【種花家務(wù)·代數(shù)】1-4-04公式分解法『數(shù)理化自學(xué)叢書6677版』

2023-09-27 16:15 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”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，其最大特點就是可用于“自學(xué)”。當(dāng)然由于本書是大半個世紀(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)教育但卻很不扎實的學(xué)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(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)打了個措手不及。

第四章因式分解?

§4-4公式分解法

1、平方差的因式分解公式

【01】在§4-1里我們曾經(jīng)看到多項式 a2－b2 可以分解成兩個因式，就是?a2－b2＝(a＋b)(a－b)……(1)? 。

【02】事實上，這里我們是反過來應(yīng)用了兩數(shù)和與差的積的公式。

【03】我們可以把(1)作為一個公式，利用它來分解由一個數(shù)的平方減去另一個數(shù)的平方所構(gòu)成的多項式的因式。平方差的因式分解公式：

????????a2－b2＝(a＋b)(a－b)（因式分解公式1）。

例1.分解因式：(1) a2－x2；(2)?x2－y2? 。

【分析】可以直接應(yīng)用公式，只要把公式里的 a,b 用有關(guān)字母代進去就可以了，公式里的 a，在(1)內(nèi)是 a；在(2)內(nèi)是 x；公式里的 b，在(1)內(nèi)是x，在(2)內(nèi)是 y? 。

【解】(1)?a2－x2＝(a＋x)(a－x)；(2)?x2－y2＝(x＋y)(x－y)? 。

例2.分解因式：(1) 4a2－9b2；(2) a?－4b?? 。

【分析】4a2＝(2a)2，9b2＝(3b)2，以 2a 和 3b 替代公式里的 a 和 b 就可以了。

【解】

(1) 4a2－9b2＝(2a)2－(3b)2＝(2a＋3b)(2a－3b)? 。

(2) a?－4b?＝(a2)2－(2b2)2＝(a2＋2b2)(a2－2b2)? 。

例3.分解因式：(1)16a1?－25b2x?；(2)36a?x1o－9b?y?? 。

【解】

(1)16a1?－25b2x?＝(4a?)2－(5bx2)2＝(4a?＋5bx2)(4a?－5bx2)；

(2)36a?x1o－9b?y?＝(6a2x?)2－(3b3y?)2＝(6a2x?＋3b3y?)(6a2x?－3b3y?)? 。

例4.分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A(1)%26%5C%3B(x-y)%5E2-z%5E2%3B%5C%5C%0A(2)%26%5C%3B4(x-y)%5E2-(a-b)%5E2%3B%5C%5C%0A(3)%26%5C%3B4(a%2Bb)%5E2-9(a-b)%5E2%3B%5C%5C%0A(4)%26%5C%3B(ax%2Bby)%5E2-1.%0A%5Cend%7Baligned%7D$

【分析】這里每一個多項式都是平方差的形式，所以都可以利用上面的公式1? 。例如，在(1)里，x－y 就相當(dāng)于公式里的 a；在(2)里，2(x－y) 相當(dāng)于公式里的 a；在(4)里，1 相當(dāng)于公式里的 b? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)(x-y)%5E%7B2%7D-z%5E%7B2%7D%5C%5C%0A%26%3D%5Cleft%5B(x-y)%2Bz%5Cright%5D%5Cleft%5B(x-y)-z%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(x-y%2Bz%5Cright)%5Cleft(x-y-z%5Cright)%3B%20%5C%5C%0A%26(2)4(x-y)%5E%7B2%7D-(a-b)%5E%7B2%7D%5C%5C%0A%26%3D%5B2(x-y)%5D%5E%7B2%7D-(%5Cboldsymbol%7Ba%7D-b)%5E%7B2%7D%20%20%5C%5C%0A%26%3D%5Cleft%5B2(x-y)%2B(a-b)%5Cright%5D%5Cleft%5B2(x-y)-(a-b)%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(2x-2y%2Ba-b%5Cright)%5Cleft(2x-2y-a%2Bb%5Cright)%20%5C%5C%0A%26(3)4(a%2Bb)%5E%7B2%7D-9(a-b)%5E%7B2%7D%5C%5C%0A%26%3D%5B2(a%2Bb)%5D%5E%7B2%7D-%5B3(a-b)%5D%5E%7B2%7D%20%5C%5C%0A%26%3D2(a%2Bb)%2B3(a-b)2(a%2Bb)-3(a-b)%20%5C%5C%0A%26%3D(2a%2B2b%2B3a-3b)(2a%2B2b-3a%2B3b)%20%5C%5C%0A%26%3D(5a-b)%5Cleft(-a%2B5b%5Cright)%5C%5C%0A%26%3D(5a-b)%5Cleft(5b-a%5Cright)%3B%20%5C%5C%0A%26(4)(ax%2Bby)%5E%7B2%7D-1%5C%5C%0A%26%3D%5Cleft%5B(ax%2Bby)%2B1%5Cright%5D%5Cleft%5B(ax%2Bby)-1%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(ax%2Bby%2B1%5Cright)%5Cleft(ax%2Bby-1%5Cright).%0A%5Cend%7Baligned%7D$

