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【种花家务·代数】1-3-04整式的加减法『数理化自学丛书6677版』

2023-09-18 17:10 作者:山嵓  | 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。

第三章整式  

§3-4整式的加减法

1、单项式的加法

【01】应用上面所讲过的合并同类项的法则,我们很容易做关于单项式的加法。举例说明如下:

例1.求下面各题里的这些单项式的和:(1)3a,-5a;(2)3a,-4b,-5a;(3)3a,-4b  。

【解】

(1) 3a 和-5a 相加,它们的和就是 3a+(-5a)  。把它写成代数和的形式,再合并同类项,得3a+(-5a)=3a-5a=-2a  。

(2) 3a+(-4b)+(-5a)=3a-4b-5a=3a-5a-4b=-2a-4b  。

(3) 3a+(-4b)=3a-4b  。

【02】从上面的例子可以看到,做单项式的加法,有下面的单项式的加法法则:几个单项式相加,只要把这些单项式连结起来写成代数和的形式,再合并同类项。

【注】把几个代数式相加时,我们把这些代数式都叫做加式。

例2.做下面的加法:

%5Cscriptsize%5Cbegin%7Baligned%7D%26(1)(-3ab)%2B(-2ab)%2B(%2B5ab)%3B%5C%5C%26(2)%5Cleft(-%5Cfrac%7B1%7D%7B2%7Dx%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B2%7Dx%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B3%7Dx%5E2%5Cright).%5Cend%7Baligned%7D

【解】

(1) (-3ab)+(-2ab)+(+5ab)=-3ab-2ab+5ab=0  。

(2)

%5Cscriptsize%5Cbegin%7Baligned%7D%26%5Cleft(-%5Cfrac%7B1%7D%7B2%7Dx%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dx%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B2%7Dx%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B3%7Dx%5E2%5Cright)%5C%5C%26%3D%5Cunderline%7B-%5Cfrac%7B1%7D%7B2%7Dx%7D-%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cunderline%7B%5Cfrac%7B1%7D%7B2%7Dx%7D%2B%5Cfrac%7B1%7D%7B3%7Dx%5E2%3D-%5Cfrac%7B1%7D%7B3%7Dx%2B%5Cfrac%7B1%7D%7B3%7Dx%5E2%3D%5Cfrac%7B1%7D%7B3%7Dx%5E2-%5Cfrac%7B1%7D%7B3%7Dx.%5Cend%7Baligned%7D

【注意】

        在(1)中合并同类项以后 ab 的系数是 0,所以结果是 0  。

        在(2)中 %5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20x%0A 和 %5Cscriptsize%5Cfrac%7B1%7D%7B3%7D%20x%5E2%0A 不是同类项,所以不能再合并。最后结果用 x 的降幂排列,这祥可以避免第一项的系发是负数。但是把 %5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20x%2B%5Cfrac%7B1%7D%7B3%7D%20x%5E2%0A 当做最后的结果也是可以的。

习题3-4(1)

1、做下面的加法:

%5Cscriptsize%5Cbegin%7Baligned%7D(1)%263a%2B(%2B7a)%3B%26(2)%26(-3a)%2B(-7a)%3B%5C%5C(3)%263x%2B(-7x)%3B%26(4)%26(-3x)%2B(%2B7x)%3B%5C%5C(5)%26%5Cfrac%7B1%7D%7B2%7Dx%2B%5Cfrac%7B1%7D%7B3%7Dx%3B%26(6)%26(-%5Cfrac%7B1%7D%7B3%7Da)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B2%7Da%5Cright).%5Cend%7Baligned%7D

2、求下列各式的和:

%5Cscriptsize%5Cbegin%7Barray%7D%7Bllll%7D%7B(1)%7D%26%7B3x%5E%7B2%7D%2C%5C%3B-5x%5E%7B2%7D%3B%7D%26%7B(2)%7D%7B-3a%5E%7B3%7D%2C%5C%3B-5a%5E%7B3%7D%3B%7D%5C%5C%7B(3)%7D%26%7B-8a%5E%7B2%7Db%5E%7B3%7D%2C%5C%3B%2B9a%5E%7B2%7Db%5E%7B3%7D%3B%7D%26%7B(4)%7D%7Bx%5E%7B2%7Dy%2C%5C%3B-x%5E%7B2%7Dy%3B%7D%5C%5C%7B(5)%7D%26%7B%5Cfrac%7B2%7D%7B3%7Da%5E%7B2%7Dx%5E%7B2%7D%2C%5C%3B-%5Cfrac%7B3%7D%7B4%7Da%5E%7B2%7Dx%5E%7B2%7D%3B%7D%26%7B(6)0.3a%5E%7B2%7Db%5E%7B3%7Dc%2C%5C%3B-%5Cfrac%7B1%7D%7B3%7Da%5E%7B2%7Db%5E%7B3%7Dc.%7D%5Cend%7Barray%7D

