【种花家务·代数】1-2-04代数式的值『数理化自学丛书6677版』

2023-09-17 13:41 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。

第二章代数式

§2-4代数式的值

【01】代数式是表示数的计算式子，如果代数式里的字母用指定的数去代替，再依照代数式里所表示的运算进行计算，所得的结果就叫做代数式的值。

例1.计算代数式－3a²b 的值：(1)当 a＝－3，b＝5；(2)当 a＝0.1，b＝8；(3)当 a＝ $%5Cscriptsize%5Cfrac%7B3%7D%7B5%7D%20$ ，b＝ $%5Cscriptsize-8%5Cfrac%7B1%7D%7B3%7D%20$ 。

【解】

(1)－3a²b＝－3(－3)²(5)＝－3·(＋9)·5＝－135；

(2)－3a²b＝－3(0.1)²(8)＝－3(0.01)(8)＝－0.24；

(3)－3a²b＝( $%5Cscriptsize%5Cfrac%7B3%7D%7B5%7D%20$ )²( $%5Cscriptsize-8%5Cfrac%7B1%7D%7B3%7D%20$ )＝－3· $%5Cscriptsize%5Cfrac%7B9%7D%7B25%7D%20$ ·( $%5Cscriptsize-%5Cfrac%7B25%7D%7B3%7D%20$ )＝＋9 。

例2.计算代数式 2x³－5x²＋3x－8 的值：(1)当 x＝1；(2)当 x＝－1；(3) x＝ $%5Cscriptsize%5Cfrac%7B1%7D%7B2%7D%20$ ；(4)当 x＝ $%5Cscriptsize-%5Cfrac%7B1%7D%7B2%7D%20$ 。

【解】

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26%20(1)2x%5E%7B3%7D-5x%5E%7B2%7D%2B3x-8%20%26%26%20%3D2(1)%5E3-5(1)%5E2%2B3(1)-8%20%20%5C%5C%0A%26%26%26%26%3D2-5%2B3-8%3D-8%3B%20%5C%5C%0A%26%26(2)2x%5E%7B3%7D-5x%5E%7B2%7D%2B3x-8%20%26%26%20%3D2(-1)%5E3-5(-1)%5E2%2B3(-1)-8%20%20%5C%5C%0A%26%26%26%26%3D2(-1)-5(%2B1)%2B3(-1)-8%20%5C%5C%0A%26%26%26%26%3D-2-5-3-8%3D-18%3B%20%5C%5C%0A%26%26(3)2x%5E%7B3%7D-5x%5E%7B2%7D-3x-8%26%26%20%3D2%5CBig(%5Cfrac%7B1%7D%7B2%7D%5CBig)%5E%7B3%7D-5%5CBig(%5Cfrac%7B1%7D%7B2%7D%5CBig)%5E%7B2%7D%2B3%5CBig(%5Cfrac%7B1%7D%7B2%7D%5CBig)-8%20%20%5C%5C%0A%26%26%26%26%3D2%5CBig(%5Cfrac%7B1%7D%7B8%7D%5CBig)-5%5CBig(%5Cfrac%7B1%7D%7B4%7D%5CBig)%2B3%5CBig(%5Cfrac%7B1%7D%7B2%7D%5CBig)-8%20%5C%5C%0A%26%26%26%26%3D%5Cfrac%7B1%7D%7B4%7D-%5Cfrac%7B5%7D%7B4%7D%2B%5Cfrac%7B3%7D%7B2%7D-8%20%5C%5C%0A%26%26%26%26%3D-1%2B1%5Cfrac%7B1%7D%7B2%7D-8%3D-7%5Cfrac%7B1%7D%7B2%7D%3B%0A%5Cend%7Beqnarray%7D$

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A(4)%26%262x%5E3-5x%5E2%2B3x-8%20%5C%5C%0A%26%26%3D2%5CBig(-%5Cfrac12%5CBig)%5E3-5%5CBig(-%5Cfrac12%5CBig)%5E2%2B3%5CBig(-%5Cfrac12%5CBig)-8%20%5C%5C%0A%26%26-2%5CBig(-%5Cfrac%7B1%7D%7B8%7D%5CBig)-5%5CBig(%2B%5Cfrac%7B1%7D%7B4%7D%5CBig)%2B3%5CBig(-%5Cfrac%7B1%7D%7B2%7D%5CBig)-8%20%5C%5C%0A%26%26%3D-%5Cfrac14-%5Cfrac54-%5Cfrac32-8%3D-%5Cfrac32-%5Cfrac32-8%3D-11.%0A%5Cend%7Beqnarray%7D$

