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# 第01讲 因式分解(上)

1．运用公式法

(1)a2-b2=(a+b)(a-b)；

(2)a2±2ab+b2=(a±b)2；

(3)a3+b3=(a+b)(a2-ab+b2)；

(4)a3-b3=(a-b)(a2+ab+b2)

(5)a2+b2+c2

(6)a3+b3+c2+b2+c2ab-bc-ca)；

(7)an-bnn-1n-3b2+…+abn-2+bn-1)其中n为正整数；

nn-12-…+abn-2-bn-1)，其中n为偶数；

nn-1-n-3b2-…-abn-2+bn-1)，其中n为奇数．

1

(1)-5n-1yn+4x3n-1yn+2-2xn-1yn+4；

(2)x3-8y3-z3-6xyz；

(3)a2+b2+c2-2bc+2ca-2ab；

(4)a7-a5b2+a2b5-b7．

=-2xn-1yn[(x2n)2-2x2ny2+(y2)2]

=-2xn-1yn(x2n-y2)2

=-2xn-1yn(xn-y)2(xn+y)2．

(2)原式=x3+(-2y)3+(-z)3-3x(-2y)(-Z)

=(x-2y-z)(x2+4y2+z2+2xy+xz-2yz)．

(3)原式=(a2-2ab+b2)+(-2bc+2ca)+c2

＝(a-b)2+2c(a-b)+c2

=(a-b+c)2．

=(a-b+c)2

(4)原式=(a7-a5b2)+(a2b5-b7)

=a5(a2-b2)+b5(a2-b2 =(a2-b2)(a5+b5)

43b+a22-4)

22b2-3+b4 例2 33+c3 (6)．

(a+b)3=a3+3a2b+3ab2+b3

a3+b3=(a+b)3-3ab(a+b)．

=［(a+b)3+c3］-3ab(a+b+c)

=(a+b+c)［(a+b)2-c(a+b)+c2]-3ab(a+b+c)

=(a+b+c)(a2+b2+c2-ab-bc-ca)．

a3+b3+c3-3abc

x16-1=(x152 所以

2．拆项、添项法

=(x3-1)-9x+9

=(x-1)(x2+x+1)-9(x-1)

=(x-1)(x2+x-8)．

=(x3-x)+(-8x+8)

=x(x+1)(x-1)-8(x-1)

=(x-1)(x2+x-8)．

-1)(x--x2．

=x3x2+x2-9x+8

2(x-1)+(x-8)(x-1)

=(x-1)(x2+x-8)．

(1)x9+x6+x3-3；

(2)(m2-1)(n2-1)+4mn；

(3)(x+1)4+(x2-1)2+(x-1)4；

(4)a3b-ab3+a2+b2+1．

=(x9-1)+(x6-1)+(x3-1) =(x3-1)(x6+x3+1)+(x3-1)(x3+1)+(x3-1) =(x3-1)(x6+2x3+3)

=(x-1)(x2+x+1)(x6+2x3+3)．

(2)将4mn拆成2mn+2mn． 原式=(m2-1)(n2-1)+2mn+2mn =m2n2-m2-n2 =(m2n2+2mn+1)-(m2-2 =(mn+1)2-2

-．

(3)将22- 原式42-2-22+(x-1)4 4(x-2+(x-1)4]-(x2-1)2 ［22]2-(x2-1)2 22-(x2-1)2=(3x2+1)(x2+3)．

+ab-ab．

3b-ab3+a2+b2+1+ab-ab =(a3b-ab3)+(a2-ab)+(ab+b2+1) =ab(a+b)(a-b)+a(a-b)+(ab+b2+1) =a(a-b)［b(a+b)+1]+(ab+b2+1) =[a(a-b)+1](ab+b2+1) =(a2-ab+1)(b2+ab+1)．

3．换元法

=(y-2)(y+5)=(x2+x-2)(x2 =(x-1)(x+2)(x2+x+5) 说明 本题也可将x2+x+1x2+x+1=u，一样可以 例7 22-90．

-90

-90

2+5x+3)(2x2+5x+2)-90．

2+5x+2，则

-90=y2+y-90

=(y+10)(y-9)

=(2x2+5x+12)(2x2+5x-7)

=(2x2+5x+12)(2x+7)(x-1)．

(x2+4x+8)2+3x(x2+4x+8)+2x2．

=(x2+6x+8)(x2+5x+8)

=(x+2)(x+4)(x2+5x+8)．

=6［(x4-2x2+1)+2x2］+7x(x2-1)- =6[(x2-1)2+2x22- =6(x2-1)2-2 =[2(x2-1)-3x2- 2-2 -1)(x+3)． 说明 x

=x2(6t2+7t-24)=x2(2t-3)(3t+8)

=x2[2(x-1/x)-3][3(x-1/x)+8]

=(2x2-3x-2)(3x2+8x-3)