3、计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)(%2B3a)%2B(-5b)%3B%5Cquad%26%26(2)(-5a%5E2)%2B(-5a)%3B%5C%5C%0A%26%26(3)(-3a%5E2b)%2B(-5ab%5E2)%3B%26%26(4)(%2B%5Cfrac12a%5E3b%5E3)%2B(%2B%5Cfrac13a%5E3b%5E2)%3B%5C%5C%0A%26%26(5)(-3a%5E2x%5E3)%2B(-5a%5E2x)%3B%26%26(6)(-3xy)%2B(%2B5ay).%0A%5Cend%7Beqnarray%7D

4、计算:

%5Cscriptsize%5Cbegin%7Baligned%7D%0A%26(1)(%2B3a)%2B(-5a)%2B(-7a)%3B%20%5C%5C%0A%26(2)(-7a%5E%7B2%7D)%2B(%2B5a%5E%7B2%7D)%2B(%2B2a%5E%7B2%7D)%3B%20%5C%5C%0A%26(3)(-7a%5E%7B2%7Db%5E%7B3%7D)%2B(-11a%5E%7B2%7Db%5E%7B3%7D)%2B(-14a%5E%7B3%7Db%5E%7B2%7D)%3B%20%5C%5C%0A%26(4)(-5a%5E2x%5E3)%2B(%2B5a%5E3x%5E2)%2B(-3a%5E3x%5E2)%3B%20%5C%5C%0A%26(5)(%2B7abx%5E%7B2%7D)%2B(-4ab%5E%7B2%7Dx)%2B(-3a%5E%7B2%7Dbx)%3B%5C%5C%0A%26(6)(-3a%5E%7B2%7Dx%5E%7B2%7Dy%5E%7B2%7D)%2B(%2B5a%5E%7B2%7Dx%5E%7B3%7Dy%5E%7B2%7D)%2B(-3a%5E%7B6%7Dx%5E%7B2%7Dy%5E%7B2%7D).%0A%5Cend%7Baligned%7D

5、计算:

%5Cscriptsize%5Cbegin%7Baligned%7D%0A%26%5Ctext%7B(1)%7D%20(%2B3a)%2B(-5a)%2B(-7a)%2B(-31a)%3B%20%20%5C%5C%0A%26%5Ctext%7B(2)%7D%20%5Cleft(%2B%5Cfrac%7B1%7D%7B2%7Dab%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dab%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B4%7Dab%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B5%7Dab%5Cright)%3B%20%20%5C%5C%0A%26%5Ctext%7B(3)%7D%20%5Cleft(%2B%5Cfrac%7B2%7D%7B3%7Dx%5E%7B2%7Dy%5Cright)%2B%5Cleft(-%5Cfrac%7B3%7D%7B2%7Dx%5E%7B2%7Dy%5Cright)%2B%5Cleft(-%5Cfrac%7B2%7D%7B3%7Dxy%5E%7B2%7D%5Cright)%2B%5Cleft(%2B%5Cfrac%7B1%7D%7B4%7Dxy%5E%7B2%7D%5Cright)%3B%20%20%5C%5C%0A%26%5Ctext%7B(4)%7D%205a%5E%7B3%7Dx%5E%7B2%7D%2B(-3a%5E%7B2%7Dx%5E%7B5%7D)%2B(%2B5a%5E%7B3%7Dx%5E%7B2%7D)%2B(-3a%5E%7B2%7Dx%5E%7B3%7D)%3B%20%20%5C%5C%0A%26%5Ctext%7B(5)%7D%20%5Cleft(-%5Cfrac%7B1%7D%7B2%7Dabx%5E%7B2%7D%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dabx%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B4%7Dabx%5E%7B2%7D%5Cright)%2B%5Cleft(-%5Cfrac%7B1%7D%7B3%7Dab%5E%7B2%7Dx%5Cright)%3B%20%20%5C%5C%0A%26%5Ctext%7B(6)%7D%20(-2a%5E%7B2%7Db%5E%7B3%7Dxy)%2B(-3a%5E%7B2%7Db%5E%7B3%7Dxy)%2B(-5a%5E%7B3%7Db%5E%7B2%7Dxy)%2B(-6a%5E%7B3%7Db%5E%7B2%7Dxy).%20%0A%5Cend%7Baligned%7D

【答案】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A4%E3%80%81%26%26(1)%5Cquad-9a%2C%5Cquad(2)%5Cquad0%2C%5Cquad(3)%5Cquad-18a%5E2b%5E3-14a%5E3b%5E3%2C%5C%5C%0A%26%26(4)%5Cquad-5a%5E2x%5E3%2B2a%5E2x%5E2%2C%5Cquad(5)%5Cquad7abx%5E2-4ab%5E2x-3a%5E2bx%2C%5C%5C%0A%26%26(6)%5Cquad-3a%5E2x%5E2y%5E3%2B5a%5E2x%5E3y%5E2-3a%5E3x%5E2y%5E2%3B%5C%5C%0A5%E3%80%81%26%26(1)%5Cquad-40a%2C%5Cquad(2)%5Cquad%5Cfrac%7B13%7D%7B60%7Dab%2C%5C%5C%0A%26%26(3)%5Cquad-%5Cfrac56x%5E2y-%5Cfrac5%7B12%7Dxy%5E2%2C%5Cquad(4)%5Cquad10a%5E3x%5E2-6a%5E2x%5E3%2C%5C%5C%0A%26%26(5)%5Cquad-%5Cfrac34abx%5E2-%5Cfrac13abx-%5Cfrac13ab%5E2x%2C%5Cquad(6)%5Cquad-5a%5E2b%5E3xy-11a%5E3b%5E2xy.%0A%5Cend%7Beqnarray%7D

