Thực hiện phép tính:

a) \(\frac{{x + 2y}}{a} + \frac{{x - 2y}}{a}\) với a là một số khác 0;

b) \(\frac{x}{{x - 1}} + \frac{1}{{1 - x}}\);

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

d) \(x + \frac{1}{{x + 1}} - 1\).

Lời giải

a) \(\frac{{x + 2y}}{a} + \frac{{x - 2y}}{a} = \frac{{x + 2y + x - 2y}}{a} = \frac{{\left( {x + x} \right) + \left( {2y - 2y} \right)}}{a} = \frac{{2x}}{a}\).

b) \(\frac{x}{{x - 1}} + \frac{1}{{1 - x}} = \frac{x}{{x - 1}} - \frac{1}{{x - 1}} = \frac{{x - 1}}{{x - 1}} = 1\).

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

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

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

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

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

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

d) \(x + \frac{1}{{x + 1}} - 1\)

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

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

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