Rút gọn các phân thức sau:

a) \(\frac{{{y^3} - {x^3}}}{{{x^3} - 3{{\rm{x}}^2}y + 3{\rm{x}}{y^2} - {y^3}}}\)

b) \(\frac{{{x^5} + x + 1}}{{{x^3} + {x^2} + x}}\)

c) \(\frac{{2{{\rm{x}}^2} - x - 3}}{{{x^2} - 4x - 5}}\).

a) Ta có:

\(\frac{{{y^3} - {x^3}}}{{{x^3} - 3{{\rm{x}}^2}y + 3{\rm{x}}{y^2} - {y^3}}}\)

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

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

b) Ta có: \(\frac{{{x^5} + x + 1}}{{{x^3} + {x^2} + x}}\)

\( = \frac{{\left( {{x^5} - {x^2}} \right) + {x^2} + x + 1}}{{x\left( {{x^2} + x + 1} \right)}}\)

\( = \frac{{{x^2}\left( {{x^3} - 1} \right) + \left( {{x^2} + x + 1} \right)}}{{x\left( {{x^2} + x + 1} \right)}}\)

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

\( = \frac{{\left( {{x^2} + x + 1} \right)\left( {{x^3} - {x^2} + 1} \right)}}{{x\left( {{x^2} + x + 1} \right)}}\)

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

c) Ta có:

\(\frac{{2{{\rm{x}}^2} - x - 3}}{{{x^2} - 4{\rm{x}} - 5}}\)

\( = \frac{{2{x^2} + 2x - 3x - 3}}{{{x^2} + x - 5x - 5}}\)

\( = \frac{{2x\left( {x + 1} \right) - 3\left( {x + 1} \right)}}{{x\left( {x + 1} \right) - 5\left( {x + 1} \right)}}\)

\( = \frac{{\left( {x + 1} \right)\left( {2x - 3} \right)}}{{\left( {x + 1} \right)\left( {x - 5} \right)}}\)

\( = \frac{{2x - 3}}{{x - 5}}\)