2、单项式的减法

【03】在有理数的减法里,我们知道,减去一个数,等于加上它的相反的数。用字母来表示,就是a-(+b)=a+(-b),a-(-b)=a+(+b)  。

【04】所以做单项式的减法,只需应用下面的单项式的减法法则:减去一个单项式,只要把这个单项式的性质符号改成相反的符号,再做加法。

【注】一个代数式减去另一个代数式,我们把第一个代数式叫做被减式,第二个代数式叫做减式。

例3.做下列减法:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)5a-(%2B3a)%3B%5Cquad%26%26(2)5a-(%2B8a)%3B%5C%5C%0A%26%26(3)(-5a)-(%2B3a)%3B%26%26(4)(-5a)-(-8a)%3B%5C%5C%0A%26%26(5)(-5a%5E3)-(-7a%5E3)%3B%26%26(6)(%2Ba%5E4b%5E3)-(-a%5E2b%5E3).%5C%0A%5Cend%7Beqnarray%7D

【解】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26%20(1)5a-(%2B3a)-5a%2B(-3a)%3D5a-3a%3D2a%3B%20%20%5C%5C%0A%26%26(2)5a-(%2B8a)%3D5a%2B(-8a)-5a-8a%3D-3a%2C%20%5C%5C%0A%26%26(3)(-5a)-(%2B3a)-(-5a)%2B(-3a)%20%3D-5a-3a%3D-8a%3B%20%5C%5C%0A%26%26(4)(-5a)-(-8a)%20%3D(-5a)%2B(%2B8a)%20%3D-5a%2B8a%3D%2B3a%3B%5C%5C%0A%26%26(5)(-5a%5E3)-(-7a%5E3)%3D(-5a%5E3)%2B(%2B7a%5E3)%3D-5a%5E3%2B7a%5E3%3D2a%5E3%3B%5C%5C%0A%26%26(6)(%2Ba%5E2b%5E3)-(-a%5E2b%5E3)%3Da%5E2b%5E3%2B(%2Ba%5E2b%5E3)%3Da%5E2b%5E3%2Ba%5E2b%5E3%3D2a%5E2b%5E3.%0A%5Cend%7Beqnarray%7D

例4.计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%26%26(1)3a-(%2B5b)%3B%5Cquad%26%26(2)5a-(-3b)%3B%5C%5C%26%26(3)3a%5E2-(%2B2a)%3B%26%26(4)3a%5E2-(-3a).%0A%5Cend%7Beqnarray%7D

【解】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26%20(1)3a-(%2B5b)%3D3a%2B(-5b)%3D3a-5b%3B%20%5C%5C%0A%26%26%20(2)5a-(-3b)%3D5a%2B(%2B3b)%3D5a%2B3b%3B%20%5C%5C%0A%26%26%20(3)3a%5E%7B2%7D-(%2B2a)-3a%5E%7B2%7D%2B(-2a)%3D3a%5E%7B2%7D-2a%3B%20%5C%5C%0A%26%26(4)3a%5E%7B2%7D-(-3a)%3D3a%5E%7B2%7D%2B(%2B3a)%3D3a%5E%7B2%7D%2B3a%20.%0A%5Cend%7Beqnarray%7D

例5.计算:

%5Cscriptsize%5Cbegin%7Baligned%7D%26(1)%5Cquad3a%5E2-5a%2B(-7a%5E2)%3B%5C%5C%26(2)%5Cquad(%2Ba%5E2b)%2B(-ab%5E2)-(-2a%5E2b)%3B%5C%5C%26(3)%5Cquad%5Cfrac12x%5E3y%5E3%2B%5Cfrac13x%5E3y%5E2-%5Cleft(-%5Cfrac13x%5E3y%5E3%5Cright)%2B%5Cleft(-%5Cfrac12x%5E3y%5E2%5Cright)%3B%5C%5C%26(4)%5Cquad(%2B5a)%2B(-3b)%2B(%2B0.3c)%2B(-3a)%2B(%2B3b)%2B0.7c.%5Cend%7Baligned%7D

