＋25b?.

65、x?＋y?＋z?－2x2y2－2x2z2－2y2z2.[提示：變成 x?＋y?＋z?－2x2y2－2x2z2－2y2z2－4x2y2]

【答案】

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【種花家務(wù)·代數(shù)】1-4-09本章復(fù)習(xí)（因式分解）『數(shù)理化自學(xué)叢書(shū)6677版』

本章提要

復(fù)習(xí)題四