Question

因式分解练习（最好简单的写一下过程）

2(x-1)^2-x+1
-a^4+2a^2b^2-b^4
(x+2y)^2-4(x-3y)^2
9(2x-y)^2-6(2x-y)+1
(m^2-6m)^2+18(m^2-6m+81)
-12a(a-b)^3+6a^2(b-a)^3

Answer 1

解：
2(x－1)²－x＋1
＝2(x－1)²－(x－1)
＝(x－1)[2(x－1)－1]
＝(x－1)(2x－2－1)
＝(x－1)(2x－3)

－a⁴＋2a²b²－b⁴
＝－(a⁴－2a²b²＋b⁴)
＝－(a²－b²)²
＝－[(a＋b)(a－b)]²
＝－(a＋b)²(a－b)²

(x＋2y)²－4(x－3y)²
＝[(x＋2y)＋2(x－3y)][(x＋2y)－2(x－3y)]
＝[x＋2y＋2x－6y][x＋2y－2x＋6y]
＝(3x－4y)(8y－x)

9(2x－y)²－6(2x－y)＋1
＝[3(2x－y)－1]²
＝(6x－3y－1)²

题目有问题，仔细看一下：
(m²－6m)²＋18(m²－6m)＋81
＝[(m²－6m)＋9]²
＝[(m－3)²]²
＝(m－3)⁴

－12a(a－b)³＋6a²(b－a)³
＝12a(b－a)³＋6a²(b－a)³
＝(b－a)³(12a＋6a²)
＝6a(b－a)³(2＋a)

Answer 2

1) 2(x-1)^2-x+1=2(x^2-2x+1)-x+1=2x^2-5x+3=(2x-3)*(x-1)
2) -a^4+2a^2b^2-b^4=-(a^2-b^2)^2
3) (x+2y)^2-4(x-3y)^2=x^2+4xy+4y^2-4(x^2-6xy+9y^2)=-3x^2+28xy-32y^2=(-3x+4y)(x-8y)
4) 9(2x-y)^2-6(2x-y)+1=9(4x^2-4xy+y^2)-12x+6y+1=36x^2-36xy-12x+6y+1+9y^2

Answer 3

2(x-1)^2-x+1=(x-1)(2x-2-1)
-a^4+2a^2b^2-b^4=-(a^2-b^2)^2
(x+2y)^2-4(x-3y)^2=(x+2y+2x-6y)(x+2y-2x+6y)=(3x-4y)(8y-x)
9(2x-y)^2-6(2x-y)+1=[3(2x-y)-1]^2
(m^2-6m)^2+18(m^2-6m)+81=(m^2-6m+9)^2=(m-3)^4
-12a(a-b)^3+6a^2(b-a)^3=6a(b-a)^3(a+2)

Answer 4

1.2(x-1)^2-x+1
=(x-1)[2(x-1)-1]
=(x-1)(2x-3)
2.-a^4+2a^2b^2-b^4
=-(a^4-2a^2b^2+b^4)
=-(a^2-b^2)^2
3.(x+2y)^2-4(x-3y)^2
=(x+2y+2x-6y)(x+2y-2x+6y)
=-(3x-4y)(x-8y)
4.9(2x-y)^2-6(2x-y)+1
=[3(2x-y)-1]^2
=(6x-3y-1)^2
5.(m^2-6m)^2+18(m^2-6m+81)
=(m^2-6m+9)^2+1377
=(m-3)^4+1377
6.-12a(a-b)^3+6a^2(b-a)^3
=12a(b-a)^3+6a^2(b-a)^3
=6a(b-a)^3(2+a)