二項展開式是依據二項式定理對(a+b)n進行展開得到的式子,由艾薩克·牛頓於1664-1665年間提出。二項展開式是高考的一個重要考點。在二項式展開式中,二項式係數是一些特殊的組合數,與術語“係數”是有區別的。二項式係數最大的項是中間項,而係數最大的項卻不一定是中間項。
基本介紹
- 中文名:二項展開式
- 外文名:Binomial expansion
- 定理:二項式定理
- 提出者:艾薩克·牛頓
- 方法:特殊值法
- 套用:粗略的分析和估計以及證明恆等式
二項式定理,理解,性質,證明,例題,某項的係數,係數最值項,指定項,
二項式定理
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理解
將
看成
個
相乘,從每個括弧中取一項 (非
即
) 相乘的所有單項式合併同類項得到的,按取
的個數分為
類 ,不取
的是
,取 1 個
的是
,..., 取
個
的是
,...,取
個
的是
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注意:
(1)選取性,二項式的兩項怎樣選取 (各取幾個) 才能構成所求的項;
(2)有序性,
的展 開式第
項是取
個
(同時取
個
), 這裡的
和
不能互換
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(3)項 、項的係數與二項式係數的區別
某項要把這一項全部寫出來;某項的係數只寫這一項的係數,不帶字母 (即把每個字母當作數 1) ;某項的二項式係數就是相應的組合數
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性質
(1)項數:n+1項
(2)第k+1項的二項式係數是
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(3)在二項展開式中,與首末兩端等距離的兩項的二項式係數相等。
(4)如果二項式的冪指數是偶數,中間的一項的二項式係數最大。如果二項式的冪指數是奇數,中間兩項的的二項式係數最大,並且相等。
(5)二項式通項:
,是第
項
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證明
這裡,採用數學歸納法對二項式定理進行證明
當
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假設二項展開式在
時成立,設
,則:
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等式也成立
結論:對於任意自然數n,等式均成立。
例題
某項的係數
求二項展開式的某項或某項的係數是高考數學的一個基本知識點,每年的高考題都有一定的題出現。
例1. 求
的展開式中
的係數
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解:
要取2個,故
的係數是
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例2. 求
的展開式中
的係數
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解:
要取4個,故
的係數是
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係數最值項
例. 求
展開式中係數最大項和最小項
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解:
通項=
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通項的係數=
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設係數
最大,則
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解得:
,因為
,所以
,故係數最大項為
和
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由於最大項在中間取得,所以最小項在兩端,計算得:
,故係數最小項為
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指定項
求二項展開式中的指定項,一般是利用通項公式進行。
例.
展開式中的常數項
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解:展開式的通項=
,令
,解得
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故常數項為:
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