【注意】把字母的指定数值代入代数式后，有些乘方或原来省略乘号的地方，需要添上括号或者乘号。

例3.计算代数式 3(3a－2b)² 的值：(1)当 a＝－2，b＝－3；(2)当 a＝－2，b＝＋3；(3)当 a＝0.4，b＝0.3；(4)当 a＝－0.4，b＝－0.3 。

【解】

(1) 3(3a－2b)²＝3[3(－2)－2(－3)]²＝3(－6＋6)²＝3·0²＝0；

(2) 3(3a－2b)²＝3[3(－2)－2(＋3)]²＝3(－6－6)²＝3(－12)²＝3(＋144)＝432；

(3) 3(3a－2b)²＝3[3(0.4)－2(0.3)]²＝3(1.2－0.6)²＝3(0.6)2＝3×0.36＝1.08；

(4)3(3a－2b)²＝3[3(－0.4)－2(－0.3)]²＝3(－12＋0.6)²＝3(＝0.6)²＝3×0.36＝1.08 。

例4.计算代数式 $%5Cscriptsize%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D$ 的值：(1)当 a＝ $%5Cscriptsize%5Cfrac%7B1%7D%7B2%7D%20$ ，b＝ $%5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20$ ；(2)当 a＝ $%5Cscriptsize1%5Cfrac%7B1%7D%7B2%7D%20$ ，b＝ $%5Cscriptsize1%5Cfrac%7B2%7D%7B3%7D%20$ ；(3)当 a＝ $%5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20$ ，b＝0.3；(4)当a＝－1.3，b＝1.5 。

【解】

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%26(1)%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D%26%3D%5Cfrac%7B%5Cfrac12%2B2%5Cleft(-%5Cfrac13%5Cright)%7D%7B2%5Cleft(%5Cfrac12%5Cright)%2B%5Cleft(-%5Cfrac13%5Cright)%7D%3D%5Cfrac%7B%5Cleft(%5Cfrac12-%5Cfrac23%5Cright)%5Ctimes6%7D%7B%5Cleft(1-%5Cfrac13%5Cright)%5Ctimes6%7D%5C%5C%26%26%3D%5Cfrac%7B3-4%7D%7B6-2%7D%3D-%5Cfrac14%2C%5C%5C%26(2)%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D%26%3D%5Cfrac%7B-1%5Cfrac12%2B2%5Cleft(1%5Cfrac23%5Cright)%7D%7B2%5Cleft(-1%5Cfrac12%5Cright)%2B%5Cleft(1%5Cfrac23%5Cright)%7D%3D%5Cfrac%7B-%5Cfrac32%2B%5Cfrac%7B10%7D3%7D%7B-3%2B%5Cfrac53%7D%5C%5C%26%26%3D%5Cfrac%7B-9%2B20%7D%7B-18%2B10%7D%3D%5Cfrac%7B11%7D%7B-8%7D%3D-1%5Cfrac38%2C%5C%5C%26(3)%7B%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D%7D%20%26%3D%5Cfrac%7B-%5Cfrac13%2B2%5Cleft(0.3%5Cright)%7D%7B2%5Cleft(-%5Cfrac13%5Cright)%2B0.3%7D%3D%5Cfrac%7B-%5Cfrac13%2B0.6%7D%7B-%5Cfrac23%2B0.3%7D%20%20%5C%5C%0A%26%26%3D%5Cfrac%7B-%5Cfrac%7B1%7D%7B3%7D%2B%5Cfrac%7B3%7D%7B5%7D%7D%7B-%5Cfrac%7B2%7D%7B3%7D%2B%5Cfrac%7B3%7D%7B10%7D%7D%3D%5Cfrac%7B-10%2B18%7D%7B-20%2B9%7D%3D%5Cfrac%7B%2B8%7D%7B-11%7D%3D-%5Cfrac%7B8%7D%7B11%7D%2C%20%5C%5C%0A%26(4)%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D%26%20%3D%5Cfrac%7B-1.3%2B2(-1.5)%7D%7B2(-1.3)%2B(-1.5)%7D%3D%5Cfrac%7B-1.3-3%7D%7B-2.6-1.5%7D%20%20%5C%5C%0A%26%26%3D%5Cfrac%7B-4.3%7D%7B-4.1%7D%3D%5Cfrac%7B43%7D%7B41%7D%3D1%5Cfrac%7B2%7D%7B41%7D.%5Cend%7Beqnarray%7D$

