Rút gọn biểu thức \(P = \frac{{{x^4}}}{{1 - x}} + {x^3} + {x^2} + x + 1\).

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

\( = \frac{{y\left( {5x + {y^2}} \right)}}{{{x^2}{y^2}}} - \frac{{x\left( {5y - {x^2}} \right)}}{{{x^2}{y^2}}}\) (Mẫu thức chung là: x²y²)

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

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

Ta có:

\(P = \frac{{{x^2} + 2x}}{{{x^3} - 1}} - \frac{1}{{{x^2} - x}} - \frac{1}{{{x^2} + x + 1}}\) (x ≠ 0, x ≠ 1)

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

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

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

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

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