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( x + 3 )( x +5)

多项式与多项式是如何相乘的?. 中观中学 吴廷飞. (a+b)(m+n). =am. +an. +bm. +bn. ( x + 3 )( x +5). =x 2. + 5x. + 3X. + 15. = x 2. + 8x. + 15. 想一想.

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( x + 3 )( x +5)

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  1. 多项式与多项式是如何相乘的? 中观中学 吴廷飞 (a+b)(m+n) =am +an +bm +bn (x + 3)( x+5) =x2 +5x +3X +15 =x2 +8x +15

  2. 想一想 • 灰太狼开了租地公司,一天他把一边长为a米的正方形土地租给慢羊羊种植.有一年他对慢羊羊说:“我把这块地的一边增加5米,另一边减少5米,再继续租给你, 你也没吃亏,你看如何?”慢羊羊一听觉得没有吃亏,就答应了.回到羊村,就把这件事对喜羊羊他们讲了,大家一听,都说道:“村长,您吃亏了!” 慢羊羊村长很吃惊…同学们,你能告诉慢羊羊这是为什么吗?

  3. 5米 (a+5)米 5米 a米 (a-5) 面积变了吗? 现在 原来 a2 (a+5)(a-5) 相等吗?

  4. 算一算,比一比,看谁算得又快又准 计算下列各题 ①(x + 4)( x-4) ②(1 + 2a)( 1-2a) ③(m+ 6n)( m-6n) ④(5y + z)(5y-z)

  5. ①(x + 4)( x-4)=x2 - 16 x2- 42 ②(1 + 2a)( 1-2a)=1-4a2 12-(2a)2 ③(m+ 6n)( m-6n)=m2 - 36n2 m2- (6n)2 (5y)2- z2 ④(5y + z)(5y-z)= 25y2 - z2 它们的结果有什么特点?

  6. 15.2.1平方差公式

  7. 平方差公式: (a+b)(a−b)= a2−b2 两数和与这两数差的积, 等于 这两数的平方差. 公式变形: 1、(a – b ) ( a + b) = a2 - b2 2、(b + a )( -b + a ) = a2 - b2

  8. 相同为a 相反为b 平方差公式 适当交换 (a+b)(a-b)=(a)2-(b)2 合理加括号 注:这里的两数可以是两个单项式也可以是两个多项式等等.

  9. 口答下列各题: (l)(-a+b)(a+b)=_________ (2)(a-b)(b+a)= __________ (3)(-a-b)(-a+b)= ________ (4)(a-b)(-a-b)= _________ b2-a2 a2-b2 a2-b2 b2-a2

  10. 1、找一找、填一填 a2-b2 (a-b)(a+b) a b (1+x)(1-x) 12-x2 1 x (-3+a)(-3-a) (-3)2-a2 a -3 a2-12 (1+a)(-1+a) a 1 (0.3x-1)(1+0.3x) ( 0.3x)2-12 0.3x 1

  11. 注意 例题 (a + b ) ( a – b ) = a2 - b2 例1、用平方差公式计算 计算:(x+2y)(x-2y) 1、先把要计算的式子与公式对照, 2、哪个是 a 哪个是 b 解:原式= x2 - (2y)2 =x2 - 4y2

  12. 试试就能行 例2运用平方差公式计算: (1) (3x+2 )( 3x-2 ) ; (2) (b+2a)(2a-b); (3) (-x+2y)(-x-2y). (2)(b+2a)(2a-b) 解:(1)(3x+2)(3x-2) =(2a+b)(2a-b) =(3x)2-22 =(2a)2-b2 =9x2-4; =4a2-b2. (3) (-x+2y)(-x-2y) =(-x)2-(2y)2 = x2-4y2

  13. 挑战自我 例3 计算: (1) 102×98; (2) (y+2) (y-2) – (y-1) (y+5) . 解: (1) 102×98 =(100+2)(100-2) = 1002-22 =1000 – 4 =9996 • (y+2)(y-2)- (y-1)(y+5) = y2-22-(y2+4y-5) = y2-4-y2-4y+5 = - 4y + 1.

  14. 练习 相信自己 我能行! 利用平方差公式计算: (1)(a+3b)(a -3b) (2)(3+2a)(-3+2a) =(a)2-(3b)2 =(2a+3)(2a-3) =a2-9b2; =(2a)2-32 =4 a2-9; (3)51×49 =(50+1)(50-1) (4)(-2x2-y)(-2x2+y) =502-12 =(-2x2 )2-y2 =2500-1 =4x4-y2. =2499 (5)(3x+4)(3x-4)-(2x+3)(3x-2) =(9x2-16) -(6x2+5x-6) =3x2-5x- 10

  15. 拓展提升 1.计算 20042 -2003×2005; 解: 20042-2003×2005 = 20042 - (2004-1)(2004+1) = 20042 - (20042-12 ) = 20042 -20042+12 知难而进 =1

  16. 2、利用平方差公式计算: (a-2)(a+2)(a2 +4) 解:原式=(a2-4)(a2+4) =a4-16

  17. 3.化简 (x4+y4) (x4+y4) (x4+y4) ( ) 知难而进

  18. 平方差公式 小结 适当交换 相同为a (a+b)(a-b)=(a)2-(b)2 合理加括号 相反为b

  19. goodbye!

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