【解】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)3a%5E%7B2%7D-5a%2B(-7a%5E%7B2%7D)%3D3a%5E%7B2%7D-5a-7a%5E%7B2%7D%3D-4a%5E%7B2%7D-5a%3B%20%5C%5C%0A%26%26(2)(%2Ba%5E%7B2%7Db)%2B(-ab%5E%7B2%7D)-(-2a%5E%7B2%7Db)%20%5C%5C%0A%26%26%5Cquad%3D%2Ba%5E%7B2%7Db-ab%5E%7B2%7D%2B2a%5E%7B2%7Db-3a%5E%7B2%7Db-ab%5E%7B2%7D%3B%20%5C%5C%0A%26%26(3)%5Cfrac12x%5E3y%5E3%2B%5Cfrac13x%5E3y%5E3-%5Cleft(-%5Cfrac13x%5E3y%5E3%5Cright)%2B%5Cleft(-%5Cfrac12x%5E3y%5E2%5Cright)%20%5C%5C%0A%26%26%5Cquad%3D%5Cfrac%7B1%7D%7B2%7Dx%5E%7B3%7Dy%5E%7B3%7D%2B%5Cfrac%7B1%7D%7B3%7Dx%5E%7B3%7Dy%5E%7B2%7D%2B%5Cfrac%7B1%7D%7B3%7Dx%5E%7B2%7Dy%5E%7B3%7D-%5Cfrac%7B1%7D%7B2%7Dx%5E%7B3%7Dy%5E%7B3%7D%20%5C%5C%0A%26%26%5Cquad%3D%5Cfrac56x%5E%7B2%7Dy%5E%7B3%7D-%5Cfrac16x%5E%7B3%7Dy%5E%7B2%7D%3B%5C%5C%0A%26%26(4)(%2B5a)%2B(-3b)%2B(%2B0.3c)%2B(-3a)%2B(%2B3b)%2B0.7c%5C%5C%26%26%5Cquad%3D5a-3b%2B0.3c-3a%2B3b%2B0.7c%3D2a%2Be.%0A%5Cend%7Beqnarray%7D

习题3-4(2)

1、计算:

%5Cscriptsize%5Cbegin%7Baligned%7D(1)%26(-5a)-(-7a)%3B%26(2)%26(%2B3a)-(%2B8a)%3B%5C%5C(3)%26(%2B5ab)-(-6ab)%3B%26(4)%26(-3a%5E3)-(-3a%5E3)%3B%5C%5C(5)%26(-%5Cfrac%7B1%7D%7B2%7Da%5E2)-(-%5Cfrac%7B1%7D%7B3%7Da%5E2)%3B%26(6)%26a%5E2b%5E3c-2a%5E2b%5E3c.%5Cend%7Baligned%7D

2、做下列减法:

%5Cscriptsize%5Cbegin%7Barray%7D%7Blll%7D(1)%26(%2B5ab)-(-6ac)%3B%26(2)(-5a)-(%2B7a%5E2)%3B%5C%5C(3)%26(-4ab%5E2)-(%2B4a%5E2b)%3B%26(4)(-9x%5E2)-(%2B9y%5E2)%3B%5C%5C(5)%26(%2B3a%5E2b%5E3)-(%2B3a%5E3b%5E2)%3B%26(6)3a%5E2b%5E2c-(-3ab%5E2c%5E2).%5Cend%7Barray%7D

3、计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)%203a%2B(%2B5a)-(-3a)%3B%20%20%5C%5C%0A%26%26(2)5a%5E2-(%2B6a%5E2)%2B(-3a%5E2)%3B%5C%5C%0A%26%26(3)3a%5E%7B2%7Db%2B2a%5E%7B2%7Db-(%2B6a%5E%7B2%7Db)%3B%20%20%5C%5C%0A%26%26(4)(-3a%5E%7B2%7D)-(%2B3a%5E%7B2%7D)-(-6a%5E%7B2%7D)%20%20%5C%5C%0A%26%26(5)(-5a%5E%7B3%7D)%2B(-3a%5E%7B2%7D)-(-3a)%3B%20%20%5C%5C%0A%26%26(6)3ab-(-5ab)%2B(-7ac).%0A%5Cend%7Beqnarray%7D

4、计算:

%5Cscriptsize%5Cbegin%7Baligned%7D%26(1)3a-(%2B5a)-(-7a)%3B%5C%5C%0A%26(2)(-5a)-(-4a)%2B(%2B7a)%3B%5C%5C%26(3)(%2B3a%5E2)%2B(-5a%5E2)-(-7a%5E2)%3B%5C%5C%26(4)%5Cfrac%7B1%7D%7B2%7Da%5E2b%5E3-%5Cleft(-%5Cfrac%7B1%7D%7B3%7Da%5E2b%5E3%5Cright)-%5Cleft(-%5Cfrac%7B1%7D%7B4%7Da%5E2b%5E3%5Cright).%5Cend%7Baligned%7D

5、计算:

%5Cscriptsize%5Cbegin%7Bgathered%7D%0A(1)(-8x%5E%7B2%7Dy)%2B(%2B10x%5E%7B2%7Dy)-(-3xy%5E%7B2%7D)%2B(-5xy%5E%7B2%7D)%3B%20%5C%5C%0A(2)(-7y%5E%7B2%7D)%2B(-4y)-(-y%5E%7B2%7D)-(%2B5y)%2B(-8y%5E%7B2%7D).%20%0A%5Cend%7Bgathered%7D

