Thực hiện phép tính \(\dfrac{2x^3-2}{x+2}.\left(\dfrac{1}{x-1}-\dfrac{x+1}{x^2+x+1}\right)\) 2 3 4 5 Hướng dẫn giải: \(\dfrac{2x^3-2}{x+2}.\left(\dfrac{1}{x-1}-\dfrac{x+1}{x^2+x+1}\right)\) \(=\dfrac{2\left(x^3-1\right)}{x+2}.\left(\dfrac{x^2+x+1-\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\right)\) \(=\dfrac{2\left(x^3-1\right)\left(x+2\right)}{\left(x+2\right)\left(x-1\right)\left(x^2+x+1\right)}\) \(=2\)
Thực hiện phép chia phân thức \(\left(\dfrac{-30x^3}{7y^2}\right):\left(-\dfrac{20x^2}{21y^2}\right)\) \(\dfrac{9x}{2}\) \(-\dfrac{9x}{2}\) \(\dfrac{9y}{2x}\) \(-\dfrac{9x}{2y}\) Hướng dẫn giải: \(\left(\dfrac{-30x^3}{7y^2}\right):\left(-\dfrac{20x^2}{21y^2}\right)\) \(=\dfrac{-30x^3}{7y^2}.\dfrac{-21y^2}{20x^2}\) \(=\dfrac{9x}{2}\)
Thực hiện phép chia phân thức \(\dfrac{5x+15}{\left(x-2\right)^2}:\dfrac{x+3}{x^2-4}\) \(\dfrac{5\left(x+2\right)}{x-2}\) \(\dfrac{-5\left(x+2\right)}{x-2}\) \(5\left(x-2\right)\) \(5\left(x+2\right)\) Hướng dẫn giải: \(\dfrac{5x+15}{\left(x-2\right)^2}:\dfrac{x+3}{x^2-4}\) \(=\dfrac{5\left(x+3\right)}{\left(x-2\right)^2}.\dfrac{\left(x-2\right)\left(x+2\right)}{x+3}\) \(=\dfrac{5\left(x+2\right)}{x-2}\)
Thực hiện phép tính \(\dfrac{8x+16}{x^2+1}:\left(x+2\right)\) \(\dfrac{8}{x^2+1}\) \(\dfrac{8}{\left(x-1\right)\left(x+1\right)}\) \(\dfrac{1}{x^2+1}\) \(\dfrac{1}{\left(x-1\right)\left(x+1\right)}\) Hướng dẫn giải: \(\dfrac{8x+16}{x^2+1}:\left(x+2\right)\) \(=\dfrac{8\left(x+2\right)}{\left(x^2+1\right).\left(x+2\right)}\) \(=\dfrac{8}{x^2+1}\)
Thực hiện phép tính \(\dfrac{x^2+2x}{6x^2-12x+6}:\dfrac{x+2}{4x-4}\) \(\dfrac{2x}{3\left(x-1\right)}\) \(-\dfrac{2x}{3\left(x-1\right)}\) \(\dfrac{2x}{3}\) \(\dfrac{2x\left(x+2\right)}{3\left(x-2\right)}\) Hướng dẫn giải: \(\dfrac{x^2+2x}{6x^2-12x+6}:\dfrac{x+2}{4x-4}\) \(=\dfrac{x\left(x+2\right)}{6\left(x-1\right)^2}.\dfrac{4\left(x-1\right)}{x+2}\) \(=\dfrac{2x}{3\left(x-1\right)}\)
Biết \(\dfrac{x^2+3x}{x-1}.Q=\dfrac{x^2-9}{x^2-x}\). Tìm phân thức Q. \(Q=\dfrac{x-3}{x^2}\) \(Q=x+3\) \(Q=\dfrac{x+3}{x}\) \(Q=-\dfrac{x+3}{x^2}\) Hướng dẫn giải: \(\dfrac{x^2+3x}{x-1}.Q=\dfrac{x^2-9}{x^2-x}\) \(\Leftrightarrow Q=\dfrac{x^2-9}{x\left(x-1\right)}:\dfrac{x\left(x+3\right)}{x-1}\) \(\Leftrightarrow Q=\dfrac{\left(x-3\right)\left(x+3\right)}{x\left(x-1\right)}.\dfrac{x-1}{x\left(x+3\right)}\) \(\Leftrightarrow Q=\dfrac{x-3}{x^2}\).
