Rút gọn x^3 - y^3 + z^3 - 3xyz / (x - y)^2 + (y - z)^2 + (z - x)^2

Đề bài: Rút gọn phân thức: $\frac{{\text{x}}^{\text{3}}{\text{ + y}}^{\text{3}}{\text{ + z}}^{\text{3}}\text{ - 3xyz}}{{\left(\text{x - y}\right)}^{\text{2}}\text{ + }{\left(\text{y - z}\right)}^{\text{2}}\text{ + }{\left(\text{z - x}\right)}^{\text{2}}}$

Trả lời

Hướng dẫn giải:

$\frac{{\text{x}}^{\text{3}}{\text{ + y}}^{\text{3}}{\text{ + z}}^{\text{3}}\text{ - 3xyz}}{{\left(\text{x - y}\right)}^{\text{2}}\text{ + }{\left(\text{y - z}\right)}^{\text{2}}\text{ + }{\left(\text{z - x}\right)}^{\text{2}}}$

$\begin{array}{l}\text{= }\frac{{\text{x}}^{\text{3}}{\text{ + y}}^{\text{3}}{\text{ + z}}^{\text{3}}\text{ - 3xyz}}{{\text{x}}^{\text{2}}{\text{ - 2xy + y}}^{\text{2}}{\text{ + y}}^{\text{2}}{\text{ - 2yz + z}}^{\text{2}}{\text{ + z}}^{\text{2}}{\text{ - 2xz + x}}^{\text{2}}}\\ \text{= }\frac{\left(\text{x + y + z}\right)\text{ . }\left({\text{x}}^{\text{2}}{\text{ + y}}^{\text{2}}{\text{ +z}}^{\text{2}}\text{ - xy - yz - zx}\right)}{\text{2 . }\left({\text{x}}^{\text{2}}{\text{ + y}}^{\text{2}}{\text{ + z}}^{\text{2}}\text{ - xy - yz - zx}\right)}\\ \text{= }\frac{\text{x + y + z}}{\text{2}}\end{array}$