【答案】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A3%E3%80%81%26%26(1)11a%2C%5Cquad(2)-4a%5E2%2C%5Cquad(3)-a%5E2b%2C%5Cquad(4)0%2C%5C%5C%26%26(5)-5a%5E3-3a%5E2%2B3a%2C%5Cquad(6)8ab-7ac%3B%5C%5C%0A4%E3%80%81%26%26(1)5a%2C%5Cquad(2)6a%2C%5Cquad(3)5a%5E2%2C%5Cquad(4)%5Cfrac%7B1.3%7D%7B1.2%7Da%5E2b%5E3%3B%5C%5C%0A5%E3%80%81%26%26(1)2x%5E2y-2xy%5E2%2C%5Cquad(2)-14y%5E2-9y.%0A%5Cend%7Beqnarray%7D

3、多项式的加法和减法

【05】根据加法结合律,我们知道:a+(b+c+d)=a+b+c+d  。根据减法的运算性质,我们还知道:a-(b+c+d)=a-b-c-d  。

【06】所以做多项式的加法和减法,可以应用下面的多项式加减法法则

        (ⅰ)加上一个多项式,可以依次加上这个多项式的各项。

        (ⅱ)减去一个多项式,可以改变减式各项的符号,把它们依次加在被减式上。

例6.做下面的加法:

%5Cscriptsize%5Cbegin%7Baligned%7D%26(1)%5Cquad(3a%5E3%2B2a%5E2%2B5a-7)%2B(-2a%5E3%2B3a%5E2-5a%2B4)%3B%5C%5C%26(2)%5Cquad(3a%2B2b-3c)%2B(5a-3b%2B2c).%5Cend%7Baligned%7D

【解】

%5Cscriptsize%5Cbegin%7Baligned%7D%0A%26(1)%20(3a%5E3%2B2a%5E2%2B5a-7)%2B(-2a%5E3%2B3a%5E2-5a%2B4)%20%20%5C%5C%0A%26%3D3a%5E%7B3%7D%2B2a%5E%7B2%7D%2B5a-7-2a%5E%7B3%7D%2B3a%5E%7B2%7D-5a%2B4%20%5C%5C%0A%26%3D3a%5E3-2a%5E3%2B2a%5E2%2B3a%5E3%2B5a-5a-7%2B4%20%5C%5C%0A%26%3Da%5E3%2B5a%5E2-3%3B%20%5C%5C%0A%26(2)(3a%2B2b-3c)%2B(5a-3b%2B2c)%20%26%20%5Ctext%7B.%7D%20%20%5C%5C%0A%26%3D3a%2B2b-3c%2B5a-3b%2B2c%20%5C%5C%0A%26%3D3a%2B5a%2B2b-3b-3c%2B2c%3D8a-b-c.%0A%5Cend%7Baligned%7D

例7.计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)%20(3a%5E%7B2%7D-5a)-(5a%5E%7B2%7D-7a)%3B%20%20%5C%5C%0A%26%26(2)%20(-3a%2B5b)-(-5a%2B7b)%3B%20%20%5C%5C%0A%26%26(3)(3a%2B2b-3c)-(5a-2b%2B3c)%3B%20%20%5C%5C%0A%26%26(4)3-(3a%5E%7B3%7D-5a-7)%3B%20%20%5C%5C%0A%26%26(5)%20(a%5E3-3a%5E2b%2B3ab%5E2-b%5E3)-(a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3)%3B%20%5C%5C%0A%26%26(6)%20(x%5E%7B3%7D-3x%5E%7B2%7Dy%2B3xy%5E%7B2%7D)-(x%5E%7B3%7D-3x%5E%7B2%7Dy-3xy%5E%7B2%7D-y%5E%7B3%7D).%20%0A%5Cend%7Beqnarray%7D

【解】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A(1)%26%26(3a%5E2-5a)-(5a%5E3-7a)%5C%5C%0A%26%26%3D3a%5E2-5a-5a%5E2%2B7a%5C%5C%26%26%3D-2a%5E2%2B2a%3B%5C%5C%0A(2)%26%26(-3a%2B5b)-(-5a%2B7b)%5C%5C%26%26%20%3D-3a%2B5b%2B5a-7b%20%20%5C%5C%0A%26%26%3D2a-2b%3B%20%5C%5C%0A(3)%26%26(3a%2B2b-3c)-(5a-2b%2B3c)%20%5C%5C%0A%26%26%3D3a%3D2h-3c-5a%2B2h-3a%20%5C%5C%0A%26%26%3D-2a%2B4b-6c%3B%20%5C%5C%0A(4)%26%263-(3a%5E2-5a-7)%5C%5C%26%26%3D3-3a%5E2%2B5a%2B7%20%5C%5C%0A%26%26%3D3a%5E2%2B5a%2B10%3B%20%5C%5C%0A(5)%26%26(a%5E3-3a%5E2b%2B3ab%5E2-b%5E3)-(a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3)%20%5C%5C%0A%26%26%3Da%5E3-3a%5E2b%2B3ab%5E2-b%5E3-a%5E3-3a%5E2b-3ab%5E2-b%5E3%20%5C%5C%0A%26%26%3D-6a%5E%7B2%7Db-2b%5E%7B3%7D%3B%20%5C%5C%0A(6)%26%26(x%5E3-3x%5E2y%2B3xy%5E2)-(x%5E3-3x%5E2y-3xy%5E2-y%5E3)%20%5C%5C%0A%26%26%3Dx%5E%7B3%7D-3x%5E%7B2%7Dy%2B2xy%5E%7B2%7D-x%5E%7B3%7D%2B3x%5E%7B2%7Dy%2B3xy%5E%7B2%7D%2By%5E%7B3%7D%20%5C%5C%0A%26%26%3D6xy%5E%7B2%7D%2By%5E%7B3%7D.%0A%5Cend%7Beqnarray%7D

