Rút gọn biểu thức \(\dfrac{2x^2+x+4}{x^3+1}+\dfrac{x-1}{x^2-x+1}-\dfrac{2}{x+1}\) \(\dfrac{x+2}{x^2-x+1}\) \(\dfrac{2}{x+1}\) \(\dfrac{x-2}{x^2+x+1}\) \(\dfrac{x+1}{x^2+x+1}\) Hướng dẫn giải: \(\dfrac{2x^2+x+5}{x^3+1}+\dfrac{x-1}{x^2-x+1}-\dfrac{2}{x+1}\) \(=\dfrac{2x^2+x+5+\left(x-1\right)\left(x+1\right)-2\left(x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{2x^2+x+5+x^2-1-2x^2+2x-2}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{x^2+3x+2}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{x+2}{x^2-x+1}\)
Rút gọn biểu thức \(\dfrac{2x^2+x+4}{x^3+1}+\dfrac{x-1}{x^2-x+1}-\dfrac{2}{x+1}\) \(\dfrac{x+2}{x^2-x+1}\) \(\dfrac{2}{x+1}\) \(\dfrac{x-2}{x^2+x+1}\) \(\dfrac{x+1}{x^2+x+1}\) Hướng dẫn giải: \(\dfrac{2x^2+x+5}{x^3+1}+\dfrac{x-1}{x^2-x+1}-\dfrac{2}{x+1}\) \(=\dfrac{2x^2+x+5+\left(x-1\right)\left(x+1\right)-2\left(x^2-x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{2x^2+x+5+x^2-1-2x^2+2x-2}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{x^2+3x+2}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{\left(x+1\right)\left(x+2\right)}{\left(x+1\right)\left(x^2-x+1\right)}\) \(=\dfrac{x+2}{x^2-x+1}\)
Tìm phân thức Q thỏa mãn điều kiện \(\dfrac{1}{x^2+x+1}-Q=\dfrac{1}{x-x^2}+\dfrac{x^2+2x}{x^3-1}\). \(-\dfrac{1}{x}\) \(\dfrac{1}{x\left(x-1\right)}\) \(\dfrac{1}{x^3-1}\) \(\dfrac{2x}{x^3-1}\) Hướng dẫn giải: \(\dfrac{1}{x^2+x+1}-Q=\dfrac{1}{x-x^2}+\dfrac{x^2+2x}{x^3-1}\) \(=Q=\dfrac{1}{x^2+x+1}-\left(\dfrac{1}{x-x^2}+\dfrac{x^2+2x}{x^3-1}\right)\) \(=Q=\dfrac{1}{x^2+x+1}-\dfrac{1}{x-x^2}-\dfrac{x^2+2x}{x^3-1}\) \(=\dfrac{1}{x^2+x+1}+\dfrac{1}{x^2-x}-\dfrac{x^2+2x}{x^3-1}\) \(=\dfrac{x^2-x+x^2+x+1-x^3-2x^2}{x\left(x^3-1\right)}=\dfrac{1-x^3}{x\left(x^3-1\right)}=-\dfrac{1}{x}\)
Làm tính nhân ta được: \(\dfrac{35x^4}{12y^2}.\dfrac{144y^4}{21x^2}\) \(20x^2y^2\) \(\dfrac{1}{7}x^2y^2\) \(\dfrac{20x^2}{y^2}\) \(20\left(x^2+y^2\right)\) Hướng dẫn giải: \(\dfrac{35x^4}{12y^2}.\dfrac{144y^4}{21x^2}\) \(=\dfrac{35x^4.144y^4}{12y^2.21x^2}\) \(=20x^2y^2\)
Thực hiện phép tính \(\dfrac{24xy}{z^2}.\left(-\dfrac{z}{8x^2y}\right)\) \(\dfrac{-3}{xz}\) \(\dfrac{3}{xz}\) \(3xy\) \(\dfrac{-3}{yz}\) Hướng dẫn giải: \(\dfrac{24xy}{z^2}.\left(-\dfrac{z}{8x^2y}\right)\) \(=\dfrac{-24xyz}{8x^2yz^2}\) \(=\dfrac{-3}{xz}\)