【注意】在第一步分解成因式時，不要省掉中括號，但以后要把這些括號內(nèi)盡量化簡，改用小括號。

習(xí)題4-4(1)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81%20a%5E2-9b%5E2.%5C%5C%0A%262%E3%80%819x%5E2-4y%5E2.%5C%5C%0A%263%E3%80%81%20a%5E4-4b%5E2.%5C%5C%0A%264%E3%80%81a%5E6-b%5E8.%5C%5C%0A%265%E3%80%8116x%5E%7B16%7D-y%5E4z%5E6.%5C%5C%0A%266%E3%80%8125a%5E2b%5E4c%5E%7B16%7D-1.%5C%5C%0A%267%E3%80%811-4x%5E2y%5E6.%5C%5C%0A%268%E3%80%81(a%2Bb)%5E2-9.%5C%5C%0A%269%E3%80%81(2x-3y)%5E2-4a%5E2.%5C%5C%0A%2610%E3%80%81(a%2B2b)%5E2-(x-3y)%5E2.%5C%5C%0A%2611%E3%80%814(a%2B2b)%5E2-25(a-b)%5E2.%5C%5C%0A%2612%E3%80%81a%5E2(a%2B2b)%5E2-9(x%2By)%5E2.%5C%5C%0A%2613%E3%80%81b%5E%7B2%7D-(a-b%2Bc)%5E%7B2%7D.%5C%5C%0A%2614%E3%80%81(a%2Bb)%5E%7B2%7D-4a%5E%7B2%7D.%5C%5C%0A%2615%E3%80%81(x-y%2Bz)%5E2-(2x-3y%2B4z)%5E2.%5C%5C%0A%2616%E3%80%814(x%2By%2Bz)%5E2-9(x-y-z)%5E2.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%269%E3%80%81(2x-3y%2B2a)(2x-3y-2a)%3B%20%5C%5C%0A%2610%E3%80%81(a%2B2b%2Bx-3y)(a%2B2b-x%2B3y)%3B%5C%5C%0A%2611%E3%80%813(7a-b)(3b-a)%20%5C%5C%0A%2612%E3%80%81(a%5E%7B2%7D%2B2ab%2B3x%2B3y)(a%5E%7B2%7D%2B2ab-3x-3y)%3B%5C%5C%0A%2613%E3%80%81(a%2Bc)(2b-a-c)%20%20%5C%5C%0A%0A%2614%E3%80%81(3a%2Bb)(b-a)%3B%5C%5C%0A%2615%E3%80%81(3x-4y%2B5z)(-x%2B2y-3z)%3B%20%5C%5C%0A%2616%E3%80%81(5x-y-z)(-x%2B5y%2B5z).%0A%5Cend%7Baligned%7D$

例5.分解因式：(1) a?－b?；(2)?a?－9b?；(3)?a?－81b?；(4)?a1?－b1?? 。

【解】

(1) a?－b?＝(a2)2－(b2)2＝(a2＋b2)(a2－b2)＝(a2＋b2)(a＋b)(a－b)? 。

【說明】a2－b2 還可以應(yīng)用公式來分解，要繼續(xù)分解到不能分解為止。但 a2＋b2 不能再分解，就把這個因式照抄下來，不要漏掉。

(2) a?－9b?＝(a2)2－(3b2)2＝(a2＋3b2)(a2－3b2)? 。

【說明】a2－3b2 不能再分解了，因為 3 不能化成一個有理數(shù)的平方的形式。〖山注||? 到無理數(shù)領(lǐng)域后可以繼續(xù)分解為 $%5Ccolor%7Bblue%7D%7B%5Cscriptsize(a-%5Csqrt%7B3%7D%20b)(a%2B%5Csqrt%7B3%7D%20b)%7D$ 〗

$%5Csmall%5Cbegin%7Baligned%7D%0A%26%5Cleft(3%5Cright)%5C%3Ba%5E%7B8%7D-81b%5E%7B8%7D%5C%5C%0A%26%20%3D(%5Cboldsymbol%7Ba%7D%5E%7B4%7D)%5E%7B2%7D-(9b%5E%7B4%7D)%5E%7B2%7D%5C%5C%0A%26%3D(a%5E%7B4%7D%2B9b%5E%7B4%7D)(a%5E%7B4%7D-9b%5E%7B4%7D)%20%20%5C%5C%0A%26%3D(a%5E4%2B9b%5E4)(a%5E2)%5E2-(3b%5E2)%5E2%20%5C%5C%0A%26%3D(a%5E4%2B9b%5E4)%5Cleft%5B%5Cleft(a%5E2%2B3b%5E2%5Cright)%5Cleft(a%5E2-3b%5E2%5Cright)%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(a%5E4%2B9b%5E4%5Cright)%5Cleft(a%5E2%2B3b%5E2%5Cright)%5Cleft(a%5E2-3b%5E2%5Cright).%5C%5C%0A%26(4)%5C%3Ba%5E%7B16%7D-b%5E%7B16%7D%5C%5C%0A%26%20%3D(a%5E8)%5E2-(b%5E8)%5E2%5C%5C%0A%26%3D(a%5E8%2Bb%5E8)(a%5E8-b%5E8)%20%20%5C%5C%0A%26%3D(a%5E%7B8%7D%2Bb%5E%7B8%7D)%5Cleft%5B(a%5E%7B4%7D)%5E%7B2%7D-(b%5E%7B4%7D)%5E%7B2%7D%5Cright%5D%20%5C%5C%0A%26%3D(a%5E%7B8%7D%2Bb%5E%7B8%7D)(a%5E%7B4%7D%2Bb%5E%7B4%7D)(a%5E%7B4%7D-b%5E%7B4%7D)%20%5C%5C%0A%26%3D(a%5E8%2Bb%5E8)%5Cleft(a%5E4%2Bb%5E4%5Cright)%5Cleft%5B(a%5E2)%5E2-(b%5E2)%5E2%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(a%5E%5Ctext%7Bs%7D%2Bb%5E%5Ctextbf%7Bs%7D%5Cright)%5Cleft(a%5E4%2Bb%5E4%5Cright)%5Cleft(a%5E2%2Bb%5E2%5Cright)%5Cleft(a%5E2-b%5E2%5Cright)%20%5C%5C%0A%26%3D%5Cleft(a%5E8%2Bb%5E8%5Cright)%5Cleft(a%5E4%2Bb%5E4%5Cright)%5Cleft(a%5E2%2Bb%5E2%5Cright)%5Cleft(a%2Bb%5Cright)%5Cleft(a-b%5Cright).%0A%5Cend%7Baligned%7D$

例6.分解因式：(1) a3－ab2；(2) a?－9a2b2；(3) a2－b2＋a－b；(4) 5(a2－b2)－a＋b? 。

【解】

(1)先提出公因式 a，再應(yīng)用平方差公式，得 a3－ab2＝a(a2－b2)＝a(a＋b)(a－b)? 。

(2)先提出公因式 a2，得 a?－9a2b2＝a2(a2－9b2)＝a2[a2－(3b)2]＝a2(a＋3b)(a－3b)? 。

(3)分成兩組，第一組應(yīng)用平方差公式，再提取公因式 (a－b)，得 a2－b2＋a－b＝(a＋b)(a－b)＋(a－b)＝(a－b)(a＋b＋1)? 。