例8.计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)(3a%2B5b)%2B(5a-4b)-(2a-3b)%3B%5C%5C%0A%26%26(2)(-a%5E3%2B3a%5E2-7a%2B5)%2B(5a%5E2-6a)-(a%5E2-4a%2B7)%3B%5C%5C%0A%26%26(3)%5Cleft(%5Cfrac13a%5E3-2a%5E2-%5Cfrac12a-6%5Cright)-%5Cleft(%5Cfrac13a%5E3-5%5Cright)-%5Cleft(%5Cfrac12a%5E3-4a-7%5Cright)%3B%5C%5C%0A%26%26(4)(0.3x%5E3-x%5E3y%2Bxy%5E2-y%5E2)-(-0.5x%5E3-x%5E2y%2B0.3xy%5E2)%2B(1.2x%5E3-1.2y%5E3).%0A%5Cend%7Beqnarray%7D

【解】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A(1)%26%26(3a%2B5b)%2B(5a-4b)-(2a-3b)%20%5C%5C%0A%26%26%3D3a%2B5b%2B5a-4b-2a%2B3b-6a%2B4b%3B%20%5C%5C%0A(2)%26%26(-a%5E3%2B3a%5E2-7a%2B5)%2B(5a%5E2-6a)-(a%5E3-5)%20%5C%5C%0A%26%26%3D-a%5E%7B3%7D%2B3a%5E%7B2%7D-7a%2B5%2B5a%5E%7B2%7D-6a-a%5E%7B3%7D%2B4a-7%20%5C%5C%0A%26%26%3D-2a%5E%7B3%7D%2B8a%5E%7B2%7D-9a%2B2%3B%5C%5C%0A(3)%26%26%5Cleft(%5Cfrac13a%5E3-2a%5E3-%5Cfrac12a-6%5Cright)-%5Cleft(%5Cfrac13a%5E2-5%5Cright)%20-%5Cleft(%5Cfrac%7B1%7D%7B2%7Da%5E%7B3%7D-4a-7%5Cright)-%5Cleft(36-69%2B96%5Cright)%5Cquad(8)%20%5C%5C%0A%26%26%3D%5Cfrac13a%5E3-2a%5E3-%5Cfrac12a-6-%5Cfrac13a%5E3%2B5-%5Cfrac12a%5E3%2B4a%2B7%20%5C%5C%0A%26%26%3D-%5Cfrac16%5Cboldsymbol%7Ba%7D%5E%7B3%7D-%5Cfrac73a%5E%7B3%7D%2B%5Cfrac72a%2B6%3B%5C%5C%0A(4)%26%26(0.3x%5E3-x%5E2y%2Bxy%5E2-y%5E3)-(-0.5x%5E3-x%5E2y%2B0.3ay%5E2)%2B(1.2x%5E3-1.2y%5E3)%20%5C%5C%0A%26%26%3D0.3x%5E3-x%5E2y%2Bxy%5E3-y%5E3%2B0.5x%5E3%2Bx%5E2y%20-0.3xy%5E2%2B1.2x%5E3-1.2y%5E3%20%5C%5C%0A%26%26%3D2x%5E3%2B0.7xy%5E3-2.2y%5E3.%0A%5Cend%7Beqnarray%7D

习题3-4(3)

1、计算:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26%5Ctext%7B(1)%7D%20(2x%5E%7B3%7D-3x%5E%7B2%7D%2B6x%2B5)%2B(x%5E%7B3%7D-6x%2B9)%3B%20%20%5C%5C%0A%26%26%5Ctext%7B(2)%7D%20(2a%5E%7B3%7D%2B5a%5E%7B2%7D%2B3a-1)%2B(3-8a%2B2a%5E%7B2%7D-6a%5E%7B3%7D)%3B%20%20%5C%5C%0A%26%26%5Ctext%7B(3)%7D%20(%2B3a%5E%7B2%7D%2Bb%5E%7B2%7D-5ab)%2B(4ab-b%5E%7B2%7D%2B7a%5E%7B2%7D)%3B%20%20%5C%5C%0A%26%26%5Ctext%7B(4%7D%20)%5Cleft(%5Cfrac%7B2%7D%7B3%7Dx%5E%7B2%7D-%5Cfrac%7B5%7D%7B6%7Dxy%2B%5Cfrac%7B3%7D%7B4%7Dy%5E%7B2%7D%5Cright)%2B%5Cleft(%5Cfrac%7B5%7D%7B12%7Dx%5E%7B2%7D-%5Cfrac%7B4%7D%7B3%7Dxy-%5Cfrac%7B7%7D%7B4%7Dy%5E%7B2%7D%5Cright)%20%2B%5Cleft(%5Cfrac%7B1%7D%7B2%7Dx%5E%7B2%7D-y%5E%7B2%7D%2B%5Cfrac%7B5%7D%7B4%7Dxy%5Cright).%0A%5Cend%7Beqnarray%7D

