【种花家务·代数】1-4-04公式分解法『数理化自学丛书6677版』

2023-09-27 16:15 作者:山嵓 0人读过 | 我要投稿

【阅前提示】本篇出自『数理化自学丛书6677版』，此版丛书是“数理化自学丛书编委会”于1963-1966年陆续出版，并于1977年正式再版的基础自学教材，本系列丛书共包含17本，层次大致相当于如今的初高中水平，其最大特点就是可用于“自学”。当然由于本书是大半个世纪前的教材，很多概念已经与如今迥异，因此不建议零基础学生直接拿来自学。不过这套丛书却很适合像我这样已接受过基础教育但却很不扎实的学酥重新自修以查漏补缺。另外，黑字是教材原文，彩字是我写的注解。

【山话嵓语】我在原有“自学丛书”系列17册的基础上又添加了1册八五人教中学甲种本《微积分初步》，原因有二：一则，我是双鱼座，有一定程度的偶双症，但“自学丛书”系列中代数4册、几何5册实在令我刺挠，因此就需要加入一本代数，使两边能够对偶平衡；二则，我认为《微积分初步》这本书对“准大学生”很重要，以我的惨痛教训为例，大一高数第一堂课，我是直接蒙圈，学了个寂寞。另外大学物理的前置条件是必须有基础微积分知识，因此我所读院校的大学物理课是推迟开课；而比较生猛的大学则是直接开课，然后在绪论课中猛灌基础高数（例如田光善舒幼生老师的力学课）。我选择在“自学丛书”17本的基础上添加这本《微积分初步》，就是希望小伙伴升大学前可以看看，不至于像我当年那样被高数打了个措手不及。

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里我们曾经看到多项式 a²－b² 可以分解成两个因式，就是 a²－b²＝(a＋b)(a－b)……(1) 。

【02】事实上，这里我们是反过来应用了两数和与差的积的公式。

【03】我们可以把(1)作为一个公式，利用它来分解由一个数的平方减去另一个数的平方所构成的多项式的因式。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】可以直接应用公式，只要把公式里的 a,b 用有关字母代进去就可以了，公式里的 a，在(1)内是 a；在(2)内是 x；公式里的 b，在(1)内是x，在(2)内是 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) a⁴－4b⁴ 。

【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) a⁴－4b⁴＝(a²)²－(2b²)²＝(a²＋2b²)(a²－2b²) 。

例3.分解因式：(1)16a¹⁶－25b²x⁴；(2)36a⁴x¹º－9b⁶y⁸ 。

【解】

(1)16a¹⁶－25b²x⁴＝(4a⁸)²－(5bx²)²＝(4a⁸＋5bx²)(4a⁸－5bx²)；

(2)36a⁴x¹º－9b⁶y⁸＝(6a²x⁵)²－(3b³y⁴)²＝(6a²x⁵＋3b³y⁴)(6a²x⁵－3b³y⁴) 。

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】这里每一个多项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 就相当于公式里的 a；在(2)里，2(x－y) 相当于公式里的 a；在(4)里，1 相当于公式里的 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时，不要省掉中括号，但以后要把这些括号内尽量化简，改用小括号。

习题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) a⁴－b⁴；(2) a⁴－9b⁴；(3) a⁸－81b⁸；(4) a¹⁶－b¹⁶ 。

【解】

(1) a⁴－b⁴＝(a²)²－(b²)²＝(a²＋b²)(a²－b²)＝(a²＋b²)(a＋b)(a－b) 。

【说明】a²－b² 还可以应用公式来分解，要继续分解到不能分解为止。但 a²＋b² 不能再分解，就把这个因式照抄下来，不要漏掉。

(2) a⁴－9b⁴＝(a²)²－(3b²)²＝(a²＋3b²)(a²－3b²) 。

【说明】a²－3b² 不能再分解了，因为 3 不能化成一个有理数的平方的形式。〖山注|| 到无理数领域后可以继续分解为 $%5Ccolor%7Bblue%7D%7B%5Cscriptsize(a-%5Csqrt%7B3%7D%20b)(a%2B%5Csqrt%7B3%7D%20b)%7D$ 〗

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

例6.分解因式：(1) a³－ab²；(2) a⁴－9a²b²；(3) a²－b²＋a－b；(4) 5(a²－b²)－a＋b 。

【解】

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

(2)先提出公因式 a²，得 a⁴－9a²b²＝a²(a²－9b²)＝a²[a²－(3b)²]＝a²(a＋3b)(a－3b) 。

(3)分成两组，第一组应用平方差公式，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 变成 a²＋a－b²－b＝a(a＋1)－b(b＋1)，就没有公因式，不能分解下去，达不到因式分解的要求。遇到这种情况，要换一种分组方法再试。

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

习题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】

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2、完全平方的因式分解公式

【04】我们计算两数和或差的平方时可以应用下面的公式：(a＋b)²＝a²＋2ab＋b²，(a－b)²＝a²－2ab＋b² 。

【05】反过来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】因为 a²＋2ab＋b² 和 a²－2ab＋b² 可以分别化成两个数的和或者两个数的差的平方，我们把它们叫做完全平方式。

