Question

Bài 3: Thực hiện phép tính:

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

b) \(\frac{x - y}{xy + y^2} - \frac{3x + y}{x^2 - xy} \cdot \frac{y - x}{x + y}\)

c) \(\frac{2x - 2y}{x + y} : \frac{x^2 - y^2}{xy} \cdot \frac{x^2 + 2xy + y^2}{x}\)

Answer 1

a: \(\dfrac{x^3+1}{x^2+2x+1}\cdot\dfrac{x^2-1}{2x^2-2x+2}\)

\(=\dfrac{\left(x+1\right)\left(x^2-x+1\right)}{\left(x+1\right)^2}\cdot\dfrac{\left(x-1\right)\left(x+1\right)}{2\left(x^2-x+1\right)}\)

\(=\dfrac{\left(x+1\right)^2}{\left(x+1\right)^2}\cdot\dfrac{x^2-x+1}{x^2-x+1}\cdot\dfrac{x-1}{2}=\dfrac{x-1}{2}\)

b: \(\dfrac{x-y}{xy+y^2}-\dfrac{3x+y}{x^2-xy}\cdot\dfrac{y-x}{x+y}\)

\(=\dfrac{x-y}{y\left(x+y\right)}+\dfrac{3x+y}{x\left(x-y\right)}\cdot\dfrac{x-y}{x+y}\)

\(=\dfrac{x-y}{y\left(x+y\right)}+\dfrac{3x+y}{x\left(x+y\right)}\)

\(=\dfrac{x\left(x-y\right)+y\left(3x+y\right)}{xy\left(x+y\right)}=\dfrac{x^2+2xy+y^2}{xy\left(x+y\right)}=\dfrac{x+y}{xy}\)

c: \(\dfrac{2x-2y}{x+y}:\dfrac{x^2-y^2}{xy}\cdot\dfrac{x^2+2xy+y^2}{x}\)

\(=\dfrac{2\left(x-y\right)}{x+y}\cdot\dfrac{xy}{\left(x-y\right)\left(x+y\right)}\cdot\dfrac{\left(x+y\right)^2}{x}\)

\(=2\cdot\dfrac{x-y}{x-y}\cdot\dfrac{\left(x+y\right)^2}{\left(x+y\right)^2}\cdot\dfrac{xy}{x}=2y\)