Question

分解因式：(2x-3y)^3+(3x-2y)^3-125(x-y)^3

Answer 1

(2x-3y)^3+(3x-2y)^3-125(x-y)^3
=(2x-3y)^3+(3x-2y)^3-(5x-5y)^3
令2x-3y=a，3x-2y=b，则5x-5y=a+b

原式=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)

Answer 2

分解因式：(2x-3y)³+(3x-2y)³-125(x-y)³
解：原式=(2x-3y)³+(3x-2y)³-(5x-5y)³=(2x-3y)³+(3x-2y)³-[(2x-3y)+(3x-2y)]³
=(2x-3y)³+(3x-2y)³-[(2x-3y)³+3(2x-3y)²(3x-2y)+3(2x-3y)(3x-2y)²+(3x-2y)³]
=-3(2x-3y)(3x-2y)[(2x-3y)+(3x-2y)]=-15(2x-3y)(3x-2y)(x-y)

Answer 3

=[(2x-3y)+(3x-2y)][(2x-3y)²-(2x-3y)(3x-2y)+(3x-2y)²]-125(x-y)³
=5(x-y)[13x²+13y²-24xy-6x²-6y²+13xy]-25×5(x-y)(x-y)²
=5(x-y)(7x²+7y²-11xy)-25×5(x-y)(x²-2xy+y²)
=5(x-y)[(7x²+7y²-11xy)-25(x²+y²-2xy)]
=5(x-y)(-18x²-18y²+39xy)
= - 15(x-y)(6x²+6y²-13xy)= -15(x-y)(2x-3y)(3x-2y)

Answer 4

∵A3+B3+C3-3ABC
=（A+B+C）（A2+B2+C2-BC-CA-AB），
若A+B+C=0，便有A3+B3+C3=3ABC，
令A=2x-3y，B=3x-2y，C=5y-5x，
则符合上述条件，易得A3+B3+C3=3ABC．
∴（2x-3y）3+（3x-2y）3-125（x-y）3
=3（2x-3y）（3x-2y）[5（y-x）]，
=15（2x-3y）（3x-2y）（y-x），
故答案为：15（2x-3y）（3x-2y）（y-x）．

Answer 5

-15(2x-3y)(3x-2y)(x-y)