【注意】如果把 a2－b2＋a－b 變成 a2＋a－b2－b＝a(a＋1)－b(b＋1)，就沒有公因式，不能分解下去，達不到因式分解的要求。遇到這種情況，要換一種分組方法再試。

(4) 5(a2－b2)－a＋b＝5(a＋b)(a－b)－(a－b)＝(a－b)[5(a＋b)－1]＝(a－b)(5a＋5b－1)? 。

習(xí)題4-4(2)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81a%5E4-x%5E4y%5E4.%5C%5C%0A%262%E3%80%81a%5E8b%5E8-1%5C%5C%0A%263%E3%80%81a%5E4-16.%5C%5C%0A%264%E3%80%8116a%5E4b%5E3-c%5E8.%5C%5C%0A%265%E3%80%81a%5E9-ab%5E2.%5C%5C%0A%266%E3%80%81a%5E2b%5E3-4a%5E2b.%5C%5C%0A%267%E3%80%81x%5E2-y%5E2%2Bx-y.%5C%5C%0A%268%E3%80%81x%5E2-y%5E2-x-y.%5C%5C%0A%269%E3%80%81x%5E2-y%5E2%2Bx%2By.%5C%5C%0A%2610%E3%80%81x%5E2-y%5E2-x%2By.%5C%5C%0A%2611%E3%80%81a%5E2-4b%5E2-a-2b.%5C%5C%0A%2612%E3%80%81a%5E2-4b%5E2-2a%2B4b.%5C%5C%0A%2613%E3%80%81a%5E3-4ab%5E2-a-2b.%5C%5C%0A%2614%E3%80%815x%5E2-5y%5E2%2Bx%2By.%5C%5C%0A%2615%E3%80%813x%5E2-3y%5E2-x-y.%5C%5C%0A%2616%E3%80%812x%5E2-2y%5E2-x%2By.%5C%5C%0A%2617%E3%80%81a%5E2%2Ba-b%5E2-b.%5C%5C%0A%2618%E3%80%81%20a%5E2%2Ba-b%5E2%2Bb.%5C%5C%0A%2619%E3%80%81a%5E3-ab%5E2%2Ba-b.%5C%5C%0A%2620%E3%80%81%20a%5E3-ab%5E2-a%5E2-ab.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81(a%5E2%2Bx%5E2y%5E2)(a%2Bxy)(a-xy)%3B%20%5C%5C%0A%262%E3%80%81(a%5E4b%5E4%2B1)(a%5E2b%5E2%2B1)(ab%2B1)(ab-1)%3B%5C%5C%0A%263%E3%80%81(a%5E2%2B4)(a%2B2)(a-2)%3B%5C%5C%0A%264%E3%80%81(4a%5E%7B2%7Db%5E%7B4%7D%2Bc%5E%7B4%7D)(2ab%5E%7B2%7D%2Bc%5E%7B2%7D)(2ab%5E%7B2%7D-c%5E%7B2%7D)%3B%5C%5C%0A%265%E3%80%81a(a%5E%7B4%7D%2Bb)(a%5E%7B4%7D-b)%3B%20%5C%5C%0A%266%E3%80%81a%5E2b(b%2B2)(b-2)%3B%5C%5C%0A%267%E3%80%81(x-y)(x%2By%2B1)%3B%5C%5C%0A%268%E3%80%81(x%2By)(x-y%5Ccdot%20%20%5C%5C%0A%269%E3%80%81(x%2By)(x-y%2B1)%3B%5C%5C%0A%2610%E3%80%81(x-y)(x%2By-1)%3B%20%5C%5C%0A%2611%E3%80%81(a%2B2b)(a-2b-1)%3B%5C%5C%0A%2612%E3%80%81(a-2b)(a%2B2b-2)%20%5C%5C%0A%2613%E3%80%81(a%2B2b)(a%5E2-2ab-1)%3B%5C%5C%0A%2614%E3%80%81(x%2By)(5x-5y%2B1)%3B%20%5C%5C%0A%2615%E3%80%81(x%2By)(3x-3y-1)%3B%5C%5C%0A%2616%E3%80%81(x-y)(2x%2B2y-1)%3B%20%5C%5C%0A%2617%E3%80%81(a-b)(a%2Bb%2B1)%3B%5C%5C%0A%2618%E3%80%81(a%2Bb)(a-b%2B1)%3B%20%5C%5C%0A%2619%E3%80%81(a-b)(a%5E2%2Bab%2B1)%3B%5C%5C%0A%2620%E3%80%81a(a%2Bb)(a-b-1).%0A%5Cend%7Baligned%7D$

2、完全平方的因式分解公式

【04】我們計算兩數(shù)和或差的平方時可以應(yīng)用下面的公式：(a＋b)2＝a2＋2ab＋b2，(a－b)2＝a2－2ab＋b2? 。

【05】反過來就得到完全平方的因式分解公式：

????????a2＋2ab＋b2＝(a＋b)2（因式分解公式2），

????????a2－2ab＋b2＝(a－b)2（因式分解公式3）。

【注】因為 a2＋2ab＋b2 和 a2－2ab＋b2 可以分別化成兩個數(shù)的和或者兩個數(shù)的差的平方，我們把它們叫做完全平方式。

例7.分解因式：(1) x2＋2x＋1；(2) x2－6ax＋9a2；(3) 4a2－12ab＋9b2；(4) a?＋2a2b3＋b?? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3Bx%5E%7B2%7D%2B2x%2B1%3Dx%5E%7B2%7D%2B2%C2%B7x%C2%B71%2B1%5E%7B2%7D%3D(x%2B1)%5E%7B2%7D%3B%20%20%5C%5C%0A%26(2)%5C%3Bx%5E%7B2%7D-6ax%2B9a%5E%7B2%7D%3Dx%5E%7B2%7D-2%C2%B7%20x%C2%B73a%2B(3a)%5E%7B2%7D%3D(x-3a)%5E%7B2%7D%3B%20%20%5C%5C%0A%26(3)%5C%3B4a%5E%7B2%7D-12ab%2B9b%5E%7B2%7D%3D(2a)%5E%7B2%7D-2%C2%B7(2a)(3b)%2B(3b)%5E%7B2%7D%20%20%3D(2a-3b)%5E2%3B%20%5C%5C%0A%26(4)%5C%3Ba%5E4%2B2a%5E2b%5E3%2Bb%5E6%3D(a%5E2)%5E2%2B2%C2%B7(a%5E2)(b%5E3)%2B(b%5E3)%5E2%20%3D(a%5E2%2Bb%5E3)%5E2.%0A%5Cend%7Baligned%7D$

