Question

Cộng trừ phân số

1)\(x+2+\frac{3}{x-2}\)

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

Answer 1

1)

\(x+2+\frac{3}{x-2}\)

\(=\frac{\left(x+2\right)\left(x-2\right)}{x-2}+\frac{3}{x-2}\)

\(=\frac{x^2-4}{x-2}+\frac{3}{x-2}\)

\(=\frac{x^2-4+3}{x-2}\)

\(=\frac{x^2-1}{x-2}\)

Answer 2

2)

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

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

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

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

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

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

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

\(=\frac{x^2y-x^2z-xy^2+y^2z+xz^2-yz^2}{x^2y-x^2z-xy^2+y^2z+xz^2-yz^2}\)

\(=1\)

Answer 3

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

\(=\frac{1}{x-y}\times\left(\frac{x^2}{x-z}-\frac{y^2}{y-z}\right)+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\left(\frac{x^2\left(y-z\right)}{\left(x-z\right)\left(y-z\right)}-\frac{y^2\left(x-z\right)}{\left(y-z\right)\left(x-z\right)}\right)+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\frac{x^2y-x^2z-xy^2+y^2z}{\left(x-z\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\frac{xy\left(x-y\right)-z\left(x^2-y^2\right)}{\left(x-z\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\frac{xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)}{\left(x-z\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\frac{\left(x-y\right)\left(xy-z\left[x+y\right]\right)}{\left(x-z\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)

\(=\frac{1}{x-y}\times\frac{\left(x-y\right)\left(xy-xz-zy\right)}{\left(x-z\right)\left(y-z\right)}+\frac{z^2}{\left(x-z\right)\left(y-z\right)}\)

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

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

= 1