设x1,x2,x3,是方程x^3+px+q的3个根,计算行列式X1x2x3X3,x1x2X2x3x1

Question

设x1,x2,x3,是方程x^3+px+q的3个根,计算行列式X1 x2 x3 X3,x1 x2 X2 x3 x1

Answer 1

由根与系数的关系,得：
x1+x2+x3=0
|x1 x2 x3| 将第2,3行都加到第1行：|x1+x2+x3 x1+x2+x3 x1+x2+x3|
|x3 x1 x2|= |x3 x1 x2 |=0
|x2 x3 x1| |x2 x3 x1 |
即行列式=0