2、做下列加法:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)(3a%5E3-5a%5E2-a%2B4)%2B(-2a%5E4-5a%5E2%2Ba-3)%2B(-3a%5E3%2B5a-7)%3B%5C%5C%0A%26%26(2)(-3a%5E2-5b%5E2%2B3a%5E2)%2B%5Cleft(%5Cfrac%7B1%7D%7B2%7Da%5E2-%5Cfrac%7B1%7D%7B3%7Db%5E2-3c%5E2%5Cright)%2B%5Cleft(1%5Cfrac%7B1%7D%7B2%7Da%5E%7B2%7D%2B%5Cfrac%7B2%7D%7B3%7Db%5E%7B2%7D%2B%5Cfrac%7B1%7D%7B4%7Dc%5E%7B2%7D%5Cright)%3B%5C%5C%0A%26%26(3)(-0.3x%5E3y%2B0.5x%5E2y%5E2%2B3.1xy%5E3%2B0.4y%5E4)%2B(-1.2x%5E4-0.13x%5E3y%2B0.36xy%5E3%2B3y%5E4)%5C%5C%26%26%2B(3.2x%5E4-3.3x%5E3y%2B2.2x%5E2y%5E2-y%5E4)%3B%5C%5C%0A%26%26(4)(3a%5E2b-5ab%5E2%2B3b%5E3)%2B(a%5E3-3a%5E2b-5ab%5E2-2b%5E3)%2B(-a%5E3%2B10ab%5E2-b%5E3).%0A%5Cend%7Beqnarray%7D

3、做下列减法:

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26(1)(6a%2B10b)-(2a%2B13b)%3B%5C%5C%0A%26%26(2)(2a-2b%2B4c)-(-2a-6b%2B4c)%3B%5C%5C%0A%26%26(3)(3a%5E2-5a%2B7)-(-3a%5E2-5a%2B7)%3B%5C%5C%0A%26%26(4)(3a%5E2-5ab%2B6b%5E2)-(3a%5E2%2B5ab-6b%5E2)%3B%5C%5C%0A%26%26(5)(ax%2Bbx%2Bay%2Bby)%3D(ax-bx-ay%2Bby)%3B%5C%5C%0A%26%26(6)(-3x%5E3-5x%5E2y%2B3xy%5E2-y%5E3)-(-2x%5E3%2B5x%5E2y-2y%5E5).%0A%5Cend%7Beqnarray%7D

4、计算:

%5Cscriptsize%5Cbegin%7Barray%7D%7Bll%7D(1)(3a%2B5b)%2B(5a-7b)-(2a-4b)%3B%5C%5C(2)(a%5E3-2a%5E2%2Ba-7)-(5a%5E2-7a%2B8)%2B(a%5E3-3a%5E2-5)%3B%5C%5C(3)(-x%5E3-3x%5E2%2B7x-8)-(3x%5E3-2x%5E3-5x%2B4)-(-x%5E3-2x%5E2%2B2)%3B%5C%5C(4)(x%5E3-3x%5E2y%2B3xy%5E2-y%5E3)-(x%5E3%2B3x%5E2y%2B3xy%5E2%2By%5E3)-(6x%5E2y).%5Cend%7Barray%7D

【答案】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A1%E3%80%81%26%26(1)3x%5E3-3x%5E2%2B14%2C%5Cquad(2)-4a%5E3%2B7a%5E2-5a%2B2%2C%20%5C%5C%0A%26%26(3)10a%5E2-ab%2C%5Cquad(4)%5Cfrac%7B19%7D%7B12%7Dx%5E2-%5Cfrac%7B11%7D%7B12%7Dxy-2y%5E2%3B%20%5C%5C%0A2%E3%80%81%26%26(1)%5Cquad-2a%5E4-10a%5E2%2B5a-6%5Ctext%7B%2C(2)%7D-a%5E2-4%5Cfrac23b%5E2%2B%5Cfrac14c%5E2%5Ctext%7B%2C%7D%20%5C%5C%0A%26%26(3)2x%5E4-3.73x%5E3y%2B2.7x%5E2y%5E2%2B3.46xy%5E3%2B2.4y%5E4%2C(4)0%3B%5C%5C%0A3%E3%80%81%26%26(1)4a-3b%2C%20%5Cquad(2)4a%2B4b%2C%5Cquad(3)6a%5E2%2C%5Cquad(4)-10ab%2B12b%5E2%2C%5C%5C%26%26(5)2bx%2B2ay%2C%20(6)-x%5E3-10x%5E2y%2B3xy%5E2%2By%5E3%3B%5C%5C%0A4%E3%80%81%26%26(1)6a%2B2b%2C%20(2)2a%5E3-10a%5E2%2B8a-20%2C%5C%5C%0A%26%26(3)-3x%5E3%2Bx%5E3%2B12x-14%5Cquad(4)-12x%5E2y-2y%5E3.%0A%0A%5Cend%7Beqnarray%7D

