分解因式:(2x-3y)的立方+(3x-2y)的立方-125(x-y)的立方

(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)

原式=（2x-3y)^3 + [(3x-2y)-5(x-y)][(3x-2y)^2+5(3x-2y)(x-y)+25(x-y)^2]
=（2x-3y)^3 + （3y-2x)[(3x-2y)^2+5(3x-2y)(x-y)+25(x-y)^2]
=(2y-3x)【(2y-3x)^2 - (2y-3x)[(3x-2y)^2+5(3x-2y)(x-y)+25(x-y)^2] 】

（2x-3y）^3+(3x-2y)^3-125(x-y)^3
=(2x-3y)^3+(3x-2y)^3-(5x-5y)^3
是a=2x-3y,b=3x-2y,a+b=5x-5y
=a^3+b^3-(a+b)^3
=a^3+b^3-a^3-3a^2b-3ab^2-b^3
=-3a^2b-3ab^2
=-3ab(a+b)
=-3(2x-3y)(3x-2y)(5x-5y)
=-15(2x-3y)(3x-2y)(x-y)