【說明】要確定能不能應(yīng)用公式2或3來分解，先要看兩個平方項，確定公式里的 a 與 b 在這里各是什么，然后看中間一項是不是相當(dāng)于＋2ab 或－2ab? 。如果是的，就可以分解成為兩數(shù)和或差的平方形式了。在初學(xué)的時候，中間這個過渡性步驟，不要省掉。

例8.看下列各式的空格處各應(yīng)該填什么，才能夠應(yīng)用上面的分解因式公式2或3? 。

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%20x%5E2%2B%5CBox%20xy%2B25y%5E2%3B%5C%5C%26(2)%5C%3B100x%5E2-%5CBox%20xy%2B49y%5E2%3B%5C%5C%26(3)%5C%3B9x%5E2-36x%2B%5CBox%3B%5C%5C%26(4)%5C%3B%5Cfrac14x%5E2y%5E2-%5CBox%2Bz%5E4%3B%5C%5C%26(5)%5C%3B36a%5E4-60a%5E2b%5E2x%2B%5CBox%3B%5C%5C%26(6)%5C%3B49a%5E2-%5CBox%2B16b%5E6.%5Cend%7Baligned%7D$

【解】

(1)這里 a 是 x，b 是 5y，∴ 2ab 應(yīng)該是 10xy，空白處是 10；

(2)這里 a 是10x，b 是 7y，∴ 2ab 應(yīng)該是 140xy，空白處是 140；

(3)這里 a 是 3x，從 36x 里分出 2·3x，得 2·3x·6，∴ b 是 6，空白處應(yīng)該是 36；

(4)這里 a 是 $%5Cscriptsize%5Cfrac12xy%20$ ，b 是 z2，空白處應(yīng)為 $%5Cscriptsize2%C2%B7%5Cfrac12xy%20%C2%B7z%5E2%3Dxyz%5E2$ ；

(5)這里 a 是 6a2，從 60a2b2x 里分出 2·6a2，得 2·6a2·5b2x，∴ b 是 5b2x，空白處應(yīng)該是 25b?x2；

(6)這里 a 是 7a，b 是 4b3，空白處應(yīng)為 2·7a·4b3＝56ab3? 。

例9.分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%26(1)%5C%3B%20a%5E3-8a%5E2%2B16a%3B%5C%5C%26(3)%5C%3B%20x%5E4a%5E2%2B2a%5E3x%5E2%2Ba%5E2%3B%5Cend%7Baligned%7D%5Cquad%5Cbegin%7Baligned%7D%26(2)%5C%3B9(a%2Bb)%5E2%2B6(a%2Bb)%2B1%3B%5C%5C%26(4)%5C%3B(x%2By)%5E2-4(x%2By)b%5E2%2B4b%5E4.%5Cend%7Baligned%7D$

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3Ba%5E%7B3%7D-8a%5E%7B2%7D%2B16a%3Da(a%5E%7B3%7D-8a%2B16)%3Da(a-4)%5E%7B2%7D%3B%20%20%5C%5C%0A%26(2)%5C%3B9(a%2Bb)%5E%7B2%7D%2B6(a%2Bb)%2B1%20%5C%5C%0A%26%3D%5B3(a%2Bb)%5D%5E2%2B2%C2%B73(a%2Bb)1%2B1%5E2%20%5C%5C%0A%26%3D%5B3(a%2Bb)%2B1%5D%5E2%3D(3a%2B3b%2B1)%5E2%3B%20%5C%5C%0A%26(3)%5C%3B%20x%5E4a%5E2%2B2a%5E2x%5E2%2Ba%5E2%3Da%5E2(x%5E4%2B2x%5E2%2B1)%20%5C%5C%0A%26%3Da%5E2%5B(x%5E2)%5E2%2B2%C2%B7%20x%5E2%C2%B71%2B1%5E2%5D%3Da%5E2(x%5E2%2B1)%5E2%3B%20%5C%5C%0A%26(4)%5C%3B(x%2By)%5E%7B2%7D-4%5Cleft(x%2By%5Cright)b%5E%7B2%7D%2B4b%5E%7B4%7D%20%20%5C%5C%0A%26%3D(x%2By)%5E2-2(x%2By)%C2%B7(2b%5E2)%5Cbecause(2b%5E2)%5C%5C%0A%26%3D%5B(x%2By)-2b%5E2%5D%5E2%3D(x%2By-2b%5E2)%5E2.%0A%5Cend%7Baligned%7D$

習(xí)題4-4(3)

分解因式(1~10)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81x%5E%7B2%7D-12x%2B36.%5C%5C%0A%262%E3%80%81x%5E2%2B8x%2B16.%20%5C%5C%0A%263%E3%80%814a%5E2-20ab%2B25b%5E2.%20%5C%5C%0A%264%E3%80%819x%5E%7B2%7D%2B12xy%2B4y%5E%7B2%7D%E3%80%82%20%5C%5C%0A%265%E3%80%81y%5E2-50xy%2B625x%5E2.%20%5C%5C%0A%266%E3%80%81x%5E2-38x%2B361.%20%5C%5C%0A%267%E3%80%819x%5E2y%5E4%2B30xy%5E2z%2B25z%5E2%20%5C%5C%0A%268%E3%80%81x%5E%7B6%7D%2B24x%5E%7B3%7D%2B144%20%5C%5C%0A%269%E3%80%811-6ab%5E%7B8%7D%2B9a%5E%7B2%7Db%5E%7B6%7D.%20%5C%5C%0A%2610%E3%80%8149a%5E2-112ab%5E2%2B64b%5E4.%20%0A%5Cend%7Baligned%7D$

