Question

求证:(x+4)(x+3)(x+2)(x+1)与(x-1)(x-2)(x-3)(x-4)-120有相同的因式。

初二竞赛试题，求详细解答过程！！

Answer 1

(x+4)(x+3)(x+2)(x+1)-(x-1)(x-2)(x-3)(x-4)+120=20x^3+100x+120=20(x^3+5x+6)=20(x+1)(x-3)(x+2)
所以有共因式(x+1)(x+2)

追问

那个三次整式是怎么分解的？

追答

(x^3+5x+6)=X^3+1+5X+5=(X+1)(X^2-X+1)+5(X+1)=(X+1)(X^2-X+1+5)=(X+1)(X^2-X+6)
我写错拉
(x+4)(x+3)(x+2)(x+1)-(x-1)(x-2)(x-3)(x-4)+120=20x^3+100x+120=20(x^3+5x+6)=20(x+1)(X^2-X+6)