Question

因式分解（1）: x的4次方+2x的3次方-3x的2次方-4x+4 （2）：2x的4次方+x的3次方+7x的2次方+4x-4

2014年3月1日晚18：00之前回答即可获得加分

Answer 1

答：

（1）
x的4次方+2x的3次方-3x的2次方-4x+4
=x^4+2x^3-3x^2-4x+4
=x^4-x^3+3x^3-3x^2-4x+4
=(x-1)x^3+3(x-1)x^2-4(x-1)
=(x-1)(x^3+3x^2-4)
=(x-1)(x^3-x^2+4x^2-4)
=(x-1)[(x-1)x^2+4(x-1)(x+1)]
=(x-1)(x-1)(x^2+4x+1)

（2）
2x的4次方+x的3次方+7x的2次方+4x-4
=2x^4+x^3+7x^2+4x-4
=(2x^4+2x^3)-(x^3+x^2)+8x^2+4x-4
=2(x+1)x^3-(x+1)x^2+4(x+1)(2x-1)
=(x+1)(2x^3-x^2+8x-4)
=(x+1)*[(2x-1)x^2+4(2x-1)]
=(x+1)*(2x-1)(x^2+4)

追问

（1）：呃呃呃，这好像有问题。。。求解释
..........
=(x-1)[(x-1)x^2+4(x-1)(x+1)]
=(x-1)(x-1)(x^2+4x+1)
应该为：
.........
=(x-1)[(x-1)x^2+4(x-1)(x+1)]
=(x-1)(x-1)(x^2+4x+4)=.........

追答

sorry，4(x+1)=4x+4，是+4，不是+1，修正如下：
（1）
x的4次方+2x的3次方-3x的2次方-4x+4
=x^4+2x^3-3x^2-4x+4
=x^4-x^3+3x^3-3x^2-4x+4
=(x-1)x^3+3(x-1)x^2-4(x-1)
=(x-1)(x^3+3x^2-4)
=(x-1)(x^3-x^2+4x^2-4)
=(x-1)[(x-1)x^2+4(x-1)(x+1)]
=(x-1)(x-1)(x^2+4x+4)
=(x-1)²(x+2)²

Answer 2

x^4+2x^3-3x^2-4x+4=x^2(x^2+2x-3)-4(x-1)=x^2(x+3)(x-1)-4(x-1)=(x-1)(x^3+3x-4)=(x-1)(x^3-1+3x-3)=(x-1)[(x-1)(x^2+x+1)-3(x-1)]=(x-1)[(x-1)^2(x+2)]=(x-1)^3(x+2)

Answer 3

x^4+2x^3-3x^2-4x+4
=x^2(x^2+2x-3)-4(x-1)
=x^2(x+3)(x-1)-4(x-1)
=(x-1)[x^2(x+3)-4]
=(x-1)(x^3+3x^2-4)
=(x-1)(x^3-x^2+4x^2-4)
=(x-1)[x^2(x-1)+4(x^2-1)]
=(x-1)[x^2(x-1)+4(x+1)(x-1)]
=(x-1)^2[x^2+4(x+1)]
=(x-1)^2(x^2+4x+4)
=(x-1)^2(x+2)^2

2x^4+x^3+7x^2+4x-4
=2x^4+x^3-x^2+8x^2+4x-4
=x^2(2x^2+x-1)+4(2x^2+x-1)
=(x^2+4)(2x^2+x-1)
=(x^2+4)(2x-1)(x+1)

Answer 4

(1)、 x^4 +2x^3 -3x^2 -4x+4
=x^2 (x^2 +2x -3)-4x+4
=x^2 (x+3)(x-1)-4(x-1)
=(x-1)[x^2 *(x+3)-4]
(2)、2x^4+x^3+7x^2+4x-4
=x^2 (2x^2 +x-1+1)+7x^2+4x-4
=x^2 (2x^2 +x-1)+x^2 +7x^2 +4x-4
=x^2 (2x^2 +x-1)+4x^2 (2x^2 +x-1)
=(2x^2 +x-1)(x^2 +4)
=(2x-1)(x+1)(x^2 +4)