Thực hiện phép tính \(\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{4}{1-x}\) \(\dfrac{2\left(x-1\right)}{x^2+x+1}\) \(\dfrac{2\left(x+1\right)}{x^2+x+1}\) \(\dfrac{7x^2+6x+2}{\left(x-1\right)\left(x^2+x+1\right)}\) \(\dfrac{x^2+6x+2}{\left(x-1\right)\left(x^2+x+1\right)}\) Hướng dẫn giải: \(\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{4}{1-x}\) \(=\dfrac{5x^2-4x+7}{x^3-1}+\dfrac{x+1}{x^2+x+1}+\dfrac{-4}{x-1}\) \(=\dfrac{5x^2-4x+7+\left(x-1\right)\left(x+1\right)-4\left(x^2+x+1\right)}{\left(x-1\right)\cdot\left(x^2+x+1\right)}\) \(=\dfrac{5x^2-4x+7+x^2-1-4x^2-4x-4}{\left(x-1\right)\left(x^2+x+1\right)}\) \(=\dfrac{2x^2-4x+2}{\left(x-1\right)\left(x^2+x+1\right)}\) \(=\dfrac{2\left(x^2-2x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\) \(=\dfrac{2\left(x-1\right)^2}{\left(x-1\right)\left(x^2+x+1\right)}\) \(=\dfrac{2\left(x-1\right)}{x^2+x+1}\)
Thực hiện phép tính \(x^2+\dfrac{x^4+1}{1-x^2}+1\) \(\dfrac{2}{1-x^2}\) \(\dfrac{x^4+x^2+1}{1-x^2}\) \(\dfrac{-2}{1-x^2}\) \(\dfrac{2x^4+1}{1-x^2}\) Hướng dẫn giải: \(x^2+\dfrac{x^4+1}{1-x^2}+1\) \(=\dfrac{x^2\left(1-x^2\right)}{1-x^2}+\dfrac{x^4+1}{1-x^2}+\dfrac{1-x^2}{1-x^2}\) \(=\dfrac{x^2-x^4+x^4+1+1-x^2}{1-x^2}\) \(=\dfrac{2}{1-x^2}\)
Thực hiện phép tính \(\dfrac{2x+1}{2x^2-x}+\dfrac{32x^2}{1-4x^2}+\dfrac{1-2x}{2x^2+x}\) . \(-8\) \(\dfrac{8\left(2x+1\right)}{2x-1}\) \(8\) \(\dfrac{8\left(2x-1\right)}{2x+1}\) Hướng dẫn giải: \(\dfrac{2x+1}{2x^2-x}+\dfrac{32x^2}{1-4x^2}+\dfrac{1-2x}{2x^2+x}\) \(=\dfrac{2x+1}{x\left(2x-1\right)}+\dfrac{32x^2}{\left(1-2x\right)\left(1+2x\right)}+\dfrac{1-2x}{x\left(2x+1\right)}\) \(=\dfrac{2x+1}{x\left(2x-1\right)}+\dfrac{-32x^2}{\left(2x-1\right)\left(1+2x\right)}+\dfrac{1-2x}{x\left(2x+1\right)}\) \(=\dfrac{\left(2x+1\right)\left(2x+1\right)-32x^2.x+\left(1-2x\right)\left(2x-1\right)}{x\left(2x-1\right)\left(2x+1\right)}\) \(=\dfrac{4x^2+4x+1-32x^3-\left(4x^2-4x+1\right)}{x\left(2x-1\right)\left(2x+1\right)}\) \(=\dfrac{8x-32x^3}{x\left(2x-1\right)\left(2x+1\right)}\) \(=\dfrac{8x\left(1-4x^2\right)}{x\left(2x-1\right)\left(2x+1\right)}\) \(=\dfrac{8x\left(1-2x\right)\left(1+2x\right)}{x\left(2x-1\right)\left(2x+1\right)}=-8\).