例7.分解因式：(1) x²＋2x＋1；(2) x²－6ax＋9a²；(3) 4a²－12ab＋9b²；(4) a⁴＋2a²b³＋b⁶ 。

【解】

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

【说明】要确定能不能应用公式2或3来分解，先要看两个平方项，确定公式里的 a 与 b 在这里各是什么，然后看中间一项是不是相当于＋2ab 或－2ab 。如果是的，就可以分解成为两数和或差的平方形式了。在初学的时候，中间这个过渡性步骤，不要省掉。

例8.看下列各式的空格处各应该填什么，才能够应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)这里 a 是 x，b 是 5y，∴ 2ab 应该是 10xy，空白处是 10；

(2)这里 a 是10x，b 是 7y，∴ 2ab 应该是 140xy，空白处是 140；

(3)这里 a 是 3x，从 36x 里分出 2·3x，得 2·3x·6，∴ b 是 6，空白处应该是 36；

(4)这里 a 是 $%5Cscriptsize%5Cfrac12xy%20$ ，b 是 z²，空白处应为 $%5Cscriptsize2%C2%B7%5Cfrac12xy%20%C2%B7z%5E2%3Dxyz%5E2$ ；

(5)这里 a 是 6a²，从 60a²b²x 里分出 2·6a²，得 2·6a²·5b²x，∴ b 是 5b²x，空白处应该是 25b⁴x²；

(6)这里 a 是 7a，b 是 4b³，空白处应为 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】

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

习题4-4(3)

分解因式(1~10)：

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

在下列各题的空白处填上适当的数字或字母，使这个式子是一个完全平方式(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】这里不能直接应用公式，但是把后面三项括成一组，先应用公式3使 a²－2ab＋b² 变成 (a－b)²，就可以应用平方差公式再进行因式分解。

【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】如果把前面两项与后面两项各分成一组，那未 x²－a²＋2ab－b²＝(x²－a²)＋(2ab－b²)＝(x＋a)(x－a)＋b(2a－b)，这样就不能再分解下去，达不到因式分解的目的。

例11.分解因式：4x²＋12xy＋9y²－16z² 。

【解】把前面三项括成一组，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】

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

习题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】从乘法公式：(a＋b)(a²－ab＋b²)＝a³＋b³ 及 (a－b)(a²＋ab＋b²)＝a³－b³，反过来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) a⁶－b⁹；(4) 8x⁶＋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 分解成为 (a＋b)³，把 a³－b³ 分解成为 (a－b)³ 。

习题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：x⁶－y⁶ 。

【解】先应用平方差公式，而后再应用公式4和5，得

x⁶－y⁶＝(x³)²－(y³)²＝(x³＋y³)(x³－y³)＝(x＋y)(x²－xy＋y²）(x－y)(x²＋xy＋y²) 。

【注】如果先应用立方差公式，那末

x⁶－y⁶＝(x²)³－(y²)³＝(x²－y²)[(x²)²＋x²y²＋(y²)²]＝(x＋y)(x－y)(x⁴＋x²y²＋y⁴) 。

下一步要把 x⁴＋x²y²＋y⁴ 再进行分解，不太容易。实际上，x⁴＋x²y²＋y⁴ 可以这样分解：

x⁴＋x²y²＋y⁴＝x⁴＋x²y²＋y⁴＋x²y²－x²y²＝x⁴＋2x²y²＋y⁴－x²y²＝(x²＋y²)²－(xy)²＝(x²＋y²＋xy)(x²＋y²－xy)＝(x²＋xy＋y²)(x²－xy＋y²)，这里要加上一个 x²y² 再减去一个 x²y²，比较复杂了。以后如果遇到平方差公式与立方差公式都可以应用时，总以先用平方差公式比较妥当。

例17.分解因式：x³＋x²＋x－y³－y²－y 。

【分析】先根据加法交换律与结合律把六项分成三组，第一组用立方差公式分解，第二组用平方差公式分解，这样可以有一个二项公因式 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 的与有 y 的两组，那末

x³＋x²＋x－y³－y²－y＝x(x²＋x＋1)－y(y²＋y＋1)，这样就不能达到分解因式的目的。

习题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】从乘法公式：(a＋b)³＝a³＋3a²b＋3ab²＋b³ 及 (a－b)³＝a³－3a²b＋3ab²－b³，反过来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：a⁶－3a⁴b＋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】

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

【说明】要应用这两个公式，可先看两个立方项，确定公式里的 a 与 b 各是什么，然后看中间两项是否刚刚是 3a²b 和 3ab²，再看符号是否对头。一定要完全合适，才能应用公式。

习题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】

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

标签：

【种花家务·代数】1-4-04公式分解法『数理化自学丛书6677版』

习题4-4(1)

习题4-4(2)

习题4-4(3)

习题4-4(4)

习题4-4(5)

习题4-4(6)

习题4-4(7)