4、多项式加减的直式运算

【07】在做多项式的加减法时,为了便于合并同类项,我们也可以用直式来进行演算。

例9.计算:(10x³-6x²+5x-4)+(9x³-2x²+4x+2)  。

【解】用直式演算:

【注意】用直式迸行加法演算时,首先要把第一个加式按照某一字母的降幂(或升幂)整理排列,其他的加式排在下面,要注意对齐同类项,这样,合并同类项时只要注意进行直行的系数的加法,写出对应的同类项来就是了。

【注意】用直式进行加法演算时,最后仍应列出横式来。

例10.计算:(8x³-6x²+5x-12)+(-4x²+3x-8)+(-5x³+5x+7)  。

【解】用直式演算:

【注意】用直式进行加法演算,如果有一些加式缺少某些幂的项时,可以空出相应地位,务使同类项上下对齐,这样才便于演算。

例11.计算:(3a²b-5a³+b³)+(6a³-7ab²-2b³)+(3a³+5a²b+7ab²)  。

【解】

例12.计算:%5Cscriptsize%5Cbegin%7Baligned%7D%5Cleft(%5Cfrac12a%5E2%2B%5Cfrac13a-%5Cfrac14%5Cright)%2B%5Cleft(%5Cfrac13a%5E2-%5Cfrac14a%2B%5Cfrac15%5Cright)%2B%5Cleft(1%5Cfrac14a%5E2-0.3a%2B1%5Cright)%5Cend%7Baligned%7D 

【解】

例13.计算:(3+x²-4x)-(5x一8十3x²)  。

【解】

【注意】同加法的直式写法一样,先要把被减式按照一个字母的降(或升)幂进行排列,再将减式的各项写在被减式的下面,使同类项互相对齐,然后在心中把减式各项的性质符号换成相反符号后,按加法合并同类项.最后还必须写出一个横式表示结果。

例14.计算:(3x⁴-2x²+x-3)-(4x³-x²+5)  。

【解】

例15.计算:(5a⁴+2a²b²+ab³-3a³b)-(5a²b-2ab²+3a²b²+b⁴)  。

【解】

习题3-4(4)

用直式进行演算:

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【答案】

%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%261.%5Cquad8x%5E4%2B2x%5E3%2B6x%5E2%2B7x-14%3B%5Cquad2.%5Cquad-14x%5E3-2x%5E2%2B8x%2B7%3B%5C%5C%0A%26%263.%5Cquad4x%5E4-x%5E3-3x-9%3B%5Cquad4.%5Cquad-3a%5E2%2Bab%2B7b%5E2%3B%5Cquad5.%5Cquad-2xy%3B%5C%5C%0A%26%266.%5Cquad%5Cfrac%7B4%7D%7B3%7Dx%5E2%2B%5Cfrac%7B1%7D%7B2%7Dxy%3B%5Cquad7.%5Cquad%5Cfrac%7B4%7D%7B3%7Dx%5E3-%5Cfrac%7B3%7D%7B4%7Dx%5E2%2Bx-5%5Cfrac%7B2%7D%7B3%7D%3B%5Cquad8.%5Cquad2a%5E4%2B12a%5E2b%5E2%2B2b%5E4%3B%5C%5C%0A%26%269.%5Cquad-a%5E3-0.2a%3B%5Cquad10.%5Cquad%5Cfrac%7B1%7D%7B30%7Da%5E2-%5Cfrac%7B1%7D%7B15%7Db%5E2%3B%5Cquad11.%5Cquad4y-6z%3B%5Cquad12.5x%5E3-x-22%3B%5C%5C%26%2613.%5Cquad12x%5E7%2B3x%5E6-8x%5E5%2B2x%5E4-11x%5E3-3x%5E2%3B%5Cquad14.%5Cquad-2ab%2B4b%5E2%3B%5C%5C%26%2615.%5Cquad6x%5E3%2B2x%5E2%2Bi%2B11%3B%5Cquad16.%5Cquad%20a%5E3%2Ba%5E2b%2B6ab%5E2%2Bb%5E3%3B%5Cquad17.%5Cquad3a%5E2b-3ab%5E2%3B%5C%5C%26%2618.%5Cquad%5Cfrac%7B29%7D%7B30%7Da%5E3%2B%5Cfrac%7B1%7D%7B10%7Da%5E2-%5Cfrac%7B3%7D%7B4%7Da%2B%5Cfrac%7B19%7D%7B18%7D.%0A%0A%5Cend%7Beqnarray%7D

【种花家务·代数】1-3-04整式的加减法『数理化自学丛书6677版』的评论 (共 条)

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