在下列各題的空白處填上適當(dāng)?shù)臄?shù)字或字母，使這個式子是一個完全平方式(11~14)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2611%E3%80%81%5Csquare%20a%5E2-6a%2B1.%5C%5C%0A%2612%E3%80%814a%5E2%2B%5CBox%20ab%2B25b%5E2.%5C%5C%0A%2613%E3%80%8164x%5E4%2B%5Csquare%2B9y%5E2.%5C%5C%0A%2614%E3%80%8149a%5E2b%5E2c%5E2-28abcd%5E2%2B%5CBox.%0A%5Cend%7Baligned%7D$

分解因式(15~20)：

$%5Csmall%5Cbegin%7Baligned%7D%0A%2615%E3%80%81a%5E3-4a%5E2b%2B4ab%5E2.%5C%5C%0A%2616%E3%80%81a%5E4x%5E2%2B4a%5E3x%5E2y%2B4x%5E2y%5E2.%5C%5C%0A%2617%E3%80%8116a%5E2b%5E4-8ab%5E3c%5E2%2Bb%5E2c%5E4.%5C%5C%0A%2618%E3%80%819(a-b)%5E2%2B6(a-b)%2B1.%5C%5C%0A%2619%E3%80%81(a%2B2b)%5E2-10(a%2B2b)%2B25.%5C%5C%0A%2620%E3%80%814x%5E2(a%2Bb)%5E2-12xy(a%2Bb)%5E2%2B9y%5E2(a%2Bb)%5E2.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81(x-6)%5E2%3B%5C%5C%0A%262%E3%80%81(x%2B4)%5E2%3B%5C%5C%0A%263%E3%80%81(2a-5b)%5E2%3B%5C%5C%0A%264%E3%80%81(3x%2B2y)%5E2%3B%5C%5C%0A%265%E3%80%81(y-25x)%5E2%3B%5C%5C%0A%266%E3%80%81(x-19)%5E2%3B%5C%5C%0A%267%E3%80%81(3xy%5E2%2B5z)%5E2%3B%5C%5C%0A%268%E3%80%81(x%5E3%2B12)%5E2%3B%5C%5C%0A%269%E3%80%81(1-3ab%5E3)%5E2%3B%5C%5C%0A%2610%E3%80%81(7a-8b%5E2)%5E2%3B%5C%5C%0A%2611%E3%80%819%3B%5C%5C%0A%2612%E3%80%8120%3B%5C%5C%0A%2613%E3%80%8148x%5E2y%3B%5C%5C%0A%2614%E3%80%814d%5E4%3B%5C%5C%0A%2615%E3%80%81a(a-2b)%5E4%3B%5C%5C%0A%2616%E3%80%81x%5E2(a%5E2%2B2y)%5E2%3B%5C%5C%0A%2617%E3%80%81b%5E2(4ab-c%5E2)%5E2%3B%5C%5C%0A%2618%E3%80%81(3a-3b%2B1)%5E2%3B%5C%5C%0A%2619%E3%80%81(a%2B2c-5)%5E2%3B%5C%5C%0A%2620%E3%80%81(a%2Bb)%5E2(2x-3y)%5E2.%0A%5Cend%7Baligned%7D$

例10.分解因式：x2－a2＋2ab－b2? 。

【分析】這里不能直接應(yīng)用公式，但是把后面三項括成一組，先應(yīng)用公式3使 a2－2ab＋b2 變成 (a－b)2，就可以應(yīng)用平方差公式再進行因式分解。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26x%5E%7B2%7D-a%5E%7B2%7D%2B2ab-b%5E%7B2%7D%5C%5C%0A%26%20%3Dx%5E2-(a%5E2-2ab%2Bb%5E2)%20%20%5C%5C%0A%26%20%3Dx%5E2-(a-b)%5E2%20%20%5C%5C%0A%26%20%3D%5Cleft%5Bx%2B(a-b)%5Cright%5D%5Cleft%5Bx-(a-b)%5Cright%5D%20%20%5C%5C%0A%26%3D%5Cleft(x%2Ba-b%5Cright)%5Cleft(x-a%2Bb%5Cright).%0A%5Cend%7Baligned%7D$

【注】如果把前面兩項與后面兩項各分成一組，那未 x2－a2＋2ab－b2＝(x2－a2)＋(2ab－b2)＝(x＋a)(x－a)＋b(2a－b)，這樣就不能再分解下去，達不到因式分解的目的。

例11.分解因式：4x2＋12xy＋9y2－16z2? 。

【解】把前面三項括成一組，得

$%5Csmall%5Cbegin%7Baligned%7D%0A%264x%5E%7B2%7D%2B12xy%2B9y%5E%7B2%7D-16z%5E%7B2%7D%5C%5C%0A%26%20%3D(4x%5E2%2B12xy%2B9y%5E2)-16z%5E2%20%20%5C%5C%0A%26%3D(2x%2B3y)%5E%7B2%7D-(4z)%5E%7B2%7D%20%5C%5C%0A%26%20%3D%5Cleft%5B%5Cleft(2x%2B3y%5Cright)%2B4z%5Cright%5D%5Cleft%5B%5Cleft(2x%2B3y%5Cright)-4z%5Cright%5D%20%20%5C%5C%0A%26%20%3D%5Cleft(2x%2B3y%2B4z%5Cright)%5Cleft(2x%2B3y-4z%5Cright).%0A%5Cend%7Baligned%7D$

例12.分解因式：2ab－a2－b2＋1? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%262ab-a%5E2-b%5E2%2B1%3D1-(a%5E2-2ab%2Bb%5E2)%5C%5C%26%3D1-(a-b)%5E2%3D%5B1%2B(a-b)%5D%5B1-(a-b)%5D%5C%5C%26%3D(1%2Ba-b)%5Cleft(1-a%2Bb%5Cright).%5Cend%7Baligned%7D$