Thực hiện phép tính \(\dfrac{x^3-27}{8x+16}.\dfrac{x^2-4}{x^2+3x+9}\) \(\dfrac{\left(x-3\right)\left(x-2\right)}{8}\) \(\dfrac{\left(x-3\right)\left(x+2\right)}{8}\) \(\dfrac{x^2+3x+6}{8}\) \(\dfrac{\left(x+2\right)\left(x-2\right)}{8}\) Hướng dẫn giải: \(\dfrac{x^3-27}{8x+16}.\dfrac{x^2-4}{x^2+3x+9}\) \(=\dfrac{\left(x-3\right)\left(x+3x+9\right).\left(x-2\right)\left(x+2\right)}{8\left(x+2\right)\left(x^2+3x+9\right)}\) \(=\dfrac{\left(x-3\right)\left(x-2\right)}{8}\)
Thực hiện phép tính \(\dfrac{6x-18}{5x-10}.\dfrac{4-2x}{x+3}\) \(\dfrac{-12\left(x-3\right)}{5\left(x+3\right)}\) \(\dfrac{12}{5}\) \(\dfrac{12\left(x+2\right)}{x-2}\) \(\dfrac{12\left(x-2\right)}{x+2}\) Hướng dẫn giải: \(\dfrac{6x-18}{5x-10}.\dfrac{4-2x}{x+3}\) \(=\dfrac{6\left(x-3\right)}{5\left(x-2\right)}.\dfrac{2\left(2-x\right)}{x+3}\) \(=\dfrac{-12\left(x-3\right)}{5\left(x+3\right)}\)
Làm tính nhân phân thức \(\dfrac{3x^2-6x+3}{4x+4}.\dfrac{2x^2-2}{3\left(x-1\right)^2}\) \(\dfrac{x-1}{4}\) \(\dfrac{x+1}{4}\) \(\dfrac{x-1}{4\left(x+1\right)}\) \(\dfrac{x-1}{12}\) Hướng dẫn giải: \(\dfrac{3x^2-6x+3}{4x+4}.\dfrac{2x^2-2}{3\left(x-1\right)^2}\) \(=\dfrac{3\left(x^2-2x+1\right)}{4\left(x+1\right)}.\dfrac{2\left(x-1\right)\left(x+1\right)}{3\left(x-1\right)^2}\) \(=\dfrac{3\left(x-1\right)^3\left(x+1\right)}{4.3\left(x+1\right)\left(x-1\right)^2}\) \(=\dfrac{x-1}{4}\)
Thực hiện phép nhân hai phân thức \(\dfrac{6x-3}{5x^2+x}.\dfrac{25x^2+10x+1}{1-8x^3}\) \(\dfrac{-3\left(5x+1\right)}{x\left(4x^2+2x+1\right)}\) \(\dfrac{3\left(5x+1\right)}{x\left(4x^2+2x+1\right)}\) \(\dfrac{-3\left(5x+1\right)}{x\left(4x^2-2x+1\right)}\) \(\dfrac{-3\left(5x-1\right)}{x\left(4x^2-2x+1\right)}\) Hướng dẫn giải: \(\dfrac{6x-3}{5x^2+x}.\dfrac{25x^2+10x+1}{1-8x^3}\) \(=\dfrac{3\left(2x-1\right)\left(5x+1\right)^2}{x\left(5x+1\right)\left(1-2x\right)\left(1+2x+4x^2\right)}\) \(=\dfrac{-3\left(5x+1\right)}{x\left(4x^2+2x+1\right)}\)
Rút gọn phân thức \(\dfrac{3x+18}{x^2-4}.\dfrac{x^2+4x+4}{x^2+9x+18}\) \(\dfrac{3}{x-2}\) \(\dfrac{3}{x+2}\) \(\dfrac{-3}{x-2}\) \(\dfrac{3}{\left(x-2\right)\left(x+2\right)}\) Hướng dẫn giải: \(\dfrac{3x+18}{x^2-4}.\dfrac{x^2+4x+4}{x^2+9x+18}\) \(=\dfrac{3\left(x+6\right)}{\left(x-2\right)\left(x+2\right)}.\dfrac{\left(x+2\right)^2}{\left(x+2\right)\left(x+6\right)}\) \(=\dfrac{3}{x-2}\)