=(2x+1)(x-2)(3x-1)(x+3)．

=u4-6u2v+9v2

=(u2-3v)2

=(x2+2xy+y2-3xy)2

=(x2-xy+y2)2．

1．分解因式：

(2)x10+x5-2

54+x35

4-2y2+y2；

(3)x32+26x+24；

(4)x4-12x+323．

3．分解因式：

(1)(2x2-3x+1)2-22x2+33x-1；

(2)x4+7x3+14x2+7x+1；

(3)(x+y)3+2xy(1-x-y)-1；

(4)(x+3)(x2-1)(x+5)-20．

1．运用公式法

(1)a2-b2=(a+b)(a-b)；

(2)a2±2ab+b2=(a±b)2；

(3)a3+b3=(a+b)(a2-ab+b2)；

(4)a3-b3=(a-b)(a2+ab+b2) (5)a2+b2+c22

(6)a3+b32+b2+c2-ab-bc-ca)；

(7)an-bnn-1+a2…n-2+bn-1)其中n为正整数；

nn-2-…+abn-2-bn-1)，其中n为偶数；

nn-1-an-2n-3b2-…-abn-2+bn-1)，其中n为奇数．

1

(1)-5n-1yn+4x3n-1yn+2-2xn-1yn+4；

(2)x3-8y3-z3-6xyz；

(3)a2+b2+c2-2bc+2ca-2ab；

(4)a7-a5b2+a2b5-b7．

=-2xn-1yn[(x2n)2-2x2ny2+(y2)2]

=-2xn-1yn(x2n-y2)2

=-2xn-1yn(xn-y)2(xn+y)2．

(2)原式=x3+(-2y)3+(-z)3-3x(-2y)(-Z)

=(x-2y-z)(x2+4y2+z2+2xy+xz-2yz)．

(3)原式=(a2-2ab+b2)+(-2bc+2ca)+c2

＝(a-b)2+2c(a-b)+c2

=(a-b+c)2．

=(a-b+c)2

(4)原式=(a7-a5b2)+(a2b5-b7)

=a5(a2-b2)+b5(a2-b2 =(a2-b2)(a5+b5)

43b+a22-4)

22b2-3+b4 例2 33+c3 (6)．

(a+b)3=a3+3a2b+3ab2+b3

a3+b3=(a+b)3-3ab(a+b)．

=［(a+b)3+c3］-3ab(a+b+c)

=(a+b+c)［(a+b)2-c(a+b)+c2]-3ab(a+b+c)

=(a+b+c)(a2+b2+c2-ab-bc-ca)．

a3+b3+c3-3abc

x16-1=(x152 所以

2．拆项、添项法

=(x3-1)-9x+9

=(x-1)(x2+x+1)-9(x-1)

=(x-1)(x2+x-8)．

=(x3-x)+(-8x+8)

=x(x+1)(x-1)-8(x-1)

=(x-1)(x2+x-8)．

-1)(x--x2．

=x3x2+x2-9x+8

2(x-1)+(x-8)(x-1)

=(x-1)(x2+x-8)．

(1)x9+x6+x3-3；

(2)(m2-1)(n2-1)+4mn；

(3)(x+1)4+(x2-1)2+(x-1)4；

(4)a3b-ab3+a2+b2+1．

=(x9-1)+(x6-1)+(x3-1) =(x3-1)(x6+x3+1)+(x3-1)(x3+1)+(x3-1) =(x3-1)(x6+2x3+3)

=(x-1)(x2+x+1)(x6+2x3+3)．

(2)将4mn拆成2mn+2mn． 原式=(m2-1)(n2-1)+2mn+2mn =m2n2-m2-n2 =(m2n2+2mn+1)-(m2-2 =(mn+1)2-2

-．

(3)将22- 原式42-2-22+(x-1)4 4(x-2+(x-1)4]-(x2-1)2 ［22]2-(x2-1)2 22-(x2-1)2=(3x2+1)(x2+3)．

+ab-ab．

3b-ab3+a2+b2+1+ab-ab =(a3b-ab3)+(a2-ab)+(ab+b2+1) =ab(a+b)(a-b)+a(a-b)+(ab+b2+1) =a(a-b)［b(a+b)+1]+(ab+b2+1) =[a(a-b)+1](ab+b2+1) =(a2-ab+1)(b2+ab+1)．

3．换元法

=(y-2)(y+5)=(x2+x-2)(x2 =(x-1)(x+2)(x2+x+5) 说明 本题也可将x2+x+1x2+x+1=u，一样可以 例7 22-90．

-90

-90

2+5x+3)(2x2+5x+2)-90．

2+5x+2，则

-90=y2+y-90

=(y+10)(y-9)

=(2x2+5x+12)(2x2+5x-7)

=(2x2+5x+12)(2x+7)(x-1)．

(x2+4x+8)2+3x(x2+4x+8)+2x2．

=(x2+6x+8)(x2+5x+8)

=(x+2)(x+4)(x2+5x+8)．

=6［(x4-2x2+1)+2x2］+7x(x2-1)- =6[(x2-1)2+2x22- =6(x2-1)2-2 =[2(x2-1)-3x2- 2-2 -1)(x+3)． 说明 x

=x2(6t2+7t-24)=x2(2t-3)(3t+8)

=x2[2(x-1/x)-3][3(x-1/x)+8]

=(2x2-3x-2)(3x2+8x-3)

=(2x+1)(x-2)(3x-1)(x+3)．

=u4-6u2v+9v2

=(u2-3v)2

=(x2+2xy+y2-3xy)2

=(x2-xy+y2)2．

1．分解因式：

(2)x10+x5-2

54+x35

4-2y2+y2；

(3)x32+26x+24；

(4)x4-12x+323．

3．分解因式：

(1)(2x2-3x+1)2-22x2+33x-1；

(2)x4+7x3+14x2+7x+1；

(3)(x+y)3+2xy(1-x-y)-1；

(4)(x+3)(x2-1)(x+5)-20．