Question

因式分解（难）

(1)分解因式： x^2 -2x-2 y^2 +4y-xy (2)分解因式：(x-2)^3-(y-2)^3-(x-y)^3 (3)分解因式： x^4 + x^3 - 3x^2 -4x-4 (4)分解因式：x(x-1)+y(y+1)-2xy (5)分解因式： x^3 +5 x^2 +3x-9 (6)已知 x^2 +x-6是多项式 2x^4 + x^3 - ax^2 +bx+a+b-1的因式，求a.b

Answer 1

(1)分解因式： x^2 -2x-2 y^2 +4y-xy=(x+y)(x-2y)-2(x-2y)=(x+y-2)(x-2y) (2)分解因式：(x-2)^3-(y-2)^3-(x-y)^3=(x-y)[(x-2)^2+(x-2)(y-2)+(y-2)^2]-(x-y)^3 =(x-y)[x^2-4x+4+xy-2x-2y+4+y^2-4y+4-x^2+2xy-y^2)=(x-y)(3xy-6x-6y+12) =3(x-y)(xy-2x-2y+4)=3(x-y)(x-2)(y-2) (3)分解因式： x^4 + x^3 - 3x^2 -4x-4=x^2(x^2+x-2)-(x^2+4x+4)=x^2(x+2)(x-1)-(x+2)^2=(x+2)(x^3-x^2-x-2) ==(x+2)(x^3-2x^2+x^2-x-2)=(x+2)(x^2(x-2)+(x+1)(x-2))=(x+2)(x-2)(x^2+x+1)= (4)分解因式：x(x-1)+y(y+1)-2xy=x^2-2xy+y^2-(x-y)=(x-y-1)(x-y) (5)分解因式： x^3 +5 x^2 +3x-9=x^3+3x^2+2x^2+3x-9=x^2(x+3)+(x+3)(2x-3)=(x+3)(x^2+2x-3)=(x+3)^2(x-1) (6)已知 x^2 +x-6是多项式 2x^4 + x^3 - ax^2 +bx+a+b-1的因式，求a.b x^2 +x-6=0有两个解x=-3，x=2 所以 x^2 +x-6是多项式 2x^4 + x^3 - ax^2 +bx+a+b-1的因式，所以x=-3，x=2也是 2x^4 + x^3 - ax^2 +bx+a+b-1=0的解代入后有 32+8-4a+2b+a+b-1=0 162-27-9a-3b+a+b-1=0 b-a+13=0 4a+b-67=0 a=16，b=3

Answer 2

1)分解因式： x^2 -2x-2 y^2 +4y-xy =(x-2y)(x+y)-2(x-2y) =(x-2y)(x+y-2) (2)分解因式： (x-2)^3-(y-2)^3-(x-y)^3 =(x-2-y+2)[(x-2)^2+(x-2)(y-2)+(y-2)^2]-(x-y)^3 =(x-y)(x^2-4x+4+xy-2x-2y+4+y^2-4y+4)-(x-y)^3 =(x-y)(x^2-6x+y^2-6y+xy+12)-(x-y)^3 =(x-y)(x^2-6x+y^2-6y+xy+12-x^2+2xy-y^2) =(x-y)(-6x-6y+3xy+12) =3(x-y)(xy-2x-2y+4) =3(x-y)[x(y-2)-2(y-2)] =3(x-y)(y-2)(x-2) (3)分解因式： x^4 + x^3 - 3x^2 -4x-4 =x^4+x^3-2x^2-(x^2+4x+4) =x^2(x^2+x-2)-(x+2)^2 =x^2(x+2)(x-1)-(x+2)^2 =(x+2)(x^3-x^2-x-2) (4)分解因式： x(x-1)+y(y+1)-2xy =x^2-x+y^2+y-2xy =(x-y)^2-(x-y) =(x-y)(x-y-1)