(共32张PPT)
17.2 用公式法分解因式
(第3课时)
第十七章 因式分解
人教版(新教材)数学八年级上册
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CONTENT
情景引入
1
合作探究
2
典例分析
3
巩固练习
4
归纳总结
5
感受中考
6
小结梳理
7
布置作业
8
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能综合运用提公因式法与公式法、多次运用公式法对多项式进行因式分解.
一
在因式分解的过程中,体会转化思想和整体思想,提升数学运算素
养和逻辑推理素养.
二
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2.公式法:平方差公式和完全平方公式.
复习引入
问题 我们学习了哪些分解因式的方法?
对于一些复杂的因式分解问题,有时需要多次运用公式法,有时还需要综合运用提公因式法和公式法.
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探究1 分解因式:
(1) x4 y4 ; (2) a3b ab .
解 (1)原式=(x2)2 (y2)2
还能继续分解因式吗?
平方差公式
=(x2+y2)(x2 y2)
=(x2+y2)(x+y)(x y).
平方差公式
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探究1 分解因式:
(1) x4 y4 ; (2) a3b ab .
(2)原式=ab(a2 1)
提取公因式
=ab(a+1)(a 1).
平方差公式
温馨提示 1.一般情况下,有公因式要先提取公因式.
2.分解因式,要进行到每一个多项式因式都不能再分解为止.
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探究2 分解因式:
(1) 3ax2+6axy+3ay2 ; (2) ax2+2a2x a3 .
解 (1)原式=3a(x2+2xy+y2)
提取公因式
=3a(x+y)2.
完全平方公式
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探究2 分解因式:
(1) 3ax2+6axy+3ay2 ; (2) ax2+2a2x a3 .
(2)原式= a(x2 2ax+a2)
提取公因式
= a(x a)2.
完全平方公式
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例1 分解因式:
(1) x2y 4y ; (2) a3 2a2+a ; (3) ax2+2a2x+a3 ;
解 (1)原式=y(x2 4)
提取公因式
=y(x+2)(x 2).
平方差公式
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例1 分解因式:
(1) x2y 4y ; (2) a3 2a2+a ; (3) ax2+2a2x+a3 ;
(2)原式=a(a2 2a+1)
提取公因式
=a(a 1)2.
完全平方公式
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例1 分解因式:
(1) x2y 4y ; (2) a3 2a2+a ; (3) ax2+2a2x+a3 ;
(3)原式=a(x2+2ax+a2)
提取公因式
=a(x+a)2.
完全平方公式
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例1 分解因式:
(4) a4+16 ; (5) 3a 6ax+3ax2 ; (6) 4bx2+8bxy 4by2 .
(4)原式=42 (a2)2
平方差公式
=(4+a2)(4 a2)
=(4+a2)(2+a)(2 a).
平方差公式
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例1 分解因式:
(4) a4+16 ; (5) 3a 6ax+3ax2 ; (6) 4bx2+8bxy 4by2 .
(5)原式=3a(1 2x+x2)
提取公因式
=3a(x 1)2.
完全平方公式
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例1 分解因式:
(4) a4+16 ; (5) 3a 6ax+3ax2 ; (6) 4bx2+8bxy 4by2 .
(6)原式= 4b(x2 2xy+y2)
提取公因式
= 4b(x y)2.
完全平方公式
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例2 分解因式:
(1) (a b)2+4ab ; (2) (p 4)(p+1)+3p .
解 (1)原式=(a2 2ab+b2)+4ab
先借助乘法公式化简,再分解因式
=a2+2ab+b2
完全平方公式
=(a+b)2.
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例2 分解因式:
(1) (a b)2+4ab ; (2) (p 4)(p+1)+3p .
(2)原式=(p2+p 4p 4)+3p
多项式乘多项式
=p2 4
平方差公式
=(p+2)(p 2).
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1. 把多项式4x2y 4xy2 x3分解因式的结果是( )
A.4xy(x y) x3 B. x(x 2y)2
C.x(4xy 4y2 x2) D. x( 4xy+4y2+x2)
B
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(1) 3a2x2+24a2x 48a2; (2) (a2+4)2 16a2.
巩固练习
解 (1)原式= 3a2(x2 8x+16)
= 3a2(x 4)2;
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(1) 3a2x2+24a2x 48a2; (2) (a2+4)2 16a2.
巩固练习
(2)原式=(a2+4)2 (4a)2
=(a2+4+4a)(a2+4 4a)
=(a+2)2(a 2)2 .
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