Thực hiện phép cộng các phân thức \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-x\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\) . 0 \(\dfrac{1}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\) \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}\) \(\dfrac{1}{\left(x-y\right)\left(z-x\right)}\) Hướng dẫn giải: \(\dfrac{1}{\left(x-y\right)\left(y-z\right)}+\dfrac{1}{\left(y-z\right)\left(z-x\right)}+\dfrac{1}{\left(z-x\right)\left(x-y\right)}\) \(=\dfrac{z-x+x-y+y-z}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}\) \(=\dfrac{0}{\left(x-y\right)\left(y-z\right)\left(z-z\right)}=0\)
Trong các phép biến đổi sau, phép biến đổi nào là đúng? \(\dfrac{A}{B}-\dfrac{C}{-D}=\dfrac{A}{B}-\dfrac{-C}{D}=\dfrac{A}{B}+\dfrac{C}{D}\) \(\dfrac{A}{B}-\dfrac{C}{-D}=\dfrac{A}{B}+\dfrac{-C}{D}=\dfrac{A}{B}+\dfrac{C}{-D}\) \(\dfrac{A}{B}-\dfrac{C}{-D}=\dfrac{A}{B}+\dfrac{-C}{D}=\dfrac{A}{B}-\dfrac{C}{D}\) \(\dfrac{A}{B}-\dfrac{C}{-D}=\dfrac{A}{B}+\dfrac{C}{D}=\dfrac{A}{B}-\dfrac{C}{D}\)
Thực hiện phép trừ hai phân thức \(\dfrac{6x-3}{2x^2y}-\dfrac{4x-3}{2x^2y}\) \(\dfrac{1}{xy}\) \(\dfrac{2x-6}{2x^2y}\) \(\dfrac{x-3}{xy}\) \(\dfrac{x+3}{xy}\) Hướng dẫn giải: \(\dfrac{6x-3}{2x^2y}-\dfrac{4x-3}{2x^2y}\) \(=\dfrac{6x-3-4x+3}{2x^2y}\) \(=\dfrac{2x}{2x^2y}\) \(=\dfrac{1}{xy}\)
Thực hiện phép tính \(\dfrac{12x}{3x-4}-\dfrac{4-6x}{4-3x}\) \(2\) \(\dfrac{6x+8}{3x-4}\) \(\dfrac{6x-8}{3x-4}\) \(\dfrac{6x}{3x-4}\) Hướng dẫn giải: \(\dfrac{12x}{3x-4}-\dfrac{8-6x}{4-3x}\) \(=\dfrac{12x}{3x-4}+\dfrac{8-6x}{3x-4}\) \(=\dfrac{6x+8}{3x-4}\)
Thực hiện phép tính \(\dfrac{4x^2+3}{3x\left(3x+1\right)}+\dfrac{-2x+1}{3x+1}\) \(\dfrac{-2x^2+3x+3}{3x\left(3x+1\right)}\) \(\dfrac{-2x^2-3x+3}{3x\left(3x+1\right)}\) \(\dfrac{1}{3x+1}\) \(\dfrac{1}{3}\) Hướng dẫn giải: \(\dfrac{4x^2+3}{3x\left(3x+1\right)}+\dfrac{-2x+1}{3x+1}\) \(\dfrac{4x^2+3}{3x\left(3x+1\right)}+\dfrac{\left(-2x+1\right).3x}{3x\left(3x+1\right)}\) \(=\dfrac{4x^2+3-6x^2+3x}{3x\left(3x+1\right)}\) \(=\dfrac{-2x^2+3x+3}{3x\left(3x+1\right)}\)
Thực hiện phép tính \(\dfrac{1}{x-5x^2}-\dfrac{25x-15}{25x^2-1}\) \(\dfrac{1-5x}{x\left(1+5x\right)}\) \(\dfrac{1+5x}{x\left(1-5x\right)}\) \(\dfrac{1}{x}\) \(\dfrac{25x^2+1}{x\left(1+5x\right)\left(1-5x\right)}\) Hướng dẫn giải: \(\dfrac{1}{x-5x^2}-\dfrac{25x-15}{25x^2-1}\) \(=\dfrac{1}{x\left(1-5x\right)}-\dfrac{25x-15}{\left(5x-1\right)\left(5x+1\right)}\) \(=\dfrac{1}{x\left(1-5x\right)}+\dfrac{25x-15}{\left(1-5x\right)\left(1+5x\right)}\) \(=\dfrac{1+5x+x\left(25x-15\right)}{x\left(1-5x\right)\left(1+5x\right)}\) \(=\dfrac{25x^2-10x+1}{x\left(1-5x\right)\left(1+5x\right)}\) \(=\dfrac{\left(1-5x\right)^2}{x\left(1-5x\right)\left(1+5x\right)}\) \(=\dfrac{1-5x}{x\left(1+5x\right)}\)
Thực hiện phép tính \(\dfrac{2x+1}{\left(x+1\right)^2}-\dfrac{1}{x-1}+\dfrac{x+2}{x^2-1}\) \(\dfrac{2x^2+2x+2}{\left(x+1\right)^2\left(x-1\right)}\) \(\dfrac{1}{x-1}\) \(\dfrac{2}{x-1}\)\(\dfrac{x^2+x+1}{\left(x+1\right)\left(x-1\right)}\) Hướng dẫn giải: \(\dfrac{2x+1}{\left(x+1\right)^2}-\dfrac{1}{x-1}+\dfrac{x+2}{x^2-1}\) \(=\dfrac{2x+1}{\left(x+1\right)^2}-\dfrac{1}{x-1}+\dfrac{x+2}{\left(x-1\right)\left(x+1\right)}\) \(=\dfrac{\left(x-1\right)\left(2x+1\right)-\left(x-1\right)\left(x+1\right)+\left(x+2\right)\left(x+1\right)}{\left(x+1\right)^2\left(x-1\right)}\) \(=\dfrac{2x^2-x-1-x^2+1+x^2+3x+2}{\left(x+1\right)^2\left(x-1\right)}\) \(=\dfrac{2x^2+2x+2}{\left(x+1\right)^2\left(x-1\right)}\)
