当m为何值时 x³+y³+z³+mxyz能被x+y+z整除
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/02 05:32:17
![当m为何值时 x³+y³+z³+mxyz能被x+y+z整除](/uploads/image/z/11805950-38-0.jpg?t=%E5%BD%93m%E4%B8%BA%E4%BD%95%E5%80%BC%E6%97%B6+x%26%23179%3B%2By%26%23179%3B%2Bz%26%23179%3B%2Bmxyz%E8%83%BD%E8%A2%ABx%2By%2Bz%E6%95%B4%E9%99%A4)
当m为何值时 x³+y³+z³+mxyz能被x+y+z整除
当m为何值时 x³+y³+z³+mxyz能被x+y+z整除
当m为何值时 x³+y³+z³+mxyz能被x+y+z整除
可以这样想,就是设法提出x+y+z出来,采用降次的方法,
x^3+y^3+z^3
=x^2(x+y+z)+y^2(x+y+z)+z^2(x+y+z)-x^2(y+z)-y^2(x+z)-z^2(x+y)
=(x^2+y^2+z^2)(x+y+z)-x^2(y+z)-y^2(x+z)-z^2(x+y)
=(x^2+y^2+z^2)(x+y+z)-[xy(x+y+z)+xz(x+y+z)+yz(x+y+z)-3xyz]
=(x^2+y^2+z^2)(x+y+z)-(xy+xz+yz)(x+y+z)+3xyz
故当m=-3时,代数式能被x+y+z整除