Thực hiện phép tính \(\dfrac{x^4-xy^3}{2xy+y^2}:\dfrac{x^3+x^2y+xy^2}{2x+y}\) \(\dfrac{x-y}{y}\) \(\dfrac{x+y}{y}\) \(\dfrac{x\left(x+y\right)}{y}\) \(\dfrac{x\left(x-y\right)}{y}\) Hướng dẫn giải: \(\dfrac{x^4-xy^3}{2xy+y^2}:\dfrac{x^3+x^2y+xy^2}{2x+y}\) \(=\dfrac{x\left(x^3-y^3\right)}{y\left(2x+y\right)}.\dfrac{2x+y}{x\left(x^2+xy+y^2\right)}\) \(=\dfrac{x-y}{y}\)
Thực hiện phép tính \(\dfrac{x^2+2x-3}{x^2+3x-10}:\dfrac{x^2+7x+12}{x^2-9x+14}\) \(\dfrac{\left(x-1\right)\left(x-7\right)}{\left(x+5\right)\left(x+4\right)}\) \(\dfrac{x-7}{x+4}\) \(\dfrac{\left(x+1\right)\left(x+7\right)}{\left(x+5\right)\left(x+4\right)}\) \(\dfrac{\left(x+1\right)\left(x+7\right)}{\left(x-5\right)\left(x-4\right)}\) Hướng dẫn giải: \(\dfrac{x^2+2x-3}{x^2+3x-10}:\dfrac{x^2+7x+12}{x^2-9x+14}\) \(=\dfrac{\left(x-1\right)\left(x+3\right)}{\left(x-2\right)\left(x+5\right)}.\dfrac{\left(x-2\right)\left(x-7\right)}{\left(x+3\right)\left(x+4\right)}\) \(=\dfrac{\left(x-1\right)\left(x-7\right)}{\left(x+5\right)\left(x+4\right)}\)
Biết \(P:\dfrac{4x^2-16}{2x-1}=\dfrac{4x^2-4x+1}{x+2}\). Xác định phân thức P. \(P=4\left(2x-1\right)\left(x-2\right)\) \(P=4\left(2x+1\right)\left(x+2\right)\) \(P=\dfrac{4\left(2x-1\right)\left(x-2\right)}{x+2}\) \(P=\dfrac{4\left(2x+1\right)\left(x+2\right)}{x-2}\) Hướng dẫn giải: \(P:\dfrac{4x^2-16}{2x-1}=\dfrac{4x^2-4x+1}{x+2}\) \(\Leftrightarrow P=\dfrac{4x^2-4x+1}{x+2}.\dfrac{4x^2-16}{2x-1}\) \(\Leftrightarrow P=\dfrac{\left(2x-1\right)^2}{x+2}.\dfrac{4\left(x-2\right)\left(x+2\right)}{2x-1}\) \(\Leftrightarrow P=4\left(2x-1\right)\left(x-2\right)\)
Biểu thức \(\dfrac{3+\dfrac{1}{x-1}}{2-\dfrac{x^2+1}{x^2-1}}\) bằng phân thức nào trong số các phân thức dưới đây: \(\dfrac{3x^2+x-2}{x^2-3}\) \(\dfrac{3x^2-x+1}{x^2-3}\) \(\dfrac{2}{x-1}\) \(\dfrac{2\left(x+1\right)}{x-1}\) Hướng dẫn giải: \(\dfrac{3+\dfrac{1}{x-1}}{2-\dfrac{x^2+1}{x^2-1}}\) \(=\dfrac{3\left(x-1\right)+1}{x-1}:\dfrac{2\left(x^2-1\right)-\left(x^2+1\right)}{x^2-1}\) \(=\dfrac{3x-2}{x-1}:\dfrac{x^2-3}{x^2-1}\) \(=\dfrac{3x-2}{x-1}.\dfrac{\left(x-1\right)\left(x+1\right)}{x^2-3}\) \(=\dfrac{\left(3x-2\right)\left(x+1\right)}{x^2-3}\) \(=\dfrac{3x^2+x-2}{x^2-3}\)
