解: 設f(x)=(x^3+x^2+x+1)(x^4+ax^3+bx^2+cx+d) =x^7+(a+1)x^6+(a+b+1)x^5+(a+b+c+1)x^4+(a+b+c+d)x^3+(b+c+d)x^2+(c+d)x+d
(1)a+b+c+d < = 10,因a+b+c+1和a+b+c+d相異,故d > = 2 (2)排除 a+1=b+c+d, a+1=c+d,a+1=d, a+b+1=c+d , a+b+1=d,a+b+c+1=d (3)a+b+c+d > b+c+d > c+d > d,a+b+c+d > a+b+c+1 > a+b+1 > a+1
依上述原則 1.d=2,(a,b,c,d)=(2,1,3,2),(2,1,4,2),(2,1,5,2),(2,2,2,2),(2,2,4,2),(2,3,2,2),(2,3,3,2),(2,4,2,2),(3,1,4,2),(3,2,1,2),(3,2,3,2),(3,3,1,2),(3,4,1,2),(4,1,1,2),(4,2,2,2),(4,3,1,2),(5,1,1,2),(5,1,2,2),(5,2,1,2),(6,1,1,2)
2.d=3,(a,b,c,d)=(5,1,1,3)
3.d > =4,無解
共21組解
|