例5.计算代数式－a² 与 (－a)² 的值：(1) a＝5，(2) a＝－5，(3) a＝0.13，(4) a＝－0.13，(5)a＝ $%5Cscriptsize-1%5Cfrac%7B2%7D%7B3%7D%20$ 。

【解】

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A(1)%20%26%26-a%5E%7B2%7D%3D-5%5E%7B2%7D%3D-25%2C%20%20%5C%5C%0A%26%26%5Cleft(-a%5Cright.%20)%5E%7B2%7D%3D(-5)%5E%7B2%7D%3D%2B25%3B%20%20%5C%5C%0A(2)%26%26%2B-a%5E%7B2%7D%3D%20-(-5)%5E%7B2%7D%3D-25%2C%20%20%5C%5C%0A%26%26%5Cleft(-a%5Cright)%20%5E2%3D%5B-(-5)%5D%5E2%3D5%5E2%3D25%3B%20%20%5C%5C%0A%20(3)%26%26-a%5E%7B2%7D%3D%20%20-(0.13)%5E2%3D-0.0169%2C%20%20%5C%5C%0A%26%26(-a)%5E%7B2%7D%20%3D(-0.13)%5E%7B2%7D%3D%2B0.0169%3B%20%20%5C%5C%0A(4)%26%26-a%5E%7B2%7D%3D%20-(-0.13)%5E%7B2%7D%3D-0.0169%2C%20%20%5C%5C%0A%26%26(-a)%5E%7B2%7D%3D%5B-(-0.13)%5D%5E%7B2%7D%3D(0.13)%5E%7B2%7D%3D0.0169%3B%20%20%5C%5C%0A%20(5)%26%26-a%5E%7B2%7D%3D%20%20-%5Cleft(-1%5Cfrac23%5Cright)%5E2%3D-%5Cleft(-%5Cfrac53%5Cright)%5E2%3D-%5Cfrac%7B25%7D9%3D-2%5Cfrac79%2C%20%20%5C%5C%0A%26%26(-a)%5E%7B2%7D%20%3D%5CBig%5B-%5CBig(-1%5Cfrac%7B2%7D%7B3%7D%5CBig)%5CBig%5D%5E%7B2%7D-%5CBig(%5Cfrac%7B5%7D%7B3%7D%5CBig)%5E%7B2%7D-%5Cfrac%7B25%7D%7B9%7D%3D2%5Cfrac%7B7%7D%7B9%7D.%20%0A%0A%5Cend%7Beqnarray%7D$

