Question

9道因式分解题

1.(x+y)^3+2xy(1-x-y)-1
2.x^3+x^2-2004*2005x
3.(x^2-x-6)(x^2+3x-4)+24
4.x^6+15x^3y+50y^2
5.（x^2+4x+6）+(x^2+6x+6)+x^2
6.(x-3)(x+4)(x-5)(x+6)+80
7.x^3-7x+6
8.x^4+y^4+(x+y)^4
9.xy(xy+1)+(xy+3)-2(x+y+1/2)-(x+y-1)^2

Answer 1

1）(x+y)^3+2xy(1-x-y)-1
立方差公式：a³-b³=(a-b)(a²+ab+b²)
(x+y)^3+2xy(1-x-y)-1
=[(x+y)³-1]-2xy(x+y-1)
=(x+y-1)[(x+y)²+(x+y)+1]- 2xy(x+y-1)
=(x+y-1)(x²+y²+x+y+1)

2）x^3+x^2-2004*2005x
=x（x²+x-2004×2005）
=x（x+2005）（x-2004）

3）(x^2-x-6)(x^2+3x-4)+24
=(x-3)(x+2)(x+4)(x-1)+24
=[(x-3)(x+4)][(x+2)(x-1)]+24
=(x²+x-12)(x²+x-2)+24
=[(x²+x)-12][(x²+x)-2]+24
=(x²+x)²-14(x²+x)+24+24
=(x²+x)²-14(x²+x)+48
=(x²+x-6)(x²+x-8)
=(x+3)(x-2)(x²+x-8）

4）x^6+15x^3y+50y^2
设x³=a，
原式
=a²+15ay+50y²
=(a+10y)(a+5y)
再代回去
=(x³+10y)(x³+5y)

5）（x^2+4x+6）+(x^2+6x+6)+x^2
设x²+5x+6=a
原式
=(a-x)(a+x)+x²
=a²
=(x²+5x+6)²
=(x+3)²(x+2)

6）(x-3)(x+4)(x-5)(x+6)+80
(x-3)(x+4)(x-5)(x+6)+80
=(x-3)(x+4)(x-5)(x+6)+80
=(x²-x-12)(x²-x-30)+80
=[(x²-x)-12][(x²-x)-30]+80
=(x²-x)²-42(x²-x)+360+80
=(x²-x)²-42(x²-x)+440
=(x²-x-20)(x²-x-22)
=(x-5)(x+4)(x²-x-22)

7）x^3-7x+6
=x³-x²+x²-x-6x+6
=(x³-x²)+(x²-x)-(6x-6)
=x²(x-1)+x(x-1)-6(x-1)
=(x-1)(x²+x-6)
=(x-1)(x+3)(x-2)

8）x^4+y^4+(x+y)^4
=(x²+y²)²-2x²y²+(x+y)^4
=[(x+y)²-2xy]²-2x²y²+(x+y)^4
=(x+y)^4-4xy(x+y)²+4x²y²-2x²y²+(x+y)^4
=2(x+y)^4-4xy(x+y)²+2x²y²
=2【[(x+y)²]²-2xy(x+y)²+(xy)²】
=2[(x+y)²-xy]²
=2(x²+xy+y²)²

9）xy(xy+1)+(xy+3)-2(x+y+1/2)-(x+y-1)^2
=x²y²+xy+xy+3-2(x+y)-1-[(x+y)²-2(x+y)+1]
=[x²y²+2xy+1]-2(x+y)+1-(x+y)²+2(x+y)-1
=(xy+1)²-(x+y)²
=(xy-x-y+1)(xy+x+y+1)
=[x(y-1)-(y-1)][y(x+1)+(y+1)]
=(x-1)(x+1)(y+1)(y-1)

【希望对你有帮助】

Answer 2

1.(x+y)^3+2xy(1-x-y)-1
=（x+y）³-1-2xy(x+y-1)
=(x+y-1)[(x+y)²+x+y+1]-2xy(x+y-1)
=(x+y-1)[(x+y)²+x+y+1-2xy]
=(x+y-1)(x²+x+y+y²+1)

2.x^3+x^2-2004*2005x
=x³-2004x²+2005x²-2004*2005x
=x²(x-2004)+2005x(x-2004)
=(x²+2005x)(x-2004)
=x(x+2005)(x-2004)

3.(x^2-x-6)(x^2+3x-4)+24
=(x+2)(x-3)(x-1)(x+4)+24(这里注意组合 x-3 x+4 和 x+2 x-1)
=（x²+x-12）（x²+x-2）+24（这里将x²+x 看成一个整体）
=（x²+x）²-14（x²+x）+48
=（x²+x-6）（x²+x-8）
=（x-2）（x+3）（x²+x-8）

4.x^6+15x^3y+50y^2
=x^6+10x^3y+5x^3y+50y^2
=x^3(x^3+10y)+5y(x^3+10y)
=(x^3+5y)(x^3+10y)

5.（x^2+4x+6）+(x^2+6x+6)+x^2
(这题符号是不是有问题)?

6.(x-3)(x+4)(x-5)(x+6)+80
同第三题注意组合
=（x²+x-12）（x²+x-30）+80
=(x²+x)²-42(x²+x)+360+80
=(x²+x-20)(x²+x-22)
=(x+5)(x-4)(x²+x-22)

7.x^3-7x+6
=x^3-1-7x+7
=(x-1)(x²+x+1)-7(x-1)
=(x-1)(x²+x-6)
=(x-1)(x-2)(x+3)
好累最后自己尝试吧 加油！

8.x^4+y^4+(x+y)^4
9.xy(xy+1)+(xy+3)-2(x+y+1/2)-(x+y-1)^2