Question

几道七年级分解因式题

（1）分解因式：（x+y)²(x-y)+(x+y)(x-y)²
（2）用平方差公式计算：2005²-2000²
（3）用平方差公式计算：100²-99²+98²-97²+96²-95²+…+2²-1²
（4）分解因式：(x+y)³（x-y)-（x+y)(x-y)³
（注：所有题都要有过程及分析）

Answer 1

（1）分解因式：（x+y)²(x-y)+(x+y)(x-y)²
（x+y)²(x-y)+(x+y)(x-y)²
=[(x+y)(x-y)][(x+y)+(x-y)] [提取(x+y)(x-y)]
=2x(x+y)(x-y)

（2）用平方差公式计算：2005²-2000²
2005²-2000²
=(2005+2000)(2005-2000)
=4005x5
=20025
参考资料 a²-b²=(a+b)(a-b)

（3）用平方差公式计算：100²-99²+98²-97²+96²-95²+…+2²-1²
100²-99²+98²-97²+96²-95²+…+2²-1²
=(100+99)(100-99)+(98+97)(98-97)+...+(2+1)(2-1) (相减的差都是1)
=100+99+98+97+...+2+1 (1到100的连加会吧)
=(1+100)100/2
=5050
参考资料 a²-b²=(a+b)(a-b) 1+2+3+..+n=(1+n)n/2

（4）分解因式：(x+y)³（x-y)-（x+y)(x-y)³
(x+y)³（x-y)-（x+y)(x-y)³
=[(x+y)(x-y)][(x+y)²-(x-y)²] (提取(x+y)(x-y))
=(x+y)(x-y)[(x+y+x-y)(x+y-x+y)] (把后面的用这个a²-b²=(a+b)(a-b))
=(x+y)(x-y)(2x*2y)
=4xy(x+y)(x-y)

Answer 2

1 （x+y)²(x-y)+(x+y)(x-y)²
＝(x+y)(x-y)(x+y)+(x+y)(x-y)(x-y) 提取公因式(x+y)(x-y)
=(x+y)(x-y)(x+y+x-y)
=2x(x+y)(x-y)

2 2005²-2000²
=(2005+2000)(2005-2000)
=4005×5=20025

3 100²-99²+98²-97² +... +2²-1²
=(100+99)(100-99) +(98+97)(98-97)+... +(2+1)(2-1)
=100+99+98+97+...+2+1
=(100+1)×100÷2=5050

4 (x+y)³（x-y)-（x+y)(x-y)³
=(x+y)²(x+y)(x-y)-(x+y)(x-y)(x-y)² 提取公因式（x+y)(x-y)
=(x+y)(x-y)[(x+y)²-(x-y)²]
=(x+y)(x-y)[(x+y+x-y)(x+y-x+y)
=(x+y)(x-y)(2x)(2y)
=4xy(x+y)(x-y)