Question

x/(x-y)(x-z)+y/(x-y)(y-z)+z/(y-z)(z-x)

 

chứng tỏ rằng khi giá trị của phân thức xác định thì giá trị đó ko phụ thuộc vào các biến x, y,z

 

Answer 1

\(\frac{x}{\left(x-y\right)\left(x-z\right)}\)  \(+\frac{y}{\left(x-y\right)\left(y-z\right)}\)\(+\frac{z}{\left(y-z\right)\left(z-x\right)}\)

\(=\)\(\frac{x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)  \(+\frac{y\left(x-z\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}-\)\(\frac{z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)

\(=\frac{x\left(y-z\right)+y\left(x-z\right)-z\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)

\(=\)\(\frac{xy-xz+xy-yz-xz+yz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\) 

\(=\)\(\frac{2xy-2xz}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)

\(=\frac{2x\left(y-z\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)

\(=\)\(\frac{2x}{\left(x-y\right)\left(x-z\right)}\)