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    因式分解方法

    因式分解方法

    因式分解方法


    因式分解是代數中的重要內容,在學習中如何進行小結與復習?按照“一提、二公式、三分組、四檢查”的步驟,效果良好。

    1. “一提”:先看多項式的各項是否有公因式,若有公因式,先提取公因式。

    2. “二公式”:若多項式的各項無公因式(或已提取公因式),第二步則看項數運用公式。如果是兩項就考慮用平方差公式,如果是三項就先考慮用完全平方公式,再考慮用型式子進行因式分解,最后考慮用十字相乘法。

    3. “三分組”:若以上兩步都不能對多項式進行因式分解,則應考慮分組分解。分組的原則是:一般先考慮分組后能運用公式(在既可用完全平方公式,又可用平方差公式時,常把能用完全平方公式的項分為一組),再考慮分組后能提取公因式。但必須確保組與組之間能繼續提取公因式或運用公式,從而達到將整個多項式分解的目的。

    4. “四檢查”:檢查多項式的每一個因式是否還能繼續分解因式,直到每一個多項式因式都不能再分解為止。用整式的乘法檢查因式分解的結果是否正確。
    本帖最近評分記錄
    • lkj258 學分 +2 感謝你的帖子,幫了我大忙 2015-5-10 19:45

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    一、分組分解因式的幾種常用方法.

    一、分組分解因式的幾種常用方法.


    1.按公因式分解
    例1 分解因式7x2-3y+xy+21x.
    分析:第1、4項含公因式7x,第2、3項含公因式y,分組后又有公因式(x-3),
    解:原式=(7x2-21x)+(xy-3y)=7x(x-3)+y(x-3)=(x-3)(7x+y).

    2.按系數分解
    例2 分解因式x3+3x2+3x+9.
    分析:第1、2項和3、4項的系數之比1:3,把它們按系數分組.
    解;原式=(x3+3x2)+(3x+9)=x2(x+3)+3(x+3)=(x+3)(x2+3).

    3.按次數分組
    例3 分解因式 m2+2m·n-3m-3n+n2.
    分析:第1、2、5項是二次項,第3、4項是一次項,按次數分組后能用公式和提取公因式.
    解:原式=(m2+2m·n+n2)+(-3m-3n)=(m+n)2-3(m+n)=(m+n)(m+n-3).

    4.按乘法公式分組
    分析:第1、3、4項結合正好是完全平方公式,分組后又與第二項用平方差公式.

    5.展開后再分組
    例5 分解因式ab(c2+d2)+cd(a2+b2).
    分析:將括號展開后再重新分組.
    解:原式=abc2+abd2+cda2十cdb2=(abc2+cda2)+(cdb2+abd2)=ac(bc+ad)+bd(bc+ad)=(bc+ad)(ac+bd).

    6.拆項后再分組
    例6 分解因式x2-y2+4x+2y+3.
    分析:把常數拆開后再分組用乘法公式.
    解:原式=x2-y2+4x+2y+4-1=(x2+4x+4)+(-y2+2y-1)=(x+2)2-(y-1)2=(x+y+1)(x-y+3).

    7.添項后再分組
    例7 分解因式x4+4.
    分析:上式項數較少,較難分解,可添項后再分組.
    解:原式=x4+4x2-4x2+4=(x2+2)2-(2x)2=(x2+2x+2)(x2-2x+2)




    二、用換元法進行因式分解
    用添加輔助元素的換元思想進行因式分解就是原式繁雜直接分解有困難,通過換元化為簡單,從而分步完成.
    例8 分解因式(x2+3x-2)(x2+3x+4)-16.
    分析:將令y=x2+3x,則原式轉化為(y-2)(y+4)-16再分解就簡單了.
    解:令y=x2+3x,則
    原式=(y-2)(y+4)-16=y2+2y-24=(y+6)(y-4).
    因此,原式=(x2+3x+6)(x2+3x-4)=(x-1)(x+4)(x2+3x+6).


    三、用求根法進行因式分解
    例9 分解因式x2+7x+2.
    分析:x2+7x+2利用上述各方法皆不好完成,但仍可以分解,可用先求該多項式對應方程的根再分解.


    四、用待定系數法分解因式.
    例10 分解因式x2+6x-16.
    分析:假設能分解,則應分解為兩個一次項式的積形式,即(x+b1)(x+b2),將其展開得
    x2+(b1+b2)x十b1·b2與x2+6x-16相比較得
    b1+b2=6,b1·b2=-16,可得b1,b2即可分解.
    解:設x2+6x-16=(x+b1)(x+b2)
    則x2+6x-16=x2+(b1+b2)x+b1·b2
    ∴x2+6x-16=(x-2)(x+8).
    本帖最近評分記錄
    • lkj258 學分 +2 感謝你 2015-5-10 19:45

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    因式分解后的結果

    應熟悉的因式分解后的結果


    (1)a^3+b^3=(a+b)(a^2-ab+b^2)
    (2)a^3-b^3=(a-b)(a^2+ab+b^2)
    (3)a^3+b^3+c^3+3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)
    a^3  :表示為a的三次方 以此類推

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    頂!!!!!!頂!!!!!頂

    TOP

    我考太厲害了,真棒,謝謝這位大哥!!!!!!
    在這混亂的古惑時代我愿做最后的悲情英雄!

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    還有十字相乘法!
    深海的冰冷覆蓋了雙魚的眼淚         寂寞的曾經淹沒了記憶的幻想

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    救救我啊

    怎么辦啊  明天期中考試  我啥也不會  救救我啊
    王巧

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    我靠,太厲害了吧.我看都看不懂啊.這是初2的嗎?強悍.看來,我也的多學點了
    ぐ..愛伱↘又怎能oo輕易dē說紛離oκ☆↘

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    哇  . 靠 ..............................看著就暈          ....想學   就是看不懂              .怎么半
    _______吥媞蕗吥泙____    ↓呮媞伱吥哘↓_____

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    可用一元二次方程分解

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    厲害厲害,頂一下.
    高手說,
    他是高手.

    TOP

    可以

    TOP

    太一般。
    h黃非非h

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    哇  . 靠 ..............................看著就暈          ....想學   就是看不懂

    TOP

    棒  棒   棒   棒

    TOP

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