【注意】－a² 要先平方再添上负号，(－a)² 要先添上负号，再行平方。

习题2-4

1、求代数式 a－b 的值（填下表）：

2、求空格内有关代数式的值，填入下表：

计算下列代数式的值(3~8)：

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%263%E3%80%81-%5Cfrac%7B2%7D%7B3%7Dab%5E2%2C%26%26%5C%5C%0A%26%26(1)%E5%BD%93%5Cfrac%7B15%7D%7B3%7Da%3D-3%2Cb%3D-2%3B%26%26(2)%E5%BD%93%5Cfrac%7B35%7D%7B3%7Da%3D0.3%2Cb%3D-0.2%3B%5C%5C%0A%26%26(3)%E5%BD%93%5Cfrac%7B35%7D%7B3%7Da%3D-2%5Cfrac%7B1%7D%7B3%7D%2Cb%3D-1%5Cfrac%7B1%7D%7B7%7D%3B%26%26(4)%E5%BD%93a%3D-0.12%2Cb%3D%2B%5Cfrac%7B5%7D%7B6%7D.%5C%5C%0A%26%264%E3%80%81-x%5E%7B8%7D%2B2x%5E%7B3%7D-3x%2B4%2C%26%26%20%5C%5C%0A%26%26(1)%E5%BD%93x%3D2%3B%26%26(2)%E5%BD%93x%3D-2%3B%20%5C%5C%0A%26%26%20(3)%E5%BD%93x%3D-0.3%3B%26%26(4)%E5%BD%93x%3D-1%5Cfrac%7B2%7D%7B3%7D.%20%5C%5C%0A%26%265%E3%80%81a%5E2-b%5E2%2C%26%26%5C%5C%0A%26%26(1)%E5%BD%93%20a%3D-3%2Cb%3D-5%3B%26%26(2)%E5%BD%93a%3D-53%2Cb%3D4.7%3B%5C%5C%0A%26%26(3)%E5%BD%93a%3D0.32%2Cb%3D-0.68%3B%26%26(4)%E5%BD%93a%3D2%5Cfrac%7B1%7D%7B3%7D%2Cb%3D-3%5Cfrac%7B1%7D%7B4%7D.%5C%5C%0A%26%266%E3%80%81(a-b)%5E%7B2%7D%2C%20%26%26%5C%5C%0A%26%26(1)%20%E5%BD%93a%3D-3%2Cb%3D-5%3B%26%26(2)%20%E5%BD%93a%3D-0.3%2Cb%3D%2B0.5%3B%20%5C%5C%0A%26%26%20(3)%20%E5%BD%93a%3D-3%5Cfrac%7B2%7D%7B3%7D%2Cb%3D5%5Cfrac%7B1%7D%7B2%7D%3B%26%26(4)%20%E5%BD%93a%3D0.7%2Cb%3D-%5Cfrac%7B2%7D%7B3%7D.%20%5C%5C%0A%26%267%E3%80%81%5Cfrac%7B(a%2Bb)%5E2%7D%7Ba%5E2%2Bb%5E2%7D%2C%26%26%5C%5C%0A%26%26(1)%E5%BD%93a%3D-2%2Cb%3D-3%3B%26%26(2)%E5%BD%93a%3D-1%5Cfrac%7B2%7D%7B3%7D%2Cb%3D2%5Cfrac%7B1%7D%7B2%7D%3B%5C%5C%0A%26%26(3)%E5%BD%93a%3D0.3%2Cb%3D-0.4%3B%26%26(4)%E5%BD%93%5Cfrac%7B32%7D%7B3%7Da%3D0.03%2Cb%3D-%5Cfrac%7B1%7D%7B30%7D.%5C%5C%0A%26%268%E3%80%81%5Cfrac%7B1%2Ba%2Ba%5E2%7D%7B1-a%2Ba%5E2%7D%2C%20%5C%5C%0A%26%26(1)%E5%BD%93a%3D-5%3B%26%26(2)%E5%BD%93a%3D-0%2C2%3B%20%5C%5C%0A%26%26(3)%E5%BD%93a%3D%2B3%5Cfrac%7B1%7D%7B3%7D%3B%20%26%26(4)%E5%BD%93a%3D-3%5Cfrac%7B1%7D%7B3%7D.%20%0A%5Cend%7Beqnarray%7D$

$%5Cscriptsize%5Cbegin%7Beqnarray%7D%0A%26%26%E3%80%90%E7%AD%94%E6%A1%88%E3%80%91%5C%5C%0A%0A%26%263%E3%80%81(1)8%2C%5Cquad(2)-0.008%2C%5Cquad(3)2%5Cfrac2%7B63%7D%2C%5Cquad(4)%5Cfrac1%7B18%7D%3B%20%5C%5C%0A%26%264%E3%80%81(1)-2%2C%5Cquad(2)26%2C%5Cquad(3)5.107%2C%5Cquad(4)19%5Cfrac%7B5%7D%7B27%7D%3B%5C%5C%0A%26%265%E3%80%81(1)-16%2C%20(2)%5E%7B-%7D6%2C%5Cquad(3)%5E%7B-%7D-0.36%2C%5Cquad(4)%5E%7B-%7D5%5Cfrac%7B17%7D%7B144%7D%3B%5C%5C%0A%26%266%E3%80%81(1)%5E%7B-%7D4%2C%5Cquad(2)%5E%7B-%7D0.64%2C%20(3)84%5Cfrac%7B1%7D%7B36%7D%2C%5Cquad(4)1%5Cfrac%7B781%7D%7B900%7D%3B%5C%5C%0A%26%267%E3%80%81(1)1%5Cfrac%7B12%7D%7B13%7D%2C%5Cquad(2)%5Cfrac%7B1%7D%7B13%7D%2C%5Cquad(3)%5Cfrac%7B1%7D%7B25%7D%2C%20%0A(4)%5Cfrac%7B1%7D%7B181%7D%3B%5C%5C%0A%26%268%E3%80%81(1)%5Cfrac%7B21%7D%7B31%7D%2C%5Cquad(2)%5Cfrac%7B21%7D%7B31%7D%2C%5Cquad(3)1%5Cfrac%7B60%7D%7B79%7D%2C%5Cquad(4)%5Cfrac%7B79%7D%7B139%7D%3B%20%0A%0A%5Cend%7Beqnarray%7D$