Question

[Trần Quang Huy]

cho x+y+z=2016 tính M=\(\frac{x^3+y^3+z^3-3xyz}{x^2+y^2+z^2+xy-yz-zx}\)

Answer 1

[Hoàng Phúc]

\(M=\frac{x^3+y^3+z^3-3xyz}{x^2+y^2+z^2-xy-yz-zx}\)

Đặt \(N=x^3+y^3+z^3-3xyz\)

\(=\left(x+y\right)^3-3x^2y-3xy^2+z^3-3xyz\)

\(=\left(x+y\right)^3+z^3-3x^2y-3xy^2-3xyz\)

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right).z+z^2\right]-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-zx-yz+z^2\right)-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-zx-yz+z^2-3xy\right)\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)\)

Vậy \(M=\frac{N}{x^2+y^2+z^2-xy-yz-zx}=\frac{\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)}{x^2+y^2+z^2-xy-yz-zx}=x+y+z=2016\)

(*) bn ghi sai đề 1 chỗ nhé:ở mẫu thức của M phải là  \(x^2+y^2+z^2-xy-yz-zx\) nhé!