Question

分解因式（2x-3y）^3+(3x-2y)^3-125(x-y)^3

1.（2x-3y）^3+(3x-2y)^3-125(x-y)^3
2.要使二次三项式x^2-2x+p在整数范围内分解因式，那么整数p的取值有几个？
3.（x^2+3x）^2-2(x^2+3x)-8
4.x^2-y^2-4x+4

Answer 1

1 (2x-3y)^3+(3x-2y)^3-125(x-y)^3
= (2x-3y)^3+(3x-2y)^3-(5x-5y)^3
= (2x-3y)^3+(3x-2y)^3-((2x-3y) +(3x-2y))^3

令m =2x-3y, n =3x-2y
上式
= m^3 +n^3 -(m+n)^3
= m^3 +n^3 -(m^3 +n^3 +3mn^2 +3m^2n)
= -3mn^2 -3m^2n
= -3mn(m+n) ....(将假设带入）
= -3(2x-3y)(3x-2y)((2x-3y)+(3x-2y))
= -3(2x-3y)(3x-2y)(5x-5y)
= -15(2x-3y)(3x-2y)(x-y)
2
3.(X^3+3X)^2-2(x^2+3X)-8
=(x^2+3x-4)(x^2+3x+2)
=(x-1)(x+4)(x+1)(x+2)
4. (x^2-4x+4)-y^2
=(x-2)^2-y^2
=(x-2-y)(x-2+y)