例13.分解因式：x2－2xy＋y2－a2－2ab－b2? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26x%5E%7B2%7D-2xy%2By%5E%7B2%7D-a%5E%7B2%7D-2ab-b%5E%7B2%7D%20%20%5C%5C%0A%26%3D(x-y)%5E2-(a%5E2%2B2ab%2Bb%5E2)%20%5C%5C%0A%26%3D(x-y)%5E2-(a%2Bb)%5E2%20%5C%5C%0A%26%3D%5Cleft%5B(x-y)%2B(a%2Bb)%5Cright%5D%5Cleft%5B(x-y)-(a%2Bb)%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(x-y%2Ba%2Bb%5Cright)%5Cleft(x-y-a-b%5Cright).%0A%5Cend%7Baligned%7D$

例14.分解因式：a2－4ab＋4b2＋6a－12b＋9? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26a%5E2%20-4ab%2B4b%5E%7B2%7D%2B6a-12b%2B9%20%20%5C%5C%0A%26%3D(a-2b)%5E2%2B2%C2%B73%C2%B7(a-2b)%2B9%20%5C%5C%0A%26%3D%5Cleft%5B(a-2b)%2B3%5Cright%5D%5E2%3D(a-2b%2B3)%5E2.%0A%5Cend%7Baligned%7D$

習(xí)題4-4(4)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81x%5E2%2B2xy%2By%5E2-9a%5E2.%5C%5C%0A%262%E3%80%814x%5E2-a%5E2-6a-9.%5C%5C%0A%263%E3%80%81x%5E2%2B4ax%2B4a%5E2-b%5E2.%5C%5C%0A%264%E3%80%819a%5E2-x%5E2%2B4x-4.%5C%5C%0A%265%E3%80%811-x%5E2%2B2xy-y%5E2.%5C%5C%0A%266%E3%80%81a%5E4-x%5E2%2B4ax-4a%5E2.%5C%5C%0A%267%E3%80%81%20a%5E2-b%5E2-x%5E2%2By%5E2-2ay%2B2bx.%5C%5C%0A%268%E3%80%81%20a%5E2%2B2ab%2Bb%5E2-2a-2b%2B1.%5C%5C%0A%269%E3%80%813a%5E2-6ab%2B3b%5E2-5a%2B5b.%5C%5C%0A%2610%E3%80%81%20a%5E2-4ab%2B4b%5E2-a%5E3%2B4ab%5E2.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81(x%2By%2B3a)(x%2By-3a)%3B%5C%5C%0A%262%E3%80%81(2x%2Ba%2B3)(2x-a-3)%3B%5C%5C%0A%263%E3%80%81(x%2B2a%2Bb)(x%2B2a-b)%3B%20%5C%5C%0A%264%E3%80%81(3a%2Bx-2)(3a-x%2B2)%3B%5C%5C%0A%265%E3%80%81(1%2Bx-y)(1-x%2By)%3B%20%5C%5C%0A%266%E3%80%81(a%5E%7B2%7D%2Bx-2a)(a%5E%7B2%7D-x%2B2a)%3B%5C%5C%0A%267%E3%80%81(a-y%2Bb%20-x)(a-y-b%2Bx)%3B%5C%5C%0A%268%E3%80%81(a%2Bb-1)%5E%7B2%7D%3B%5C%5C%0A%269%E3%80%81(a-b)(3a-3b-5)%3B%20%20%5C%5C%0A%2610%E3%80%81(a-2b)(a-2b-a%5E%7B2%7D-2ab).%0A%5Cend%7Baligned%7D$

3、立方和或立方差的因式分解法

【06】從乘法公式：(a＋b)(a2－ab＋b2)＝a3＋b3 及 (a－b)(a2＋ab＋b2)＝a3－b3，反過來就得到立方和或立方差的因式分解公式：

????????a3＋b3＝(a＋b)(a2－ab＋b2)（因式分解公式4），

????????a3－b3＝(a－b)(a2＋ab＋b2)（因式分解公式5）。

例15.分解因式：(1) a3＋8b3；(2) 27a3－1；(3) a?－b?；(4) 8x?＋27y12? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26(1)%5C%3B%20a%5E3%2B8b%5E3%5C%5C%0A%26%3Da%5E3%2B(2b)%5E3%5C%5C%0A%26%3D(a%2B2b)%5Cleft%5Ba%5E2-a%C2%B7(2b)%2B(2b)%5E2%5Cright%5D%5C%5C%0A%26%3D(a%2B2b)%C2%B7(a%5E2-2ab%2B4b%5E2)%3B%5C%5C%0A%26(2)%5C%3B27a%5E3-1%5C%5C%0A%26%3D(3a)%5E3-1%5C%5C%26%3D(3a-1)%5Cleft%5B(3a)%5E2%2B8a%C2%B71%2B1%5E4%5Cright%5D%5C%5C%0A%26%3D(3a-1)%5Cleft(9a%5E2%2B3a%2B1%5Cright)%3B%5C%5C%0A%26(3)%5C%3B%20a%5E6-b%5E9%5C%5C%0A%26%3D(a%5E2)%5E3-(b%5E3)%5E3%5C%5C%0A%26%3D(a%5E3-b%5E3)%5Cleft%5B(a%5E2)%5E2%2B(a%5E2)%5Cleft(b%5E3%5Cright)%2B(b%5E3)%5E2%5Cright%5D%5C%5C%0A%26%3D(a%5E3-b%5E3)%5Cleft(a%5E4%2Ba%5E2b%5E3%2Bb%5E6%5Cright)%3B%5C%5C%0A%26(4)%5C%3B8x%5E6%2B27y%5E%7B12%7D%5C%5C%0A%26%3D(2x%5E2)%5E3%2B(3y%5E4)%5E3%5C%5C%0A%26%3D(2x%5E3%2B3y%5E4)%5Cleft%5B(2x%5E2)%5E2-(2x%5E2)(3y%5E4)%2B(3y%5E4)%5E2%5Cright%5D%5C%5C%0A%26%3D(2x%5E2%2B3y%5E4)%5Cleft(4x%5E4-6x%5E2y%5E4%2B9y%5E8%5Cright).%0A%5Cend%7Baligned%7D$

