【種花家務(wù)·代數(shù)】1-2-04代數(shù)式的值『數(shù)理化自學(xué)叢書6677版』

2023-09-17 13:41 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(diǎn)就是可用于“自學(xué)”。當(dāng)然由于本書是大半個(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(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(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?；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(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，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(gè)措手不及。

第二章代數(shù)式??

§2-4代數(shù)式的值

【01】代數(shù)式是表示數(shù)的計(jì)算式子，如果代數(shù)式里的字母用指定的數(shù)去代替，再依照代數(shù)式里所表示的運(yùn)算進(jìn)行計(jì)算，所得的結(jié)果就叫做代數(shù)式的值。

例1.計(jì)算代數(shù)式－3a2b 的值：(1)當(dāng) a＝－3，b＝5；(2)當(dāng) a＝0.1，b＝8；(3)當(dāng) 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.計(jì)算代數(shù)式 2x3－5x2＋3x－8 的值：(1)當(dāng) x＝1；(2)當(dāng) x＝－1；(3) x＝ $%5Cscriptsize%5Cfrac%7B1%7D%7B2%7D%20$ ；(4)當(dāng) 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】把字母的指定數(shù)值代入代數(shù)式后，有些乘方或原來省略乘號的地方，需要添上括號或者乘號。

例3.計(jì)算代數(shù)式 3(3a－2b)2 的值：(1)當(dāng) a＝－2，b＝－3；(2)當(dāng) a＝－2，b＝＋3；(3)當(dāng) a＝0.4，b＝0.3；(4)當(dāng) 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.計(jì)算代數(shù)式 $%5Cscriptsize%5Cfrac%7Ba%2B2b%7D%7B2a%2Bb%7D$ 的值：(1)當(dāng) a＝ $%5Cscriptsize%5Cfrac%7B1%7D%7B2%7D%20$ ，b＝ $%5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20$ ；(2)當(dāng) a＝ $%5Cscriptsize1%5Cfrac%7B1%7D%7B2%7D%20$ ，b＝ $%5Cscriptsize1%5Cfrac%7B2%7D%7B3%7D%20$ ；(3)當(dāng) a＝ $%5Cscriptsize-%5Cfrac%7B1%7D%7B3%7D%20$ ，b＝0.3；(4)當(dāng)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.計(jì)算代數(shù)式－a2?與 (－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 要先平方再添上負(fù)號，(－a)2 要先添上負(fù)號，再行平方。

習(xí)題2-4

1、求代數(shù)式 a－b 的值（填下表）：

2、求空格內(nèi)有關(guān)代數(shù)式的值，填入下表：

計(jì)算下列代數(shù)式的值(3~8)：

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

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