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2021年利用平方差公式进行因式分解教育方案设计.pdf

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1、利用平方差公式进行因式分解教学目标：知识与技能：1.理解平方差公式的本质:结构的不变性,字母的可变性.2.会用平方差公式进行因式分解.3.使学生了解提公因式法是因式分解首先考虑的方法,再考虑用公式法分解.过程与方法：经历探索利用平方差公式进行因式分解的过程,发展学生的逆向思维,渗透数学的互逆、换元、整体的思想,感受数学知识的完整性.情感态度与价值观：在探究的过程中培养学生独立思考的习惯,在交流的过程中学会向别人清晰地表达自己的思维和想法,在解决问题的过程中让学生深刻感受到数学的价值.教学重点：掌握运用平方差公式分解因式的方法.教学难点：用平方差公式分解因式;培养学生多步骤分解因式的能力.教学过

2、程一、新课导入导入一:【问题】填空.(1)(x+5)(x-5)=;(2)(3x+y)(3x-y)=;(3)(3m+2n)(3m-2n)=.它们的结果有什么共同特征?尝试将它们的结果分别写成两个因式的乘积:名师归纳总结|大肚有容，容学习困难之事，学业有成，更上一层楼第 1 页，共 5 页(1)x2-25=;(2)9x2-y2=;(3)9m2-4n2=.设计意图学生通过观察、对比,把整式乘法中的平方差公式进行逆向应用,发展学生的观察能力与逆向思维能力.导入二:在前两节课中我们学习了因式分解的定义,即把一个多项式分解成几个整式的积的形式,还学习了提公因式

3、法分解因式,即如果一个多项式的各项都含有公因式,那么就可以把这个公因式提出来,从而将多项式化成几个因式乘积的形式.如果一个多项式的各项不都含有相同的因式,是否就不能分解因式了呢?当然不是,只要我们记住因式分解是整式乘法的逆过程,就能利用这种关系找到新的因式分解的方法,本节课我们就来学习另外一种因式分解的方法公式法.设计意图复习之前学过的知识后,提出疑问,直接引入新课,开门见山,激发学生的学习兴趣.二、新知构建1、用平方差公式分解因式请看乘法公式:(a+b)(a-b)=a2-b2.(1)左边是整式乘法,右边是一个多项式,把这个等式反过来就是:a2-b2=(a+b)(a-b).(2)左边是一个多

4、项式,右边是整式的乘积.大家判断一下,第二个式子从左边到右边是否为因式分解?符合因式分解的定义,因此是因式分解.名师归纳总结|大肚有容，容学习困难之事，学业有成，更上一层楼第 2 页，共 5 页文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1

5、HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 Z

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8、1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8

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10、文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2等式(1)是整式乘法中的平

11、方差公式,等式(2)可以看做是因式分解中的平方差公式.a2-b2是一个二项式,每项都可以化成整式的平方,整体来看是两个整式的平方差.如果一个二项式,它能够化成两个整式的平方差的形式,那么就可以用平方差公式分解因式,将多项式分解成两个整式的和与差的积.如:x2-16=x2-42=(x+4)(x-4);9m2-4n2=(3m)2-(2n)2=(3m+2n)(3m-2n).设计意图让学生通过自己的归纳找到因式分解中平方差公式的特征,并能利用相关结论进行实例练习.2、例题讲解过渡语同学们,前面我们学习了用平方差公式分解因式,下面我们通过几个例题来巩固所学的知识.(教材例 1)把下列各式因式分解:(

12、1)25-16x2;(2)9a2-b2.解:(1)25-16x2=52-(4x)2=(5+4x)(5-4x).(2)9a2-b2=(3a)2-=3a+b3a-b.(教材例 2)把下列各式因式分解:(1)9(m+n)2-(m-n)2;(2)2x3-8x.解:(1)9(m+n)2-(m-n)2=3(m+n)2-(m-n)2=3(m+n)+(m-n)3(m+n)-(m-n)=(3m+3n+m-n)(3m+3n-m+n)=(4m+2n)(2m+4n)名师归纳总结|大肚有容，容学习困难之事，学业有成，更上一层楼第 3 页，共 5 页文档编码:CZ6X1A6D9

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15、2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6

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17、9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D

18、3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1

19、A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2=4(2m+n)(m+2n).(2)2x3-8x=2x(x2-4)=2x(x+2)(x-2).说明:教材例 1 是把一个多项式的两项都化成两个单项式的平方,利用平方差公式分解因式;教材例2 的(1)是把一个二项式化成两个多项式的平方差,然后用平方差公式分解因式,教材例2 的(2)是先提取公因

20、式,然后再用平方差公式分解因式,由此可知,当一个题中既要用提公因式法,又要用公式法分解因式时,首先要考虑提公因式法,再考虑公式法.设计意图教师讲解例题,明确思维方法,给出书写范例.三、课堂小结平方差公式:a2-b2=(a+b)(a-b).我们已学习过的因式分解的方法有提公因式法和平方差公式法.如果多项式各项含有公因式,那么第一步是提公因式,然后看是否符合平方差公式的结构特点,若符合则继续进行.分解因式以后,若所含的多项式还可以继续分解,则需要进一步分解因式,直到每个多项式都不能分解为止.四、检测反馈1.下列因式分解正确的是()A.x2+y2=(x+y)(x-y)B.x2-y2=(x+y)(x

21、-y)C.x2+y2=(x+y)2D.x2-y2=(x-y)2解析:x2+y2不能在有理数范围内因式分解,x2-y2=(x+y)(x-y).故选 B.2.分解因式:a3-4a=.解析:a3-4a=a(a2-4)=a(a+2)(a-2).故填 a(a+2)(a-2).3.(2015恩施中考)因式分解:9bx2y-by3=.名师归纳总结|大肚有容，容学习困难之事，学业有成，更上一层楼第 4 页，共 5 页文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3

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25、H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:

26、CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 H

27、Y1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL

28、10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2解析:原式=by(9x2-y2)=by(3x+y)(3x-y).故填 by(3x+y)(3x-y).4.已知 x2-y2=69,x+y=3,则 x-y=.解析:因为 x2-y2=69,所以(x+y)(x-y)=69,因为 x+y=3,所以 3(x-y)=69,所以 x-y=23.故填 23.5.分解因式:(3a-2b)2-(2a+3b)2.解:(3a-2b)2-(2a+3b)2=(3a-2b)+(2a+3b)(3

29、a-2b)-(2a+3b)=(3a-2b+2a+3b)(3a-2b-2a-3b)=(5a+b)(a-5b).五、布置作业【必做题】教材第 100 页随堂练习的1,2 题.【选做题】教材第 100 页习题 4.4 的 1,2 题.六、板书设计公式法（利用平方差公式进行因式分解）一、用平方差公式分解因式a2-b2=(a+b)(a-b)二、例题讲解名师归纳总结|大肚有容，容学习困难之事，学业有成，更上一层楼第 5 页，共 5 页文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5

30、T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5

31、D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X

32、1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8

33、N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3

34、I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ

35、6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2文档编码:CZ6X1A6D9V1 HY1H8N5T9I8 ZL10H3I5D3I2

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关键词：: 2021 年利平方公式进行因式分解教育方案设计

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