【注意】切勿把 a3＋b3 分解成為 (a＋b)3，把 a3－b3?分解成為 (a－b)3? 。

習(xí)題4-4(5)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81a%5E3-125b%5E3.%5C%5C%0A%262%E3%80%818x%5E3%2B27.%5C%5C%0A%263%E3%80%81x%5E6%2By%5E9.%5C%5C%0A%264%E3%80%81x%5E6%2By%5E6.%5C%5C%0A%265%E3%80%81a%5E%7B12%7D%2Bb%5E%7B12%7D.%5C%5C%0A%266%E3%80%8164a%5E3-1.%5C%5C%0A%267%E3%80%81%5Cfrac18x%5E3-%5Cfrac1%7B27%7Dy%5E3.%5C%5C%0A%268%E3%80%81(x%2By)%5E3%2B8.%5C%5C%0A%269%E3%80%81343m%5E3-125n%5E6.%5C%5C%0A%2610%E3%80%811-8(a%2Bb)%5E3.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%265%E3%80%81(a%5E4%2Bb%5E4)(a%5E8-a%5E4b%5E4%2Bb%5E8)%3B%20%5C%5C%0A%266%E3%80%81(4a-1)(16a%5E2%2B4a%2B1)%3B%20%5C%5C%0A%267%E3%80%81%5Cleft(%5Cfrac12x-%5Cfrac13y%5Cright)%5Cbiggl(%5Cfrac14x%5E2%2B%5Cfrac16xy%2B%5Cfrac19y%5E2%5Cbiggr)%3B%20%5C%5C%0A%268%E3%80%81(x%2By%2B2)(x%5E2%2B2xy%2By%5E2-2x-2y%2B4)%3B%20%5C%5C%0A%269%E3%80%81(7m-5n%5E2)(49m%5E2%2B35mn%5E2%2B25n%5E4)%3B%20%5C%5C%0A%2610%E3%80%81(1-2a-2b)(1%2B2a%2B2b%2B4a%5E%7B2%7D%2B8ab%2B4b%5E%7B2%7D).%0A%5Cend%7Baligned%7D$

例16.分解因式：x?－y?? 。

【解】先應(yīng)用平方差公式，而后再應(yīng)用公式4和5，得?

x?－y?＝(x3)2－(y3)2＝(x3＋y3)(x3－y3)＝(x＋y)(x2－xy＋y2）(x－y)(x2＋xy＋y2)? 。

【注】如果先應(yīng)用立方差公式，那末

x?－y?＝(x2)3－(y2)3＝(x2－y2)[(x2)2＋x2y2＋(y2)2]＝(x＋y)(x－y)(x?＋x2y2＋y?)? 。

下一步要把 x?＋x2y2＋y? 再進行分解，不太容易。實際上，x?＋x2y2＋y? 可以這樣分解：

x?＋x2y2＋y?＝x?＋x2y2＋y?＋x2y2－x2y2＝x?＋2x2y2＋y?－x2y2＝(x2＋y2)2－(xy)2＝(x2＋y2＋xy)(x2＋y2－xy)＝(x2＋xy＋y2)(x2－xy＋y2)，這里要加上一個 x2y2 再減去一個?x2y2，比較復(fù)雜了。以后如果遇到平方差公式與立方差公式都可以應(yīng)用時，總以先用平方差公式比較妥當(dāng)。

例17.分解因式：x3＋x2＋x－y3－y2－y? 。

【分析】先根據(jù)加法交換律與結(jié)合律把六項分成三組，第一組用立方差公式分解，第二組用平方差公式分解，這樣可以有一個二項公因式 x－y? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26x%5E%7B3%7D%20%2Bx%5E%7B2%7D%2Bx-y%5E%7B3%7D-y%5E%7B2%7D-y%20%20%5C%5C%0A%26%3D(x%5E3-y%5E3)%2B(x%5E2-y%5E2)%2B(x-y)%20%5C%5C%0A%26%3D(x-y)%5Cleft(x%5E2%2Bxy%2By%5E2%5Cright)%2B(x%2By)%5Cleft(x-y%5Cright)%2B(x-y)%20%5C%5C%0A%26%3D(x-y)%5Cleft%5B(x%5E2%2Bxy%2By%5E2)%2B(x%2By)%2B1%5Cright%5D%20%5C%5C%0A%26%3D%5Cleft(x-y%5Cright)%5Cleft(x%5E2%2Bxy%2By%5E2%2Bx%2By%2B1%5Cright).%0A%5Cend%7Baligned%7D$

【注意】如果把原式直接分成有?x 的與有 y 的兩組，那末

x3＋x2＋x－y3－y2－y＝x(x2＋x＋1)－y(y2＋y＋1)，這樣就不能達到分解因式的目的。

習(xí)題4-4(6)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81a%5E%7B6%7D-64b%5E%7B6%7D%5C%5C%0A%26%202%E3%80%81a%5E%7B12%7D-b%5E%7B12%7D.%20%20%5C%5C%0A%263%E3%80%81x%5E%7B3%7D%2B6x%2By%5E%7B3%7D%2B6y%5C%5C%0A%26%204%E3%80%81x%5E%7B3%7D-y%5E%7B3%7D-x%5E%7B2%7D%2B2xy-y%5E%7B2%7D.%20%20%5C%5C%0A%265%E3%80%81%20x%5E3-x%5E2-x-y%5E3%2By%5E2%2By%5C%5C%0A%26%206%E3%80%81a%5E3-a%5E2-a%2Bb-b%5E2%2B2ab-b%5E3.%20%20%5C%5C%0A%267%E3%80%81a%5E3%2Ba%5E2%2Bb%5E3%2Bb%5E2%2B2ab.%5C%5C%0A%26%208%E3%80%81a%5E%7B3%7D%2Ba%5E%7B2%7D%2Bb%5E%7B3%7D-b%5E%7B2%7D%2Ba%2Bb.%20%20%5C%5C%0A%269%E3%80%81a%5E%7B3%7D%2B8b%5E%7B3%7D%2B2a%2B4b.%5C%5C%0A%26%2010%E3%80%81a%5E%7B6%7D%2Ba%5E%7B2%7D%2Bb%5E%7B6%7D%2Bb%5E%7B2%7D%20%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81(a%2B2b)(a%5E2-2ab%2B4b%5E2)(a-2b)(a%5E2%2B2ab%2B4b%5E2)%3B%5C%5C%0A%262%E3%80%81(a%5E2%2Bb%5E2)(a%5E4-a%5E2b%5E2%2Bb%5E4)(a%2Bb)(a%5E2-ab%2Bb%5E2)(a-b)(a%5E2%2Bab%2Bb%5E2)%3B%5C%5C%0A%263%E3%80%81(x%2By)(x%5E2-xy%2By%5E2%2B6)%3B%5C%5C%0A%264%E3%80%81(x-y)(x%5E2%2Bxy%2By%5E2-x%2By)%3B%5C%5C%0A%265%E3%80%81(x-y)(x%5E2%2Bxy%2By%5E2-x-y-1)%3B%5C%5C%0A%266%E3%80%81(a-b)(a%5E2%2Bab%2Bb%5E2-a%2Bb-1)%3B%5C%5C%0A%267%E3%80%81(a%2Bb)(a%5E2-ab%2Bb%5E2%2Ba%2Bb)%3B%5C%5C%0A%268%E3%80%81(a%2Bb)(a%5E2-ab%2Bb%5E2%2Ba-b%2B1)%3B%5C%5C%0A%269%E3%80%81(a%2B2b)(a%5E2-2ab%2B4b%5E2%2B2)%3B%5C%5C%0A%2610%E3%80%81(a%5E2%2Bb%5E2)(a%5E4-a%5E2b%5E2%2Bb%5E4%2B1).%0A%5Cend%7Baligned%7D$

4、完全立方的因式分解法

【07】從乘法公式：(a＋b)3＝a3＋3a2b＋3ab2＋b3 及?(a－b)3＝a3－3a2b＋3ab2－b3，反過來可得完全立方的因式分解公式：

????????a3＋3a2b＋3ab2＋b3＝(a＋b)3（因式分解公式6），

????????a3－3a2b＋3ab2－b3＝(a－b)3（因式分解公式7）。

例18.分解因式：a?－3a?b＋3a2b2－b3? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%26a%5E6-3a%5E4b%2B3a%5E2b%5E2-b%5E3%5C%5C%26%3D(a%5E2)%5E3-3%5Cleft(a%5E2%5Cright)%5E2b%2B3%5Cleft(a%5E2%5Cright)b%5E2-b%5E3%5C%5C%26%3D(a%5E2-b)%5E3.%5Cend%7Baligned%7D$

例19.分解因式：a3＋6a2b＋12ab2＋8b3? 。

【解】

$%5Csmall%5Cbegin%7Baligned%7D%0A%26a%5E3%2B6a%5E2b%2B12ab%5E2%2B8b%5E3%5C%5C%0A%26%3Da%5E3%2B3%C2%B7a%5E2(2b)%2B3%C2%B7a(2b)%5E2%2B(2b)%5E3%5C%5C%0A%26%3D(a%2B2b)%5E3.%0A%5Cend%7Baligned%7D$

【說明】要應(yīng)用這兩個公式，可先看兩個立方項，確定公式里的 a 與 b 各是什么，然后看中間兩項是否剛剛是 3a2b 和 3ab2，再看符號是否對頭。一定要完全合適，才能應(yīng)用公式。

習(xí)題4-4(7)

分解因式：

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%818a%5E%7B3%7D-12a%5E%7B2%7Db%2B6ab%5E%7B2%7D-b%5E%7B3%7D.%5C%5C%0A%262%E3%80%8127x%5E%7B3%7D%2B54x%5E%7B2%7Dy%2B36xy%5E%7B2%7D%2B8y%5E%7B3%7D.%5C%5C%0A%263%E3%80%8127x%5E3-108x%5E2y%2B144xy%5E2-64y%5E3.%5C%5C%0A%264%E3%80%811%2B12x%5E2y%5E2%2B48x%5E4y%5E4%2B64x%5E6y%5E6.%5C%5C%0A%265%E3%80%81x%5E6-6x%5E4y%2B12x%5E2y%5E2-8y%5E3.%5C%5C%0A%266%E3%80%8127x%5E3-9x%5E2y%2Bxy%5E2-%5Cfrac1%7B27%7Dy%5E3.%5C%5C%0A%267%E3%80%81a%5E4-3a%5E3%2B3a%5E2-a.%5C%5C%0A%268%E3%80%811-3(x-y)%2B3(x-y)%5E2-(x-y)%5E3.%5C%5C%0A%269%E3%80%81x%5E6-3x%5E4y%5E2%2B3x%5E2y%5E4-y%5E6.%5C%5C%0A%2610%E3%80%811-12a%5E2b%5E2%2B48a%5E4b%5E4-64a%5E5b%5E8.%0A%5Cend%7Baligned%7D$

【答案】

$%5Csmall%5Cbegin%7Baligned%7D%0A%261%E3%80%81(2a-b)%5E3%3B%5C%5C%0A%262%E3%80%81(3x%2B2y)%5E3%3B%5C%5C%0A%263%E3%80%81(3x-4y)%5E3%3B%5C%5C%0A%264%E3%80%81(1%2B4x%5E2y%5E2)%5E3%3B%5C%5C%0A%265%E3%80%81(x%5E2-2y)%5E3%3B%5C%5C%0A%266%E3%80%81(3x-%5Cfrac13y)%5E3%3B%5C%5C%0A%267%E3%80%81%20a(a-1)%5E3%3B%5C%5C%0A%268%E3%80%81(1-x%2By)%5E3%3B%5C%5C%0A%269%E3%80%81(x%2By)%5E3(x-y)%5E3%3B%5C%5C%0A%2610%E3%80%81(1%2B2ab)%5E3(1-2ab)%5E3.%0A%5Cend%7Baligned%7D$

標(biāo)簽：

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【種花家務(wù)·代數(shù)】1-4-04公式分解法『數(shù)理化自學(xué)叢書6677版』

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